Graded Rings and Graded Grothendieck Groups

R. Hazrat

Centre for Research in Mathematics

University of Western Sydney Australia

October 13, 2013

... inside of a book is there whether you read it or not. Even if nobody ever reads it, it’s there, doing its work.

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Contents

1 Graded Rings And Graded Modules  7
1.1 Graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Graded modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3 Grading on matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4 Graded division rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5 Strongly graded rings and Dade’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 46
1.6 Grading on graph algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.7 The graded IBN and graded type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.8 The graded stable rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2 Graded Morita Theory  64
2.1 First instance of the graded Morita equivalence . . . . . . . . . . . . . . . . . . . . . 64
2.2 Graded generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3 General graded Morita equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Graded Grothendieck Groups  75
3.1 The graded Grothendieck group K_{0}^{gr} . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2 K_{0}^{gr} from idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3 Relative K_{0}^{gr}-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4 K_{0}^{gr} of non-unital rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5 K_{0}^{gr} is a pre-ordered module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.6 K_{0}^{gr} of graded division rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.7 K_{0}^{gr} of graded local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.8 K_{0}^{gr} of Leavitt path algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.9 Symbolic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.10 K_{1}^{gr}-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4 Graded Picard Groups  112
4.1 Pic^{gr} of a graded commutative ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2 Pic^{gr} of a graded noncommutative ring . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5 Graded Ultramatricial Algebras, Classification via K_{0}^{gr}  120
5.1 Graded matricial algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2 Graded ultramatricial algebras, classification via K_{0}^{gr} . . . . . . . . . . . . . . . . . . 124
6 Graded Versus Non-graded (Higher) K-Theory  129
6.1 K_{∗}^{gr} of positively graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.2 The fundamental theorem of K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.3 Relating K_{∗}^{gr} to K_{∗} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Introduction

A bird’s eye view of the graded module theory over a graded ring gives an impression of the module theory with the added adjective “graded” to all its statements. Once the grading is considered to be trivial, the graded theory reduces to the usual module theory. So from this perspective, the graded module theory can be considered as an extension of the module theory. However, one aspect that could be easily missed from such a panoramic view is that, the graded module theory comes equipped with a shifting, thanks to being able to partition the structures and rearranging these partitions. This adds an extra layer of structure (and complexity) to the theory. An sparkling example of this is the theory of graded Grothendieck groups which is the main focus of this monograph.

The construction of the Grothendieck group of a ring is easy. Starting from a ring A with identity, the isomorphism classes of finitely generated projective (right) A-modules equipped with the direct sum forms an abelian monoid, denoted by 𝒱(A). The group completion of this monoid is denoted by K_{0}(A) and is called the Grothendieck group of A. For a Γ-graded ring A, the graded isomorphism classes of graded finitely generated projective modules equipped with the direct sum forms an abelian monoid, denoted by 𝒱^{gr}(A). The shifting of graded modules (see §1.2.1) induces a Γ-module structure on 𝒱^{gr}(A). The group completion of this monoid is denoted by K_{0}^{gr}(A) and is called the graded Grothendieck group of A which has, consequently, a natural ℤ[Γ]-module structure. As we will see throughout this note, this extra structure carries a substantial information about the graded ring A.

Starting from a graded ring A, the relation between the category of graded A-modules, Gr-A, the category of A-modules, Mod- A, and the category of modules over the ring of the zero homogeneous part, Mod- A_{0}, is one of the main forces to study the graded theory. In the same way, the relations between K_{0}^{gr}(A), K_{0}(A) and K_{0}(A_{0}) are of interest.

The motivation to write this note came from the recent activities which adopt the graded Grothendieck group as an invariant to classify the Leavitt path algebras [46, 47, 48, 76]. Surprisingly, not much is recorded about the graded version of the Grothendieck group in the literature, despite the fact that K_{0} has been used in many occasions as a crucial invariant, and there is a substantial amount of information about the graded version of other invariants such as (co)homology groups, Brauer groups, etc. The other surge of interest on this group stems from the recent activities on (graded) representation theory of Hecke algebras. In particular for a quiver Hecke algebra, its graded Grothendieck group is closely related to its corresponding quantised enveloping algebra. For this line of research see the survey [56].

This note tries to fill this gap, by systematically developing the theory of graded Grothendieck groups. In order to do this, we have to carry over and work out the details of known results in non-graded case to the graded setting, and gather important results scattered in research papers on the graded theory. The emphasis is on using the graded Grothendieck groups as invariants for classifications.

The group K_{0} has been successfully used in the theory of C^{∗}-algebras to classify certain class of C^{∗}-algebras. Building on the work of Brattelli, Elliott in [34] used the the pointed ordered K_{0}- groups (called dimension groups) as a complete invariant for AF C^{∗}-algebras. Another cornerstone of using K-groups for the classifications of a wider range of C^{∗}-algebras was the work of Kirchberg and Phillips [77], who showed that K_{0} and K_{1}-groups together are complete invariant for a certain type of C^{∗}-algebras. The Grothendieck group considered as a module induced by a group action was used by Handelman and Rossmann [43] to give a complete invariant for the class of direct limits of finite dimensional, representable dynamical systems. Krieger [57] introduced (past) dimension groups as a complete invariant for the shift equivalence of topological Markov chains (shift of finite types) in symbolic dynamics. Surprisingly as we will see, Krieger’s groups can be naturally expressed by graded Grothendieck groups (§3.9).

For two reasons we develop the theory for rings graded by abelian groups rather than an arbitrary groups, although most of the results could be carried over to non-abelian graded rings. One reason is that, by using the abelian grading, the presentation and proofs are much more transparent. Furthermore, in most applications of graded K_{0}, the ring has an abelian grading (often times ℤ-grading). A brief outline. Let Γ be an (abelian) group and A be a Γ-graded ring. For any subgroup Ω ⊆ Γ, one can consider the natural Γ/Ω-grading on A (see Example 1.1.8). In the same manner, any Γ-graded A-module can be considered as Γ/Ω-graded module. These induce functors (see §1.2.8)

U : Gr^{Γ}-A ⟶ Gr^{Γ/Ω}-A,
(−)_{Ω} : Gr^{Γ}-A ⟶ Mod- A_{Ω},

where Gr^{Γ}-A is the category of Γ-graded (right) A-modules, Gr^{Γ/Ω}-A is the category of Γ/Ω-graded A-modules and Mod- A_{Ω} is the category of A_{Ω}-modules.

One aspect of the graded theory of rings is to investigate how these categories are related, and what properties can be lifted from one category to another. In particular when the subgroup Ω happen to be Γ or trivial group, respectively, we obtain the (forgetful) functors

U : Gr^{Γ}-A ⟶ Mod- A,
(−)_{0} : Gr^{Γ}-A ⟶ Mod- A_{0}.

One aspect of this note is to study the interrelation between these categories. This has been done in Chapters 1 and 2.

The rest of the note is devoted to the Grothendieck group, denoted by K_{0}, of these category. The category of graded finitely generated projective A-modules, Pgr^{Γ}-A, is an exact category. Thus using Quillen’s machinery of K-theory [79], one can define graded K-groups,

K_{i}^{gr}(A) := K_{i}(Pgr^{Γ}-A),

for i ∈ ℕ. Since the shifting of modules induces an auto-equivalence functor on Gr^{Γ}-A (§1.2.2), these K-groups carry a Γ-module structure. This extra structure is an emphasis of this note. In many important examples, in fact this shifting is all the difference between the graded Grothendieck group and the usual Grothendieck group, i.e.,

K_{0}^{gr}(A)/⟨[P] − [P (α)]⟩ ≅ K_{0}(A),

where P is a graded projective A-module, α ∈ Γ and P (α) is a shift module (§6, see Corollary 6.3.2).

In Chapter 3 we compute K_{0}^{gr} for certain graded rings, such as graded local rings and (Leavitt) path algebras. We study the pre-ordering available on K_{0}^{gr} and determine the action of Γ on this group. In Chapter 4 the graded Picard groups are studied and in Chapter 5 we prove that for the so called graded ultramatricial algebras, the graded Grothendieck group is a complete invariant. Finally in Chapter 6, we explore the relations between (higher) K_{i}^{gr} and K_{i}, for the class of ℤgraded rings. We describe a generalisation of the Quillen and van den Bergh theorems. The latter theorem uses the techniques employed in the proof of the fundamental theorem of K-theory, where the graded K-theory appears. For this reason we present a proof of the fundamental theorem in this chapter.

Conventions. Throughout this note, unless it is explicitly stated, all rings have identities, homomorphisms preserve the identity and all modules are unitary. Furthermore, all modules are considered right modules. For a set Γ by ⨁_{Γ} ℤ or ℤ^{Γ}, we mean ⨁_{γ∈Γ} ℤ_{γ}, where ℤ_{γ} = ℤ for each γ ∈ Γ. We denote the cyclic group ℤ/nℤ with n elements by ℤ_{n}.

Acknowledgement. I learned about the graded techniques in algebra from Adrian Wadsworth. Judith Millar worked with me to study the graded K-theory of Azumaya algebras. Gene Abrams was a source of encouragement that the graded techniques would be fruitful in the study of Leavitt path algebras. Andrew Mathas told me how the graded Grothendieck groups are relevant in the representation theory and pointed me to the relevant literature. To all of them, I am grateful.

Chapter 1

Graded Rings And Graded Modules

This note deals with graded rings and the category of graded modules over a graded ring. This category is an abelian category (in fact a Grothendieck category). Many of the classical invariants constructed for the category of modules can be constructed, mutatis mutandis, starting from the category of graded modules. The general viewpoint of this note is, once a ring has a natural graded structure, the graded invariants would capture more information than the non-graded counterparts.

In this chapter we give a quick introduction to the graded ring theory. We introduce grading on matrices, study graded division rings and introduce grading on graph algebras which would be a source of many interesting examples.

1.1 Graded rings

1.1.1

A ring A is called a Γ-graded ring, or simply a graded ring, if A = _{⨁γ∈Γ}A_{γ}, where Γ is an (abelian) group, each A_{γ} is an additive subgroup of A and A_{γ}A_{δ} ⊆ A_{γ+δ} for all γ, δ ∈ Γ.

The set A^{h} = ⋃_{γ∈Γ} A_{γ} is called the set of homogeneous elements of A. The additive group A_{γ} is called the γ-component of A and the non-zero elements of A_{γ} are called homogeneous of degree γ. We write deg(a) = γ if a ∈ A_{γ}\{0}. We call the set

Γ_{A} = {γ ∈ Γ | A_{γ} ≠ 0}

the support of A. We say the Γ-graded ring A has a trivial grading, or A is concentrated in degree zero if the support of A is the trivial group, i.e., A_{0} = A and A_{γ} = 0 for γ ∈ Γ\{0}.

For Γ-graded rings A and B, a Γ-graded ring homomorphism f : A → B is a ring homomorphism such that f(A_{γ}) ⊆ B_{γ} for all γ ∈ Γ. A graded homomorphism f is called a graded isomorphism if f is bijective and, when such a graded isomorphism exists, we write A ≅_{gr} B. Notice that if f is a graded ring homomorphism which is bijective, then its inverse f^{−1} is also a graded ring homomorphism.

Proposition 1.1.1. Let A = ⨁_{γ∈Γ} A_{γ} be a Γ-graded ring. Then

(1) 1_{A} is homogeneous of degree 0;

(2) A_{0} is a subring of A; (3) Each A_{γ} is an A_{0}-bimodule;

(4) For an invertible element a ∈ A_{γ}, its inverse a^{−1} is homogeneous of degree −γ, i.e., a^{−1} ∈ A_{−γ}.

Proof. (1) Suppose 1_{A} = ∑_{γ∈Γ} a_{γ} for a_{γ} ∈ A_{γ}. Let b ∈ A_{δ}, δ ∈ Γ, be an arbitrary non-zero homogeneous element. Then b = b1_{A} = ∑_{γ∈Γ} ba_{γ}, where ba_{γ} ∈ A_{δ+γ} for all γ ∈ Γ. Since the decomposition is unique, ba_{γ} = 0 for all γ ∈ Γ with γ ≠ 0. But as b was arbitrary, this holds for all b ∈ A (not necessarily homogeneous), and in particular 1_{A}a_{γ} = a_{γ} = 0 if γ ≠ 0. Thus 1_{A} = a_{0} ∈ A_{0}.

(2) This follows since A_{0} is an additive subgroup of A with A_{0}A_{0} ⊆ A_{0} and 1 ∈ A_{0}.

(3) This is immediate.

(4) Let b = ∑_{δ∈Γ} b_{δ}, with deg(b_{δ}) = δ, be the inverse of a ∈ A_{γ}, so that 1 = ab = ∑_{δ∈Γ} ab_{δ}, where ab_{δ} ∈ A_{γ+δ}. By (1), since 1 is homogeneous of degree 0 and the decomposition is unique, it follows that ab_{δ} = 0 for all δ ≠ −γ. Since a is invertible, b_{−γ} ≠ 0, so b = b_{−γ} ∈ A_{−γ} as required.

The proof of Proposition 1.1.1(4), in fact, shows that if a ∈ A_{γ} has a left (or right) inverse then that inverse is in A_{−γ}. In Theorem 1.6.8, we characterise ℤ-graded rings such that A_{1} has a left (or right) invertible element.

Example 1.1.2. \underline{Group rings} For a group Γ, the group ring ℤ[Γ] has a natural Γ-grading

ℤ[Γ] = ⨁_{γ∈Γ} ℤ[Γ]_{γ}, where ℤ[Γ]_{γ} = ℤγ.

In §1.1.3, we study the crossed-products which are graded rings and are generalisations of group rings and skew groups rings.

Example 1.1.3. \underline{Tensor algebras as positively graded rings}

Let A be a commutative ring and M be an A-module. Denote by T_{n}(M), n ≥ 1, the tensor product of n copies of M over A. Set T_{0}(M) = A. Then the natural A-module isomorphism T_{n}(M) ⊗_{A} T_{m}(M) → T_{n+m}(M), induces a ring structure on

T (M) := ⨁_{n∈ℕ} T_{n}(M).

The A-algebra T (M) is called the tensor algebra of M. Setting T (M)_{n} := T_{n}(M), makes T (M) a ℤ-graded ring with support ℕ. From the definition, we have T (M)_{0} = A.

If M is a free A-module, then T (M) is a free algebra over A, generated by a basis of M. Thus free rings are ℤ-graded rings with the generators being homogeneous elements of degree 1. We will systematically study the grading of free rings in §1.6.1.

Example 1.1.4. \underline{Formal matrix rings as graded rings}

Let R and S be rings, M be a R − S-bimodule and N be a S − R-bimodule. Consider the set T consisting of 2 × 2-matrices

(\begin{matrix}r & m \\ n & s \end{matrix}) ,

where r ∈ R, s ∈ S, m ∈ M and n ∈ N. Suppose that there are bimodule homomorphisms ϕ : M ⊗_{S} N → R and ψ : N ⊗_{R} M → S such that (mn)m′ = n(nm′), where we set ϕ(m, n) = mn and ψ(n, m) = nm. One can then check that T with the matrix addition and multiplication forms a ring with identity. The ring T is called the formal matrix ring and denoted also by

T = (\begin{matrix}R & M \\ N & S \end{matrix}) .

Considering

T_{0} = (\begin{matrix}R & 0 \\ 0 & S \end{matrix}) , T_{1} = (\begin{matrix}0 & M \\ N & 0 \end{matrix}) ,

it is easy to check that T becomes a ℤ_{2}-graded ring. On the other hand, considering further T_{i} = 0 for i ≠ 0, 1, makes T also a ℤ-graded ring. In the cases that the images of ϕ and ψ are zero, these rings have been extensively studied (see [58] and references therein). When N = 0, the ring T is called a formal triangular matrix ring. In this case there is no need of considering the homomorphisms ϕ and ψ.

One specific example of such grading on (subrings of) formal triangular matrix ring is used in representation theory. Recall that for a field K, a finite dimensional K-algebra R is called Frobenius algebra if R ≅ R^{∗} as a right R-module, where R^{∗} := Hom_{K}(R, K). Note that R^{∗} has a natural R-bimodule structure.

Starting from a finite dimensional K-algebra R, one constructs the trivial extension of R which is a Frobenius algebra and has a natural ℤ-graded structure as follows. Consider A := R ⨁ R^{∗}, with addition defined component-wise and multiplication defined as

(r_{1}, q_{1})(r_{2}, q_{2}) = (r_{1}r_{2}, r_{1}q_{1} + q_{2}r_{2}),

where r_{1}, r_{2} ∈ R and q_{1}, q_{2} ∈ R^{∗}. Clearly A is a Frobenius algebra with identity (1, 0). Furthermore, setting

A_{0} = R ⊕ 0,
A_{1} = 0 ⊕ R^{∗},
A_{i} = 0, otherwise,

makes A into a ℤ-graded ring with support {0, 1}. In fact this ring is a subring of formal triangular matrix ring

T_{0} = (\begin{matrix}R & R^{∗} \\ 0 & R \end{matrix}) ,

consisting of elements (\begin{matrix}a & q \\ 0 & a \end{matrix}) .

These rings appear in representation theory (see [44, §2.2]).

Example 1.1.5. \underline{The graded ring A as A_{0}-module}

Let A be a Γ-graded ring. Then A can be considered as a A_{0}-bimodule. In many cases A is a projective A_{0}-module, for example in the case of group rings (Example 1.1.2) or when A is a strongly graded ring (see §1.1.2 and §1.1.2). Here is an example that this is not the case in general. Consider the ring T of formal matrix ring

T = (\begin{matrix}R & M \\ 0 & 0 \end{matrix}) ,

where M is a left R-module, which is not projective R-module. Then by Example 1.1.4, T is a ℤ-graded ring with T_{0} = R and T_{1} = M. Now T as a T_{0}-module is R ⊕ M as R-module. Since M is not projective, R ⊕ M is not R-module projective. We also get that T_{1} is not T_{0}-projective.

Example 1.1.6. \underline{Tensor product of graded rings}

Let A and B be a Γ-graded rings. Then A⊗_{ℤ} B has a natural Γ-graded ring structure as follows. Since each of A_{γ} and B_{γ}, γ ∈ Γ, are ℤ-module then A ⊗_{ℤ} B can be decomposed as a direct sum A ⊗_{ℤ} B = ⨁_{γ∈Γ}(A ⊗ B)_{γ}, where

(A ⊗ B)_{γ} = { ∑_{i} a_{i} ⊗ b_{i} | a_{i} ∈ A^{h}, b_{i} ∈ A^{h}, deg(a_{i}) + deg(b_{i}) = γ}.

It is easy to see that with this decomposition A ⊗_{ℤ} B becomes a Γ-graded ring. We give specific examples of this construction in Example 1.1.7. One can replace ℤ by a field K, if A and B are K-algebras and A_{γ}, B_{γ} are K-modules.

Example 1.1.7. Let A be a ring with identity and Γ be a group. We consider A as Γ-graded ring concentrated in degree zero. Then by Example 1.1.6, A[Γ] ≅ A⊗_{ℤ}ℤ[Γ] has a Γ-graded structure, i.e, A[Γ] = _{⨁γ∈Γ}Aγ. If A itself is a (nontrivial) Γ-graded ring A = _{⨁γ∈Γ}A_{γ}, then by Example 1.1.6, A[Γ] has also a Γ-grading

A[Γ] = _{⨁γ∈Γ}A^{γ}, where A^{γ} = _{⨁γ=ζ+ζ_{′}}A_{ζ}ζ′.  (1.1)

An specific example is when A is a positively graded ℤ-graded ring. Then A[x] ≅ A ⊗ ℤ[x] is a ℤ-graded ring with support ℕ, where

A[x]_{n} = ⨁_{i+j=n} A_{i}x^{j}.

This graded ring will be used in §6.2.2 when we prove the fundamental theorem of K-theory.

Example 1.1.8. \underline{Partitioning of the graded rings} Let A be a Γ-graded ring and Ω be a subgroup of Γ.

Subgroup grading: The ring A_{Ω} := ⨁_{γ∈Ω} A_{γ} forms a Ω-graded ring. In particular, A_{0} corresponds to the trivial subgroup of Γ.

Quotient grading: Considering

A = ⨁_{Ω+α∈Γ/Ω} A_{Ω+α},

where

A_{Ω+α} := ⨁_{ω∈Ω} A_{ω+α},

makes A a Γ/Ω-graded ring. (Note that if Γ is not abelian, then for this construction, Ω needs to be a normal subgroup.) Notice that with this grading, A_{0} = A_{Ω}. If Γ_{D} ⊆ Ω, then A considered as Γ/Ω-graded ring, is concentrated in degree zero.

This construction induces a functor from the category of Γ-graded rings to the category of Γ/Ω-graded rings. If Ω = Γ, this gives the obvious forgetful functor from the category of Γ-graded rings to the category of rings. We give a specific example of this construction in Example 1.1.9 and another in Example 1.6.1.

Example 1.1.9. Let A be a Γ × Γ-graded ring. Define a Γ-grading on A as follows. For γ ∈ Γ, set

A′_{γ} = ∑_{α∈Γ} A_{γ−α,α}.

It is easy to see that A = ⨁_{γ∈Γ} A′_{γ} is a Γ-graded ring. When A is a ℤ × ℤ-graded, then the ℤgrading on A is obtained from considering all the homogeneous components on a diagonal together as the following figure shows.

x y−3 −2 −1 1 2 3 A′_{0} A′_{1} A′_{2} A′_{−1} −3 −2 −1 1 2 3A_{1,1} A_{−1,0} A_{1,−2} A_{−2,1} A_{1,−1} A_{4,0} A_{−1,1} A_{1,2} A_{−2,2} A_{1,3} A_{0,2} A_{0,3} A_{1,1} A_{−1,2} A_{−1,3} A_{2,1} A_{3,1} A_{2,2} A_{−2,3} A_{0,1} A_{2,−1} A_{3,−1} A_{3,−2} A_{2,−2} A_{0,−1} A_{0,0} A_{1,0} A_{2,0} A_{3,0}

In fact this example follows from the general construction given in Example 1.1.8. Consider the homomorphism Γ × Γ → Γ, (α, β) ↦ α + β. Let I be the kernel of this map. Clearly Γ × Γ/I ≅ Γ. One can check that the Γ × Γ/I-graded ring A gives the graded ring constructed in this example (Remark 1.1.26).

Example 1.1.10. \underline{The direct limit of graded rings}

Let A_{i}, i ∈ I, be a direct system of Γ-graded rings, i.e., I is a directed partially ordered set and for i ≤ j, there is a graded homomorphism ϕ_{ij} : A_{i} → A_{j} which is compatible with the ordering. Then A := \underrightarrow{lim} A_{i} is a Γ-graded ring with homogeneous components A_{α} = \underrightarrow{lim} A_{i}_{α}. For a detailed construction of such direct limits see [22, II, §11.3, Remark 3].

As an example, the ring A = ℤ[x_{i} | i ∈ ℕ], where A = \underrightarrow{lim}_{i∈ℕ} ℤ[x_{1}, …, x_{i}], with deg(x_{i}) = 1 is a ℤ-graded ring with support ℕ. We give another specific example of this construction in Example 1.1.11.

We will study in detail one type of these graded rings, i.e., graded ultramatricial algebras (§5, Definition 5.2.1) and will show that the graded Grothendieck group (§3) will classify these graded rings completely.

Example 1.1.11. Let A = _{⨁γ∈Γ}A_{γ} and B = _{⨁γ∈Γ}B_{γ} be Γ-graded rings. Then A × B has a natural grading given by A × B = _{⨁γ∈Γ}(A × B)_{γ} where (A × B)_{γ} = A_{γ} × B_{γ}.

Example 1.1.12. \underline{Localisation of graded rings}

Let S be a central multiplicative closed subsets of Γ-graded ring A, consisting of homogeneous elements. Then S^{−1}A has a natural a Γ-graded structure. Namely, for a ∈ A^{h}, define deg(a/s) = deg(a) − deg(s) and for γ ∈ Γ,

(S^{−1}A)_{γ} = {a/s | a ∈ A^{h}, deg(a/s) = γ}.

It is easy to see that this is well-defined and make S^{−1}A a Γ-graded ring.

Many rings have a ‘canonical’ graded structure, among them, crossed products (group rings, skew group rings, twisted group rings), edge algebras, path algebras, incidence rings, etc. (see [55] for a review of these ring constructions). We will study some of these rings in this note.

Remark 1.1.13. \underline{Rings graded by a category}

There is a generalised notion of groupoid graded rings as follows. Recall that a groupoid is a small category with the property that all morphisms are isomorphisms. Let Γ be a groupoid and A be a ring. A is called Γ-groupoid graded ring, if A = _{⨁γ∈Γ}A_{γ}, where γ is a morphism of Γ, each A_{γ} is an additive subgroup of A and A_{γ}A_{δ} ⊆ A_{γδ} if the morphism γδ is defined and A_{γ}A_{δ} = 0, otherwise. For a group Γ, considering it as a category with one element and Γ as the set of morphisms, we recover the Γ-group graded ring A. One can develop the theory of groupoid graded rings parallel and similar to the group graded rings. See [65] for this approach. For a general notion of a ring graded by a category see [1, §2], where it is shown that the category of graded modules (graded by a category) is a Grothendieck category.

Remark 1.1.14. \underline{Rings graded by a semigroup}

In the definition of a graded ring (§1.1.1), one can replace the group grading with a semigroup. With this setting, the tensor algebras of Example 1.1.3 are ℕ-graded rings. A number of results on the group graded rings can also be established in the more general setting of rings graded by cancellative monoids or semigroups (see for example [22, §11]). However, in this note we only consider group graded rings.

Remark 1.1.15. \underline{Graded rings without identity}

For a ring without identity, one defines the concept of the graded ring exactly as when the ring has an identity. The concept of the strongly graded is defined similarly. In several occasions in this note we construct graded rings without identity. For example, the Leavitt path algebras arising from infinite graphs are graded rings without identity §1.6.3. See also §1.6.1, the graded free rings.

1.1.2

Let A be a Γ-graded ring. By Proposition 1.1.1, 1 ∈ A_{0}. This implies A_{0}A_{γ} = A_{γ} and A_{γ}A_{0} = A_{γ} for any γ ∈ Γ. If these equality holds for any two arbitrary elements of Γ, we call the ring a strongly graded ring. Namely, a Γ-graded ring A = _{⨁γ∈Γ}A_{γ} is called a strongly graded ring if A_{γ}A_{δ} = A_{γ+δ} for all γ, δ ∈ Γ. A graded ring A is called a crossed-product if there is an invertible element in every homogeneous component A_{γ} of A; that is, A^{∗} ∩ A_{γ} ≠ ∅ for all γ ∈ Γ, where A^{∗} is the group of all invertible elements of A. We define the support of homogeneous elements of A as

Γ_{A}^{∗} = {γ ∈ Γ | A_{γ}^{∗} ≠ ∅},  (1.2)

where A_{γ}^{∗} := A^{∗} ∩ A_{γ}. It is easy to see that Γ_{A}^{∗} is a subgroup of Γ_{A} (see Proposition 1.1.1(4)). Clearly A is a crossed-product if and only if Γ_{A}^{∗} = Γ.

Proposition 1.1.16. Let A = ⨁_{γ∈Γ} A_{γ} be a Γ-graded ring. Then

(1) A is strongly graded if and only if 1 ∈ A_{γ}A_{−γ} for any γ ∈ Γ;

(2) If A is strongly graded then the support of A is Γ;

(3) Any crossed-product ring is strongly graded.

Proof. (1) If A is strongly graded, then 1 ∈ A_{0} = A_{γ}A_{−γ} for any γ ∈ Γ. For the converse, the assumption 1 ∈ A_{γ}A_{−γ} implies that A_{0} = A_{γ}A_{−γ} for any γ ∈ Γ. Then for σ, δ ∈ Γ,

A_{σ+δ} = A_{e}A_{σ+δ} = (A_{σ}A_{−σ})A_{σ+δ} = A_{σ}(A_{−σ}A_{σ+δ}) ⊆ A_{σ}A_{δ} ⊆ A_{σ+δ}

proving A_{σδ} = A_{σ}A_{δ}, so A is strongly graded.

(2) By (1), 1 ∈ A_{γ}A_{−γ} for any γ ∈ Γ. This implies A_{γ} ≠ 0 for any γ, i.e., Γ_{A} = Γ.

(3) Let A be a crossed-product ring. By definition, for γ ∈ Γ, there exists a ∈ A^{∗} ∩ A_{γ}. So a^{−1} ∈ A_{−γ} by Proposition 1.1.1(4) and 1 = aa^{−1} ∈ A_{γ}A_{−γ}. Thus A is strongly graded by (1).

The converse of (3) in Proposition 1.1.16 does not hold. One can prove that if A is strongly graded and A_{0} is a local ring, then A is a crossed-product algebra (see [73, Theorem 3.3.1]). In §1.6 we give examples of strongly graded algebra A such that A is crossed-product but A_{0} is not a local ring. We also give an example such that A_{0} is not local and A is not crossed-product (Example 1.6.18).

If Γ is a finitely generated group, generated by the set {γ_{1}, …, γ_{n}}, then (1) in Proposition 1.1.16 can be simplified to the following: A is strongly graded if and only if 1 ∈ A_{γ_{i}}A_{−γ_{i}} and 1 ∈ A_{−γ_{i}}A_{γ_{i}}, where 1 ≤ i ≤ n. Thus if Γ = ℤ, in order that A to be strongly graded, we only need to have 1 ∈ A_{1}A_{−1} and 1 ∈ A_{−1}A_{1}. This will be used, for example, in Proposition 1.6.6 to show that certain corner skew Laurent polynomial rings (§1.6.2) are strongly graded.

Example 1.1.17. \underline{Constructing strongly graded rings via tensor products}

Let A and B be Γ-graded rings. Then by Example 1.1.6, A ⊗_{ℤ} B is a Γ-graded ring. If one of the rings is strongly graded then A ⊗_{ℤ} B is strongly graded. Indeed, suppose B is strongly graded. Then by Proposition 1.1.16(1) for any γ ∈ Γ, 1 ∈ B_{γ}B_{−γ}. So

1 ⊗ 1 ∈ A_{0} ⊗ B_{γ}B_{−γ} ⊆ (A_{0} ⊗ B_{γ})(A_{0} ⊗ B_{−γ}) ⊆ (A ⊗_{ℤ} B)_{γ}(A ⊗_{ℤ} B)_{−γ}.

Again, Proposition 1.1.16(1) implies that A ⊗_{ℤ} B is strongly graded.

As an specific case, suppose A is a ℤ-graded ring. Then A[x, x^{−1}] = A ⊗ ℤ[x, x^{−1}] is a strongly graded ring. Notice that with this grading, A[x, x^{−1}]_{0} ≅ A.

Example 1.1.18. \underline{Strongly graded as a Γ/Ω-graded ring}

Let A be a Γ-graded ring. Using Proposition 1.1.16, it is easy to see that if A is a strongly Γ-graded ring, then it is also strongly Γ/Ω-graded ring, where Ω is a subgroup of Γ. However the strongly graded is not a “closed” property, i.e, if A is strongly Γ/Ω-graded ring and A_{Ω} is strongly Ω-graded ring, it does not follow that A is strongly Γ-graded. For example, for a field K, consider the ℤ-graded ring K[x^{3}, x^{−3}] and the subgroup 2ℤ.

1.1.3 Crossed-product

Natural examples of strongly graded rings are crossed-product algebras (see Proposition 1.1.16(3)). They cover, as special cases, the skew group rings and twisted groups rings. We briefly describe the construction here.

Let A be a ring, Γ a group (as usual we use the additive notation), and let ϕ : Γ → Aut(A) and ψ : Γ × Γ → A^{∗} be maps such that for any α, β, γ ∈ Γ and a ∈ A,

(i) ^{α}(^{β}a) = ψ(α, β)^{α+β}a ψ(α, β)^{−1},

(ii) ψ(α, β)ψ(α + β, γ) = ^{α}ψ(β, γ) ψ(α, β + γ),

(iii) ψ(α, 0) = ψ(0, α) = 1

Here for α ∈ Γ and a ∈ A, ϕ(α)(a) is denoted by ^{α}a. The map ψ is called a 2-cocycle map. Denote by A_{ψ}^{ϕ}[Γ] the free A-module with the basis Γ, and define the multiplication by

(aα)(bβ) = a^{α}b ψ(α, β)(α + β).

One can show that this is a ring, and is Γ-graded ring with homogeneous components Aγ, γ ∈ Γ. In fact γ ∈ Aγ is invertible, so A_{ψ}^{ϕ}[Γ] is a crossed-product algebra [73, Proposition 1.4.1].

On the other hand any crossed-product algebra is of this form (see [73, §1.4]): for any γ ∈ Γ choose u_{γ} ∈ A^{∗} ∩ A_{γ} and define ϕ : Γ → Aut(A_{0}) by ϕ(γ)(a) = u_{γ}au_{γ}^{−1} for γ ∈ Γ and a ∈ A_{0}. Furthermore define the cocycle map ψ : Γ × Γ → A_{0}^{∗} by ψ(ζ, η) = u_{ζ}u_{η}u_{ζ+η}^{−1}. Then A = A_{0}_{ψ}^{ϕ}[Γ] = ⨁γ∈ΓA_{0}γ with multiplication (aζ)(bη) = a^{ζ}bψ(ζ, η)(ζ + η), where ^{ζ}b is defined as ϕ(ζ)(b).

Note that when Γ is cyclic, one can choose u_{i} = u_{1}^{i} for u_{1} ∈ A^{∗} ∩ A_{1} and thus the cocycle map ψ is trivial, ϕ is a homomorphism and the crossed-product is a skew group ring denoted by A_{0} ⋆_{ϕ} ℤ. Moreover if u_{1} is in the centre of A, then ϕ is the identity map and the crossed-product reduces to the group ring A_{0}[Γ].

If ψ : Γ × Γ → A^{∗} is a trivial map, i.e., ψ(α, β) = 1 for all α, β ∈ Γ, then Conditions (ii) and (iii) trivially hold, and Condition (i) reduces to ^{α}(^{β}a) = ^{α+β}a which means that ϕ : Γ → Aut(A) is a group homomorphism. In this case A_{ψ}^{ϕ}[Γ], denoted by A ⋆_{ϕ} Γ, is a skew group ring with multiplication

(aα)(bβ) = a^{α}b (α + β).  (1.3)

If ϕ : Γ → Aut(A) is trivial, i.e., ϕ(α) = 1_{A} for all α ∈ Γ, then Condition (i) implies that ψ(α, β) ∈ Z(A) ∩ A^{∗} for any α, β ∈ Γ. In this case A_{ψ}^{ϕ}[Γ], denoted by A_{ψ}[Γ], is a twisted group ring with multiplication

(aα)(bβ) = abψ(α, β)(α + β).  (1.4)

A well-known theorem in the theory of central simple algebras states that if D is a central simple F -algebra with a maximal subfield L such that L/F is a Galois extension and [A : F] = [L : F]^{2}, then D is a crossed-product, with Γ = Gal(L/F) and A = L (see [33, §12, Theorem 1]).

1.1.4

Some of the graded rings we are dealing with in this note are of the form K[x, x^{−1}], where K is a field. This is an example of a graded field.

A Γ-graded ring A = ⨁_{γ∈Γ} A_{γ} is called a graded division ring if every non-zero homogeneous element has a multiplicative inverse. If A is also a commutative ring, then A is called a graded field.

Let A be a Γ-graded division ring. It follows from Proposition 1.1.1(4) that Γ_{A} is a group, so we can write A = ⨁_{γ∈Γ_{A}} A_{γ}. Then, as a Γ_{A}-graded ring, A is a crossed-product and it follows from Proposition 1.1.16(3) that A is strongly Γ_{A}-graded. Note that if Γ_{A} ≠ Γ, then A is not strongly Γ-graded. Also note that if A is a graded division ring, then A_{0} is a division ring.

Theorem 1.1.19. Let A = ⨁_{γ∈Γ} A_{γ} be a Γ-graded ring with A_{0} a field. Then A is a graded division ring if and only if A is isomorphic to a twisted group ring A_{0}_{ψ}[Γ], where ψ : Γ × Γ → A^{∗} is a 2-cocycle.

Proof. Suppose A_{0}_{ψ}[Γ] is a twisted group ring. For any non-zero homogeneous element dγ of A_{0}_{ψ}[Γ], by (1.4), we have

(dγ)(d^{−1}ψ(γ, −γ)^{−1}(−γ)) = 1(0) = 1_{A_{0}_{ψ}[Γ]}.

Thus any non-zero homogeneous element of A_{1}_{ψ}[Γ] is invertible. This shows that A_{1}_{ψ}[Γ] is a graded division ring.

Conversely, suppose A is a graded division ring. Observe that for any α ∈ Γ, A_{α} is a A_{0}-vector space of dimension 1. For 0 ∈ Γ set a_{0} = 1 ∈ A_{0} and for any other α ∈ Γ, fix a non-zero a_{α} ∈ A_{α}. Then

a_{α}a_{β} = ψ(α, β)a_{α+β},  (1.5)

for some ψ(α, β) ∈ A_{0}. From (1.5) it easily follows that for any α ∈ Γ, ψ(α, 0) = ψ(0, α) = 1. Furthermore, the associativity of A shows that

ψ(α, β)ψ(α + β, γ) = ψ(β, γ)ψ(α, β + γ).

Thus ψ is a 2-cocycle. Define the map

A_{0}_{ψ}[Γ] ⟶ A
∑_{α∈Γ} r_{α}α ⟼ ∑_{α∈Γ} r_{α}a_{α}.

This map is a graded surjective homomorphism. Since A_{0}_{ψ}[Γ] is graded simple (i.e., has no two sided graded ideals, see §1.1.5), this map is a graded isomorphism. This completes the proof.

In the following we give some concrete examples of graded division rings.

Example 1.1.20. \underline{The Veronese subring}

Let A = ⨁_{i∈ℤ} A_{i} be a ℤ-graded ring. The nth-Veronese subring of A is defined as A^{(n)} = ⨁_{i∈ℤ} A_{in}. This is a ℤ-graded ring. It is easy to see that the support of A is ℤ if and only if the support of A^{(n)} is ℤ. Note that if A is strongly graded, so is A^{(n)}.

Let D be a division ring and let A = D[x, x^{−1}] be the Laurent polynomial rings. The elements of A consist of finite sums ∑_{i∈ℤ} a_{i}x^{i}, where a_{i} ∈ D. Then A is a ℤ-graded division ring with A = ⨁_{i∈ℤ} A_{i}, where A_{i} = {ax^{i} | a ∈ D}. Consider the nth-Veronese subring A^{(n)} which is the ring D[x^{n}, x^{−n}]. The elements of A^{(n)} consist of finite sums ∑_{i∈ℤ} a_{i}x^{in}, where a_{i} ∈ D. Then A^{(n)} is a ℤ-graded division ring, with A^{(n)} = ⨁_{i∈ℤ} A_{in}. Here both A and A^{(n)} are strongly graded rings.

There is also another way to consider the ℤ-graded ring B = D[x^{n}, x^{−n}] such that it becomes a graded subring of A = D[x, x^{−1}]. Namely, we define B = ⨁_{i∈ℤ} B_{i}, where B_{i} = Dx^{i} if i ∈ nℤ and B_{i} = 0 otherwise. This way B is a graded division ring and a graded subring of A. The support of B is clearly the subgroup nℤ of ℤ.

Example 1.1.21. \underline{Different gradings on a graded division ring}

Let ℍ = ℝ⊕ℝi⊕ℝj ⊕ℝk be the real quaternion algebra, with multiplication defined by i^{2} = −1, j^{2} = −1 and ij = −ji = k. It is known that ℍ is a noncommutative division ring with centre ℝ. We give ℍ two different graded division ring structures, with grade groups ℤ_{2} and ℤ_{2} × ℤ_{2}, respectively as follows.

ℤ_{2}-grading: Let ℍ = ℂ_{0} ⊕ ℂ_{1}, where ℂ_{0} = ℝ ⊕ ℝi and ℂ_{1} = ℂj = ℝj ⊕ ℝk. One can check that ℂ_{0}ℂ_{0} = ℂ_{0}, ℂ_{0}ℂ_{1} = ℂ_{1}ℂ_{0} = ℂ_{1} and ℂ_{1}ℂ_{1} = ℂ_{0}. This makes ℍ a strongly ℤ_{2}-graded division ring.

ℤ_{2} × ℤ_{2}-grading: Let ℍ = R_{(0,0)} ⊕ R_{(1,0)} ⊕ R_{(0,1)} ⊕ R_{(1,1)}, where

R_{(0,0)} = ℝ, R_{(1,0)} = ℝi, R_{(0,1)} = ℝj, R_{(1,1)} = ℝk.

It is routine to check that ℍ forms a strongly ℤ_{2} × ℤ_{2}-graded division ring.

We give here another example, which generalises the above example of ℍ as a ℤ_{2} × ℤ_{2}-graded ring.

Example 1.1.22. \underline{Symbol algebras} Let F be a field, ξ be a primitive n-th root of unity and let a, b ∈ F^{∗}. Let

A = ⨁_{i=0}^{n−1} ⨁_{j=0}^{n−1} F x^{i}y^{j}

be the F -algebra generated by the elements x and y, which are subject to the relations x^{n} = a, y^{n} = b and xy = ξyx. By [33, Thm. 11.1], A is an n^{2}-dimensional central simple algebra over F. We will show that A forms a graded division ring. Clearly A can be written as a direct sum

A = ⨁_{(i,j)∈ℤ_{n}⊕ℤ_{n}} A_{(i,j)}, where A_{(i,j)} = F x^{i}y^{j}

and each A_{(i,j)} is an additive subgroup of A. Using the fact that ξ^{−kj}x^{k}y^{j} = y^{j}x^{k} for each j, k, with 0 ≤ j, k ≤ n − 1, we can show that A_{(i,j)}A_{(k,l)} ⊆ A_{([i+k],[j+l])}, for i, j, k, l ∈ ℤ_{n}. A non-zero homogeneous element fx^{i}y^{j} ∈ A_{(i,j)} has an inverse

f^{−1}a^{−1}b^{−1}ξ^{−ij}x^{n−i}y^{n−j},

proving A is a graded division ring. Clearly the support of A is ℤ_{n} × ℤ_{n}, so A is strongly ℤ_{n} × ℤ_{n}- graded.

These examples can also be obtained from graded free rings (see Example 1.6.3).

Example 1.1.23. \underline{A good counter-example}

In the theory of graded rings, in many instances it has been established that if the graded group Γ is finite (or in some cases, finitely generated), then the graded property descends to the nongraded setting. For example, one can prove that if a ℤ-graded ring is graded Artinian (Noetherian), then the ring is Artinian (Noetherian). One good example which provides counter-examples to such phenomena is the following graded field.

Let K be a field and A = K[x_{1}^{±1}, x_{2}^{±1}, x_{3}^{±1}, …] be a Laurent polynomial ring with infinite variables. This ring is a graded field with its ‘canonical’ ⨁_{∞} ℤ-grading.

1.1.5 Graded ideals

Let A be a Γ-graded ring. A two-sided ideal I of A is called a graded ideal (or homogeneous ideal) if

I = ⨁_{γ∈Γ}(I ∩ A_{γ}).

There are similar notions of a graded subring, a graded left and graded right ideal.

When I is a graded ideal of A, the quotient ring A/I forms a graded ring, with

A/I = ⨁_{γ∈Γ}(A/I)_{γ}, where (A/I)_{γ} = (A_{γ} + I)/I.  (1.6)

With this grading (A/I)_{0} ≅ A_{0}/I_{0}, where I_{0} = A_{0} ∩ I. It is not difficult to show that an ideal I of A is a graded ideal if and only if I is generated as a two-sided ideal of A by homogeneous elements. Also, for a two sided ideal I of A, if (1.6) induces a grading on A/I, then I has to be a graded ideal. It is also easy to see that if A is a strongly graded or a crossed product, so is the graded quotient ring A/I.

Example 1.1.24. \underline{Symmetric algebras and exterior algebras}

Recall from Example 1.1.3 that for a commutative ring A and an A-module M, the tensor algebra T (M) is a ℤ-graded ring with support ℕ. The symmetric algebra S(M) is defined as the quotient of T (M) by the ideal generated by elements x ⊗ y − y ⊗ x, x, y ∈ M. Since these elements are homogeneous of degree two, S(M) is a ℤ-graded commutative ring.

Similarly, the exterior algebra of M, denoted by ⋀ M, is defined as the quotient of T (M) by the ideal generated by homogeneous elements x ⊗ x, x ∈ M. So ⋀ M is a ℤ-graded ring.

A nonzero graded ring A is said to be graded simple if the only graded two-sided ideals of A are {0} and A. The structure of graded simple Artinian rings are known (see Remark 1.4.8).

Recall that for Γ-graded rings A and B, a Γ-graded ring homomorphism f : A → B is a ring homomorphism such that f(A_{γ}) ⊆ B_{γ} for all γ ∈ Γ. It can easily be shown that ker(f) is a graded ideal of A and im(f) is a graded subring of B. It is also easy to see that f is injective (surjective/bijective) if and only if for any γ ∈ Γ, the restriction of f on A_{γ} is injective (surjective/bijective).

Note that if Γ is an abelian group, then the centre of a graded ring A, Z(A), is a graded subring of A.

Example 1.1.25. \underline{The centre of the graded ring}

If a group Γ is not abelian, then the centre of a Γ-graded ring may not be a graded subring. For example, let Γ = S_{3} = {e, a, b, c, d, f} be the symmetric group of order 3, where

a = (23), b = (13), c = (12), d = (123), f = (132).

Let A be a ring, and consider the group ring R = A[Γ], which is a Γ-graded ring by Example 1.1.2. Let x = 1d + 1f ∈ R, where 1 = 1_{A}, and we note that x is not homogeneous in R. Then x ∈ Z(R), but the homogeneous components of x are not in the centre of R. As x is expressed uniquely as the sum of homogeneous components, we have x ∉ ⨁_{γ∈Γ}(Z(R) ∩ R_{γ}).

This example can be generalised by taking a non-abelian finite group Γ with a subgroup Ω which is normal and non-central. Let A be a ring and consider the group ring R = A[Γ] as above. Then x = ∑_{ω∈Ω} 1ω is in the centre of R, but the homogeneous components of x are not all in the centre of R.

Remark 1.1.26. Let Γ and Λ be two groups. Let A be a Γ-graded ring and B be a Λ-graded ring. Suppose f : A → B is a ring homomorphism and g : Γ → Λ a group homomorphism such that for any γ ∈ Γ, f(A_{γ}) ⊆ B_{g(γ)}. Then f is called a Γ − Λ-graded homomorphism. In case Γ = Λ and g = id, we recover the usual definition of a Γ-graded homomorphism. For example, if Ω is a subgroup of Γ, then the identity map 1_{A} : A → A is a Γ − Γ/Ω-graded homomorphism, where A is considered as Γ and Γ/Ω-graded rings, respectively (see Example 1.1.8).

Throughout this note, we fix a given group Γ and we work with the Γ-graded category and all our considerations are within this category (See Remark 2.3.12 for references to literature where mixed grading is studied).

Graded prime and maximal ideals

A graded ideal P of Γ-graded ring A is called a graded prime ideal of A if P ≠ A and if for two graded ideals I, J, IJ ⊆ P, then I ⊆ P or J ⊆ P. If A is commutative, we obtain the familiar formulation that P is a graded prime ideal if and only if for x, y ∈ A^{h}, xy ∈ P implies that x ∈ P or y ∈ P. Note that a graded prime ideal is not necessarily a prime ideal.

A graded maximal ideal of a Γ-graded ring A is defined to be a proper graded ideal of A which is maximal among the set of proper graded ideals of A. Using Zorn’s lemma, one can show that graded maximal ideals exist, and it is not difficult to show that a graded maximal ideal is a graded prime. For a graded commutative ring, a graded ideal is maximal if and only if its quotient ring is graded field. There are similar notions of graded maximal left and right ideals.

Parallel to the non-graded setting, for a Γ-graded ring A, the graded Jacobson radical, J^{gr}(A), is defined as the intersection of all graded left maximal ideals of A. This coincides with the intersection of all graded right maximal ideals and so J^{gr}(A) is a two-sided ideal (see [73, Proposition 2.9.1]). We denote by J(A) the usual Jacobson radical.

1.1.6 Graded von Neumann regular rings

The von Neumann regular rings constitute an important class of rings. A unital ring A is von Neumann regular, if for any a ∈ A, we have a ∈ aAa. There are several equivalent module theoretical definitions, such as A is von Neumann regular if and only if any module over A is flat. This gives a comparison with the class of semisimple rings, which are characterised by the property that any module is projective. Goodearl’s book [38] is devoted to the class of von Neumann regular rings. The definition extends to non-unital ring in an obvious manner.

If a ring has a graded structure, one defines the graded version of regularity in a natural way: the graded ring A is called a graded von Neumann regular, if for any homogeneous element a ∈ A, we have a ∈ aAa. This means, for any homogeneous element a ∈ A, one can find a homogeneous element b ∈ A such that a = aba. As an example, a direct sum of graded division rings is a graded von Neumann regular ring. Many of the module theoretic properties established for von Neumann regular rings can be extended to the graded setting; For example, A is graded regular if and only if any graded module is (graded) flat. We refer the reader to [72, C, I.5] for a treatment of such rings and [10, §2.2] for a concise survey. Several of the graded rings we construct in this note are graded von Neumann regular, such as Leavitt path algebras (Corollary 1.6.13) and corner skew Laurent series (Proposition 1.6.7).

In this section, we briefly give some of the properties of graded von Neumann regular rings. The following proposition is the graded version of [38, Theorem 1.1] which has a similar proof.

Proposition 1.1.27. Let A be a Γ-graded ring. The following statements are equivalent.

(1) A is a graded von Neumann regular ring;

(2) Any finitely generated right (left) graded ideal of A is generated by one homogeneous idempotent.

Proof. (1) ⇒ (2) First we show that any principal graded ideal is generated by a homogeneous idempotent. So consider the principal ideal xA, where x ∈ A^{h}. By the assumption, there is y ∈ A^{h} such that xyx = x. This immediately implies xA = xyA. Now note that xy is homogeneous idempotent.

Next we will prove the claim for graded ideals generated by two elements. The general case follows by an easy induction. So let xA + yA be a graded ideal generated by two homogeneous elements x, y. By the previous paragraph, xA = eA for a homogeneous idempotent e. Note that y − ey ∈ A^{h} and y − ey ∈ xA + yA. Thus

xA + yA = eA + (y − ey)A.  (1.7)

Again, the previous paragraph gives us a homogeneous idempotent f such that (y − ey)A = fA. Let g = f − fe ∈ A_{0}. Notice that ef = 0 which implies that e and g are orthogonal idempotents. Furthermore, fg = g and gf = f. It then follows gA = fA = (y − ey)A. Now from (1.7), we get

xA + yA = eA + gA = (e + g)A.

(2) ⇒ (1) Let x ∈ A^{h}. Then xA = eA for some homogeneous idempotent e. Thus x = ea and e = xy for some a, y ∈ A^{h}. Then x = ea = eea = ex = xyx.

Recall that a (graded) ring is called a (graded) semi-prime if for any (graded) ideal I in A, I^{n} ⊆ A, n ∈ ℕ, implies I ⊆ A.

Proposition 1.1.28. Let A be a Γ-graded von Neumann regular ring. Then

(1) Any graded right (left) ideal of A is idempotent;

(2) Any graded ideal is graded semi-prime;

(3) Any finitely generated right (left) graded ideal of A is a projective module.

Furthermore, if A is a ℤ-graded regular ring then,

(4) J(A) = J^{gr}(A) = 0.

Proof. The proofs of (1)-(3) are similar to the non-graded case [38, Corollary 1.2]. We provide the easy proofs here.

(1) Let I be a graded right ideal. For any homogeneous element x ∈ I, there is y ∈ A^{h} such that x = xyx. Thus x = (xy)x ∈ I^{2}. It follows that I^{2} = I.

(2) This follows immediately from (1).

(3) By Proposition 1.1.27, any finitely generated right ideal is generated by a homogeneous idempotent. However this latter ideal is a direct summand of the ring, and so is a projective module.

(4) By Bergman’s observation, for a ℤ-graded ring A, J(A) is a graded ideal and J(A) ⊆ J^{gr}(A) (see [18]). By Proposition 1.1.27, J^{gr}(A) contains an idempotent, which then forces J^{gr}(A) = 0.

Later in the note, in Corollary 1.5.9, we show that if A is a strongly graded ring, then A is graded von Neumann regular if and only if A_{0} is a von Neumann regular ring. The proof uses the equivalence of suitable categories over the rings A and A_{0}. An element-wise proof of this fact can also be found in [95, Theorem 3].

1.2 Graded modules

1.2.1

Let A be a Γ-graded ring. A graded right A-module M is defined to be a right A-module M with a direct sum decomposition M = ⨁_{γ∈Γ} M_{γ}, where each M_{γ} is an additive subgroup of M such that M_{λ} · A_{γ} ⊆ M_{γ+λ} for all γ, λ ∈ Γ.

For Γ-graded right A-modules M and N, a Γ-graded module homomorphism f : M → N is a module homomorphism such that f(M_{γ}) ⊆ N_{γ} for all γ ∈ Γ. A graded homomorphism f is called a graded module isomorphism if f is bijective and, when such a graded isomorphism exists, we write M ≅_{gr} N. Notice that if f is a graded module homomorphism which is bijective, then its inverse f^{−1} is also a graded module homomorphism.

1.2.2 Shifting of modules

For δ ∈ Γ, we define the δ-suspended, or δ-shifted module, A-module, M(δ) as

M(δ) = ⨁_{γ∈Γ} M(δ)_{γ}, where M(δ)_{γ} = M_{γ+δ}.

This shifting plays a pivotal role in the theory of graded rings. For example, if M is a ℤ-graded A-module, then the following table shows how the shifting like “the tick of the clock” moves the homogeneous components of M to the left.

degrees -3  -2  -1  0  1  2  3
M  M_{−1} M_{0} M_{1} M_{2}
M(1)  M_{−1} M_{0} M_{1} M_{2}
M(2)  M_{−1} M_{0} M_{1} M_{2}

Let M be a Γ-graded right A-module. A submodule N of M is called a graded submodule if

N = ⨁_{γ∈Γ}(N ∩ M_{γ}).

Example 1.2.1. Let A be a Γ-graded ring and a ∈ A be a homogeneous element of degree ω. Then aA is a graded right A-module with α ∈ Γ homogeneous component defined as (aA)_{α} := aA_{α−ω} ⊆ A_{α}. With this grading aA is a graded submodule (and graded right ideal) of A.

However, note that defining the grading on aA as (aA)_{α} := aA_{α} ⊆ A_{α+ω} makes aA a graded submodule of A(ω).

There are similar notions of graded left and graded bi-submodules (see §1.2.5). When N is a graded submodule of M, the factor module M/N forms a graded A-module, with

M/N = ⨁_{γ∈Γ}(M/N)_{γ}, where (M/N)_{γ} = (M_{γ} + N)/N.  (1.8)

Example 1.2.2. Let A be a Γ-graded ring. Define a grading on the matrix ring 𝕄_{n}(A) as follows. For α ∈ Γ, 𝕄_{n}(A)_{α} = 𝕄_{n}(A_{α}) (for a general theory of grading on matrix rings see §1.3). Let e_{ii} ∈ 𝕄_{n}(A), 1 ≤ 1 ≤ n, be a matrix with 1 in (i, i) position and zero everywhere else, and consider e_{ii} 𝕄_{n}(A). By Example 1.2.1, e_{ii} 𝕄_{n}(A) is a graded 𝕄_{n}(A)-module and ⨁_{i=1}^{n} e_{ii} 𝕄_{n}(A) = 𝕄_{n}(A). This shows that the graded module e_{ii} 𝕄_{n}(A) is a projective module. This is an example of a graded projective module (see §1.2.9).

Example 1.2.3. Let A be ring. Consider the matrix ring 𝕄_{n}(A) as a ℤ-graded ring concentrated in degree zero. Furthermore, consider 𝕄_{n}(A) as a graded 𝕄_{n}(A)-module with the grading defined as follows: 𝕄_{n}(A)_{i} = e_{ii} 𝕄_{n}(A) for 1 ≤ i ≤ n and zero otherwise. Note that all non-zero homogeneous elements of this module are zero-divisiors. We will use this example to show that a free module which is graded is not necessarily a graded free module (§1.2.4).

1.2.3 The Hom group

For two graded right A-modules M and N, a graded A-module homomorphism of degree δ is an A-module homomorphism f : M → N, such that f(M_{γ}) ⊆ N_{γ+δ} for any γ ∈ Γ. Let Hom_{A}(M, N)_{δ} denote the subgroup of Hom_{A}(M, N) consisting of all graded A-module homomorphisms of degree δ, i.e.,

Hom_{A}(M, N)_{δ} = {f ∈ Hom_{A}(M, N) | f(M_{γ}) ⊆ N_{γ+δ}, γ ∈ Γ}.

Clearly a graded module homomorphism defined in §1.2.1 is a graded homomorphism of degree 0. By Gr-A (or Gr^{Γ}-A to emphasis the graded group of A), we denote a category consists of Γ-graded right A-modules as objects and graded homomorphisms as the morphisms, i.e.,

Hom_{Gr-A}(M, N) = Hom_{A}(M, N)_{0}.

Furthermore, for α ∈ Γ, one can write

Hom_{Gr-A} (M(−α), N) = Hom_{Gr-A} (M, N(α)) = Hom_{A}(M, N)_{α}.  (1.9)

A full subcategory of Gr-A, consisted of graded finitely generated A-modules is denoted by gr- A.

For α ∈ Γ, the α-suspension functor or shift functor

𝒯_{α} : Gr-A ⟶ Gr-A,
M ⟼ M(α),

is an isomorphism with the property 𝒯_{α}𝒯_{β} = 𝒯_{α+β}, α, β ∈ Γ.

Remark 1.2.4. Let A be a Γ-graded ring and Ω be a subgroup of Γ such that Γ_{A} ⊆ Ω ⊆ Γ. Then the ring A can be considered naturally as a Ω-graded ring. Similarly, if A, B are Γ-graded rings and f : A → B is a Γ-graded homomorphism and Γ_{A}, Γ_{B} ⊆ Ω ⊆ Γ, then the homomorphism f can be naturally considered as a Ω-graded homomorphism. In this case, to make a distinction, we write Gr^{Γ}-A for the category of Γ-graded A-modules and Gr^{Ω}-A for the category of Ω-graded A-modules.

Theorem 1.2.5. For graded right A-modules M and N, if M is finitely generated, then Hom_{A}(M, N) has a natural decomposition

Hom_{A}(M, N) = ⨁_{γ∈Γ} Hom_{A}(M, N)_{γ}.  (1.10)

Furthermore, Hom_{A}(M, M) is a Γ-graded ring.

Proof. Let f ∈ Hom_{A}(M, N). For λ ∈ Γ, define f_{λ} ∈ Hom_{A}(M, N) as follows: for m ∈ M,

f_{λ}(m) = ∑_{γ∈Γ} f(m_{γ−λ})_{γ},  (1.11)

where m = ∑_{γ∈Γ} m_{γ}. If m ∈ M_{α}, α ∈ Γ, then (1.11) reduces to f_{λ}(m) = f(m)_{α+λ} ⊆ M_{α+λ}. This shows that f_{λ} ∈ Hom_{A}(M, N)_{λ}. Also, f_{λ}(m) is zero for all but a finite number of λ ∈ Γ. Furthermore, ∑_{λ} f_{λ}(m) = ∑_{λ} f(m)_{α+λ} = f(m). Now since M is finitely generated, there is a finite number of homogeneous elements which generate any m ∈ M. The above argument shows only a finite number of f_{λ}(m) are not zero and f = ∑_{λ} f_{λ}. This shows Hom_{A}(M, N) = ∑_{γ∈Γ} Hom_{A}(M, N)_{γ}. Finally, it is easy to see that Hom_{A}(M, N)_{γ}, γ ∈ Γ, constitute a direct sum.

For the second part, replacing N by M in (1.10), we get Hom_{A}(M, M) = ⨁_{γ∈Γ} Hom_{A}(M, M)_{γ}. Furthermore, if f ∈ Hom_{A}(M, M)_{γ} and g ∈ Hom_{A}(M, M)_{λ} then fg ∈ Hom_{A}(M, M)_{γ+λ}. This shows that when M is finitely generated, the ring Hom_{A}(M, M) is a Γ-graded ring. This complete the proof.

Let M be a graded finitely generated right A-module. Then the usual dual of M, i.e., M^{∗} = Hom_{A}(M, A), is a left A-module. Furthermore, using Theorem 1.2.5, one can check that M^{∗} is a graded left A-module. Since

Hom(M, N)(α) = Hom(M(−α), N) = Hom(M, N(α)),

we have

M(α)^{∗} = M^{∗}(−α).  (1.12)

This should also make sense as if we “push forward” M with α, then we “pull back” the dual M^{∗} by α.

When M is a free module, Hom_{A}(M, M) can be represented by a matrix ring over A. Next we define graded free modules. In §1.3 we will see that if M is a graded free module, the graded ring Hom_{A}(M, M) can be represented by matrix ring over A with a very concrete grading.

1.2.4 Graded free modules

A graded (right) A-module F is called a graded free A-module if F is a free right A-module with a homogeneous base. Clearly a graded free module is a free module but the converse is not correct (see Example 1.2.3).

Consider a Γ-graded A-module ⨁_{i∈I} A(δ_{i}), where I is an indexing set and δ_{i} ∈ Γ. Note that for each i ∈ I, the element e_{i} of the standard basis (i.e., 1 in the ith component and zero elsewhere) is homogeneous of degree −δ_{i}. The set {e_{i}}_{i∈I} forms a base for ⨁_{i∈I} A(δ_{i}), which by definition makes this a graded free A-module. On the other hand, a graded free A-module F with a homogeneous base {b_{i}}_{i∈I}, where deg(b_{i}) = −δ_{i} is graded isomorphic to ⨁_{i∈I} A(δ_{i}). Indeed one can easily observe that the map induced by

φ : ⨁_{i∈I} A(δ_{i}) ⟶ F
e_{i} ⟼ b_{i}

is a graded A-module isomorphism.

1.2.5 Graded bimodules

The notion of the graded left A-modules is developed similarly. The category of graded left Amodules with graded homomorphisms is denoted by A -Gr. In a similar manner for Γ-graded rings A and B, we can consider the graded A − B bimodule M. Namely, M is a A − B-bimodule and additionally M = ⨁_{γ∈Γ} M_{γ} is a graded left A-module and a graded right B-module, i.e., A_{α}M_{γ}B_{β} ⊆ M_{α+γ+β}, α, γ, β ∈ Γ. The category of graded A-bimodules is denoted by Gr-A -Gr.

1.2.6 Tensor product of graded modules

Let A be a Γ-graded ring and M_{i}, i ∈ I, be a direct system of Γ-graded A-modules, i.e., I is a directed partially ordered set and for i ≤ j, there is a graded A-homomorphism ϕ_{ij} : M_{i} → M_{j} which is compatible with the ordering. Then M := \underrightarrow{lim} M_{i} is a Γ-graded A-module with homogeneous components M_{α} = \underrightarrow{lim} M_{i}_{α} (see Example 1.1.10 for the similar construction for rings).

In particular, let {M_{i} | i ∈ I} be Γ-graded right A-modules. Then ⨁_{i∈I} M_{i} has a natural graded A-module given by (⨁_{i∈I} M_{i})_{α} = ⨁_{i∈I} M_{i}_{α}, α ∈ Γ.

Let M be a graded right A-module and N be a graded left A-modules. We will observe that the tensor product M ⊗_{A} N has a natural graded ℤ-module structure. Since each of M_{γ}, γ ∈ Γ, is a right A_{0}-module and similarly N_{γ}, γ ∈ Γ, is a left A_{0}-module, then M ⊗_{A_{0}} N can be decomposed as a direct sum M ⊗_{A_{0}} N = ⨁_{γ∈Γ}(M ⊗ N)_{γ}, where

(M ⊗ N)_{γ} = { ∑_{i} m_{i} ⊗ n_{i} | m_{i} ∈ M^{h}, n_{i} ∈ N^{h}, deg(m_{i}) + deg(n_{i}) = γ}.

Now note that M ⊗_{A} N ≅ (M ⊗_{A_{0}} N)/J, where J is a subgroup of M ⊗_{A_{0}} N generated by the homogeneous elements {ma ⊗ n − m ⊗ an | m ∈ M^{h}, n ∈ N^{h}, a ∈ A^{h}}. This shows that M ⊗_{A} N is also a graded module. It is easy to check that, for example, if N is a graded A-bimodule, then M ⊗_{A} N is a graded right A-module.

Observe that for a graded right A-module M, the map

M ⊗_{A} A(α) ⟶ M(α),  (1.13)
m ⊗ a ⟼ ma,

is a graded isomorphism. In particular, for any α, β ∈ Γ, there is a graded A-bimodule isomorphism

A(α) ⊗_{A} A(β) ≅_{gr} A(α + β).

Example 1.2.6. \underline{Graded formal matrix rings}

The construction of formal matrix rings (Example 1.1.4) can be carried over to the graded setting as follows. Let A and B be Γ-graded rings, M be a graded R − S-bimodule and N be a graded S −R-bimodule. Suppose that there are graded bimodule homomorphisms ϕ : M ⊗_{S} N → R and ψ : N ⊗_{R} M → S such that (mn)m′ = n(nm′), where we set ϕ(m, n) = mn and ψ(n, m) = nm. Consider the ring

T = (\begin{matrix}R & M \\ N & S \end{matrix}) ,

and define, for any γ ∈ Γ,

T_{γ} = (\begin{matrix}R_{γ} & M_{γ} \\ N_{γ} & S_{γ} \end{matrix}) .

One can easily check that T is a Γ-graded ring, called a graded formal matrix ring. One specific type of such rings is a Morita ring which appears in graded Morita theory (§2.3).

1.2.7 Forgetting the grading

Most forgetful functors in algebra tend to have left adjoints, which have a ‘free’ constructions. One such example is the forgetful functor from the category of abelian groups to abelian monoids that we will study in §3 in relation with the Grothendieck groups. However, some of the forgetful functors in the graded setting naturally have right adjoints as we will see below.

Consider the forgetful functor

U : Gr-A → Mod- A,  (1.14)

which simply assigns to any graded module M in Gr-A, its underlying module M in Mod- A, ignoring the grading. Similarly, the graded homomorphisms are sent to the same homomorphisms, disregarding their graded compatibilities.

There is a functor F : Mod- A → Gr-A which is a right adjoint to U. The construction is as follows: let M be an A-module. Consider the abelian group F (M) := ⨁_{γ∈Γ} M_{γ}, where M_{γ} is a copy of M. Furthermore, for a ∈ A_{α} and m ∈ M_{γ} define m.a = ma ∈ M_{α+γ}. This defines a graded A-module structure on F (M) and makes F an exact functor from Mod- A to Gr-A. One can prove that for any M ∈ Gr-A and N ∈ Mod- A, we have a bijective map

Hom_{Mod- A} (U(M), N) ⟶^{ϕ} Hom_{Gr-A} (M, F (M)),
f ⟼ ϕ_{f},

where ϕ_{f}(m_{α}) = f(m_{α}) ∈ N_{α}.

Remark 1.2.7. One can prove that for any M ∈ Gr-A, F U(M) ≅_{gr} ⨁_{γ∈Γ} M(γ). We also note that if Γ is finite, then F is also a left adjoint functor of U. Further, if U has a left adjoint functor, then one can prove that Γ is finite (see [73, §2.5] for details).

1.2.8 Partitioning of the graded modules

Let Γ be a group and Ω a subgroup of Γ. Recall from Example 1.1.8 that the functor from the category of Γ-graded rings to the category of Γ/Ω-graded rings which gives the natural forgetful functor when Ω = Γ. This functor has also a right adjoint functor (see [73, Proposition 1.2.2]).

There is a canonical ‘forgetful’ functor U : Gr^{Γ}-A → Gr^{Γ/Ω}-A, such that when Ω = Γ, it gives the functor 1.14. The construction is as follows. Let M = ⨁_{α∈Γ} M_{α} be a Γ-graded A-module. Write

M = ⨁_{Ω+α∈Γ/Ω} M_{Ω+α},  (1.15)

where

M_{Ω+α} := ⨁_{ω∈Ω} M_{ω+α}.  (1.16)

One can easily check that M is a Γ/Ω-graded A-module. Furthermore, U(M(α)) = M(Ω + α). We will use this functor to relate the Grothendieck groups of these categories in Example 3.1.10.

Let A be a Γ-graded ring and Ω be a subgroup of Γ. The above construction motivates the following which establishes a relation between the categories Gr^{Γ}-A and Gr^{Ω}-A.

Consider the quotient group Γ/Ω and fix a complete set of coset representative {α_{i}}_{i∈I}. Let β ∈ Γ and consider the permutation map ρ_{β}

ρ_{β} : Γ/Ω ⟶ Γ/Ω
Ω + α_{i} ⟼ Ω + α_{i} + β = Ω + α_{j}.

This defines a bijective map (called ρ_{β} again) ρ_{β} : {α_{i}}_{i∈I} → {α_{i}}_{i∈I}. Furthermore, for any α_{i}, since

Ω + α_{i} + β = Ω + α_{j} = Ω + ρ_{β}(α_{i}),

there is a unique w_{i} ∈ Ω such that

α_{i} + β = ω_{i} + ρ_{β}(α_{i}).  (1.17)

Define the functor

𝒫 : Gr^{Γ}-A ⟶ ⨁_{Γ/Ω} Gr^{Ω}-A_{Ω},
M ⟼ ⨁_{Ω+α_{i}∈Γ/Ω} M_{Ω+α_{i}}

(1.18)

where

M_{Ω+α_{i}} = ⨁_{ω∈Ω} M_{ω+α_{i}}.

Since M_{Ω+α}, α ∈ Γ, defined in (1.16), can be naturally considered as Ω-graded A_{Ω}-module, where

(M_{Ω+α})_{ω} = M_{ω+α},  (1.19)

the functor 𝒫 defined in 1.18 is well-defined. Note that the homogeneous components defined in (1.19), depend on the coset representation, thus choosing another complete set of coset representative gives a different functor between these categories.

For any β ∈ Γ, define a shift functor

\overline{ρ}_{β} : ⨁_{Γ/Ω} Gr^{Ω}-A_{Ω} ⟶ ⨁_{Γ/Ω} Gr^{Ω}-A_{Ω},
⨁_{Ω+α_{i}∈Γ/Ω} M_{Ω+α_{i}} ⟼ ⨁_{Ω+α_{i}∈Γ/Ω} M(ω_{i})_{Ω+ρ_{β}(α_{i})},  (1.20)

where ρ_{β}(α_{i}) and ω_{i} are defined in (1.17). The \overline{ρ}_{β} for morphisms are defined accordingly.

We are in a position to prove the next theorem.

Theorem 1.2.8. Let A be a Γ-graded ring and Ω a subgroup of Γ. Then for any β ∈ Γ, the following diagram is commutative,

\xymatrix{Gr^{Γ}-A \ar[r]^{𝒫} \ar[d]_{𝒯_{β}} & ⨁_{Γ/Ω} Gr^{Ω}-A_{Ω} \ar[d]^{_{\overline{ρ}_{β}}} \\ Gr^{Γ}-A \ar[r]^{𝒫} & ⨁_{Γ/Ω} Gr^{Ω}-A_{Ω},}  (1.21)

where the functor 𝒫 is defined in (1.18). Furthermore, if Γ_{A} ⊆ Ω, then the functor 𝒫 induces an equivalence of categories.

Proof. We first show that Diagram 1.21 is commutative. Let β ∈ Γ and M a Γ-graded A-module. As in (1.17), let {α_{i}} be a fixed complete set of coset representative and

α_{i} + β = ω_{i} + ρ_{β}(α_{i}).

Then

𝒫(𝒯_{β}(M)) = 𝒫(M(β)) = ⨁_{Ω+α_{i}∈Γ/Ω} M(β)_{Ω+α_{i}}.  (1.22)

But

M(β)_{Ω+α_{i}} = ⨁_{ω∈Ω} M(β)_{ω+α_{i}} = ⨁_{ω∈Ω} M_{ω+α_{i}+β} = ⨁_{ω∈Ω} M_{ω+ω_{i}+ρ_{β}(α_{i})} =

⨁_{ω∈Ω} M(ω_{i})_{ω+ρ_{β}(α_{i})} = M(ω_{i})_{Ω+ρ_{β}(α_{i})}.

Replacing this into equation 1.22 we have

𝒫(𝒯_{β}(M)) = ⨁_{Ω+α_{i}∈Γ/Ω} M(ω_{i})_{Ω+ρ_{β}(α_{i})}.  (1.23)

On the other hand by (1.20),

\overline{ρ}_{β}𝒫(M) = \overline{ρ}_{β}( ⨁_{Ω+α_{i}∈Γ/Ω} M_{Ω+α_{i}}) = ⨁_{Ω+α_{i}∈Γ/Ω} M(ω_{i})_{Ω+ρ_{β}(α_{i})}.  (1.24)

Comparing (1.23) and (1.24) shows that the Diagram 1.21 is commutative.

For the last part, suppose Γ_{A} ⊆ Ω. We construct a functor 𝒫′ : ⨁_{Γ/Ω} Gr^{Ω}-A_{Ω} → Gr^{Γ}-A, which depends on the coset representative {α_{i}}_{i∈I} of Γ/Ω. First note that any α ∈ Γ can be written uniquely as α = α_{i} + ω, where ω ∈ Ω. Now let

⨁_{Ω+α_{i}∈Γ/Ω} N_{Ω+α_{i}} ∈ ⨁_{Γ/Ω} Gr^{Ω}-A_{Ω},

where N_{Ω+α_{i}} is Ω-graded A_{Ω}-module. Define a Γ-graded A-module N as follows: N = ⨁_{α∈Γ} N_{α}, where N_{α} := (N_{Ω+α_{i}})_{ω} and α = α_{i} + ω. We check that N is a Γ-graded A-module, i.e, N_{α}A_{γ} ⊆ N_{α+γ}, for α, γ ∈ Γ. If γ ∉ Ω, since Γ_{A} ⊆ Ω, A_{γ} = 0 and thus 0 = N_{α}A_{γ} ⊆ N_{α+γ}. Let γ ∈ Ω. Then

N_{α}A_{γ} = (N_{Ω+α_{i}})_{ω}A_{γ} ⊆ (N_{Ω+α_{i}})_{ω+γ} = N_{α+γ},

as α + γ = α_{i} + ω + γ.

We define 𝒫′(⨁_{Ω+α_{i}∈Γ/Ω} N_{Ω+α_{i}}) = N and accordingly for the morphisms. It is now not difficult to check that 𝒫′ is an inverse of the functor 𝒫. This finishes the proof.

The above theorem will be used to compare the graded K-theories with respect to Γ and Ω (see Example 3.1.11).

Corollary 1.2.9. Let A be a Γ-graded ring concentrated in degree zero. Then

Gr-A ≈ ⨁_{Γ} Mod- A.

Proof. This follows by replacing Ω by a trivial group in Theorem 1.2.8.

The following corollary, which is a more general case of Corollary 1.2.9 with a similar proof, will be used in the proof of Lemma 6.1.6.

Corollary 1.2.10. Let A be a Γ × Ω graded ring which is concentrated in Ω. Then

Gr^{Γ×Ω}-A ≅ ⨁_{Γ} Gr^{Ω}-A.

The action of Γ × Ω on ⨁_{Γ} Gr^{Ω}-A described in (1.20) reduces to the following: for (γ, ω) ∈ Γ × Ω,

\overline{ρ}_{(γ,ω)}( ⨁_{α∈Γ} M_{α}) = ⨁_{α∈Γ} M_{α+γ}(ω).

1.2.9 Graded projective modules

In this note, the graded projective modules play a crucial role. They will appear in the graded Morita theory §2 and will be used to define the graded Grothendieck groups §3. Furthermore, the graded higher K-theory is constructed from the exact category consisted of graded finitely generated projective modules §6. In this section we define the graded projective modules and give several equivalent criteria for a module to be graded projective. As before, unless stated otherwise, we work in the category of (graded) right modules.

A graded A-module P is called a graded projective module if it is a projective object in the abelian category Gr-A. More concretely, P is graded projective if for any diagram of graded A-module homomorphisms

\xymatrix{ & P \ar[d]^{j} \\ M \ar[r]_{g} & N \ar[r] & 0,}

there is a graded A-module homomorphism h : P → M with gh = j.

In Proposition 1.2.12 we give some equivalent characterisations of graded projective modules, including the one that shows an A-module is graded projective if and only if it is graded and projective as an A-module. By Pgr-A (or Pgr^{Γ}-A to emphasis the graded group of A) we denote a full subcategory of Gr-A, consisting of graded finitely generated projective right A-modules. This is the primary category we are interested in. The graded Grothendieck group (§3) and higher K-groups (§6) are constructed from this category.

We need the following lemma, which says if a graded map factors into two maps, with one being graded, then one can replace the other one with a graded map as well.

Lemma 1.2.11. Let P, M, N be graded A-modules, with A-module homomorphisms f, g, h

\xymatrix{ & M \ar[dr]^{g} \\ P \ar[ur]^{h} \ar[rr]_{f} &  & N}

such that f = gh, where f is a graded A-module homomorphism. If g (resp. h) is a graded Ahomomorphism then there exists a graded homomorphism h′ : P → M (resp. g′ : M → N) such that f = gh′ (resp. f = g′h).

Proof. Suppose g : M → N is a graded A-module homomorphism. Define h′ : P → M as follows: for p ∈ P_{α}, α ∈ Γ, let h′(p) = h(p)_{α} and extend this linearly to all elements of P. One can easily see that h′ : P → M is a graded A-module homomorphism. Furthermore, for p ∈ P_{α}, α ∈ Γ, we have

f(p) = gh(p) = g( ∑_{γ∈Γ} h(p)_{γ}) = ∑_{γ∈Γ} g(h(p)_{γ}).

Since f and g are graded homomorphisms, comparing the degrees of the homogeneous elements of each side of the equation, we get f(p) = g(h(p)_{α}) = gh′(p). Using the linearity of f, g, h′ we get that f = gh′ which proves the lemma for the case g. The other case is similar.

We are in a position to give equivalent characterisations of graded projective modules.

Proposition 1.2.12. Let A be a Γ-graded ring and P be a graded A-module. Then the following are equivalent:

(1) P is graded and projective;

(2) P is graded projective;

(3) Hom_{Gr-A}(P, −) is an exact functor in Gr-A;

(4) Every short exact sequence of graded A-module homomorphisms

0 ⟶ L ⟶^{f} M ⟶^{g} P ⟶ 0

splits via a graded map;

(5) P is graded isomorphic to a direct summand of a graded free A-module.

Proof. (1) ⇒ (2) Consider the diagram

\xymatrix{ & P \ar[d]^{j} \\ M \ar[r]_{g} & N \ar[r] & 0,}

where g and j are graded homomorphisms and g is surjective. Since P is projective, there is an A-module homomorphism h : P → M with gh = j. By Lemma 1.2.11, there is a graded A-module homomorphism h′ : P → M with gh′ = j. This gives that P is a graded projective module.

(2) ⇒ (3) In exactly the same way as the non-graded setting (see [52, Thm. IV.4.2]) we can show that Hom_{Gr-A}(P, −) is left exact. Then it follows immediately from (2) that it is right exact.

(3) ⇒ (4) Let

0 ⟶ L ⟶^{f} M ⟶^{g} P ⟶ 0  (1.25)

be a short exact sequence. Since Hom_{Gr-A}(P, −) is exact,

Hom_{Gr-A}(P, M) ⟶ Hom_{Gr-A}(P, P)
h ⟼ gh

is an epimorphism. In particular, there is graded homomorphism h such that gh = 1, i.e., the short exact sequence (1.25) is spilt.

(4) ⇒ (5) First note that P is a homomorphic image of a graded free A-module as follows: Let {p_{i}}_{i∈I} be a homogeneous generating set for P, where deg(p_{i}) = δ_{i}. Let ⨁_{i∈I} A(−δ_{i}) be the graded free A-module with standard homogeneous basis {e_{i}}_{i∈I} where deg(e_{i}) = δ_{i}. Then there is an exact sequence

0 ⟶ ker(g) ⟶^{⊆} ⨁_{i∈I} A(−δ_{i}) ⟶^{g} P ⟶ 0,  (1.26)

since the map g : ⨁_{i∈I} A(−δ_{i}) → P; e_{i} ↦ p_{i} is a surjective graded A-module homomorphism. By the assumption, there is a graded A-module homomorphism h : P → ⨁_{i∈I} A(−δ_{i}) such that gh = id_{P}.

Since the exact sequence 1.26 is, in particular, a split exact sequence of A-modules, we know from the non-graded setting [66, Prop. 2.5] that there is an A-module isomorphism

θ : P ⊕ ker(g) ⟶ ⨁_{i∈I} A(−δ_{i})
(p, q) ⟼ h(p) + q.

Clearly this map is also a graded A-module homomorphism, so P ⊕ ker(g) ≅_{gr} ⨁_{i∈I} A(−δ_{i}).

(5) ⇒ (1) Graded free modules are free, so P is isomorphic to a direct summand of a free A-module. From the non-graded setting, we know that P is projective.

The proof of Proposition 1.2.12 (see in particular (4) ⇒ (5) and (5) ⇒ (1)) shows that a graded A-module P is a graded finitely generated projective A-module if and only if

P ⊕ Q ≅_{gr} A^{n}(\overline{α}),  (1.27)

for some \overline{α} = (α_{1}, …, α_{n}), α_{i} ∈ Γ. This fact will be used frequently throughout this note.

Theorem 1.2.13 (The dual basis lemma). Let A be a Γ-graded ring and P be a graded Amodule. Then P is graded projective if and only if there exists p_{i} ∈ P^{h} with deg(p_{i}) = δ_{i} and f_{i} ∈ Hom_{Gr-A}(P, A(−δ_{i})), for some indexing set I, such that

(1) for every p ∈ P, f_{i}(p) = 0 for all but a finite subset of i ∈ I, (2) for every p ∈ P, ∑_{i∈I} f_{i}(p)p_{i} = p.

Proof. Since P is graded projective, by Proposition 1.2.12(5), there is a graded module Q such that P ⊕ Q ≅_{gr} ⨁_{i} A(−δ_{i}). This gives two graded maps ϕ : P → ⨁_{i} A(−δ_{i}) and π : ⨁_{i} A(−δ_{i}) → P, such that πϕ = 1_{P}. Let π_{i} : ⨁_{i} A(−δ_{i}) → A(−δ_{i}), {a_{i}}_{i∈I} ↦ a_{i} be the projection on the ith component. So if a = {a_{i}}_{i∈I} ∈ ⨁_{i} A(−δ_{i}) then ∑_{i} π_{i}(a)e_{i} = a, where {e_{i}}_{i∈I} are the standard homogeneous basis of ⨁_{i} A(−δ_{i}). Now let p_{i} = π(e_{i}) and f_{i} = π_{i}ϕ. Note that deg(p_{i}) = δ_{i} and f_{i} ∈ Hom_{Gr-A}(P, A(−δ_{i})). Clearly f_{i}(p) = π_{i}ϕ(p) is zero for all but a finite number of i ∈ I. This gives (1). Furthermore,

∑_{i} p_{i}f_{i}(p) = ∑_{i} p_{i}π_{i}ϕ(p) = ∑ π(e_{i})π_{i}ϕ(p) = π( ∑_{i} e_{i}π_{i}ϕ(p)) = πϕ(p) = p.

This gives (2).

Conversely, suppose that there exists a dual basis {p_{i}, f_{i} | i ∈ I}. Consider the maps ϕ : P → ⨁_{i} A(−δ_{i}), p ↦ {f_{i}(p)}_{i∈I} and π : ⨁_{i} A(−δ_{i}) → P, {a_{i}}_{i∈I} ↦ ∑_{i} p_{i}a_{i}. One can easily see that ϕ and π are graded right A-module homomorphism and πϕ = 1_{P}. Therefore the exact sequence

0 ⟶ ker(π) ⟶ ⨁_{i} A(−δ_{i}) ⟶^{π} P ⟶ 0

splits. Thus P is a direct summand of the graded free module ⨁_{i} A(−δ_{i}). By Proposition 1.2.12, P is a graded projective.

Remark 1.2.14. \underline{Graded injective modules}

Proposition 1.2.12 shows that an A-module P is graded projective if and only if P is graded and projective. However the similar statement is not valid for graded injective modules. Recall that a graded A-module I is called a graded injective module if for any diagram of graded A-module homomorphisms

\xymatrix{0 \ar[r] & N \ar[r]^{g} \ar[d]_{j} & M \\  & I}

there is a graded A-module homomorphism h : M → I with hg = j.

Using Lemma 1.2.11 one can show that a graded module which is injective, is also graded injective. However a graded injective module is not necessarily injective. The reason for this difference between projective and injective modules is that the forgetful functor U is a left adjoint functor (see Remark 1.2.7). In details, consider a graded projective module P and the diagram

\xymatrix{ & P \ar[d]^{j} \\ M \ar[r]_{g} & N \ar[r] & 0.}

Since the diagram below is commutative

\xymatrix{Hom_{Mod- A}(U(P), M) \ar[r]^{≅} \ar[d] & Hom_{Gr-A}(P, F (M)) \ar[d] \\ Hom_{Mod- A}(U(P), N) \ar[r]^{≅} & Hom_{Gr-A}(P, F (N))}

and there is a graded homomorphism h′ : P → F (M) such that the diagram

\xymatrix{ & P \ar[d]^{j′} \ar[dl]_{h′} \\ F (M) \ar[r]_{F (g)} & F (N) \ar[r] & 0,}

is commutative, there is a homomorphism h : P → M such that gh = j. So P is projective (see Proposition 1.2.12 for another proof).

If the graded group is finite, then the forgetful functor is right adjoint as well (see Remark 1.2.7) and a similar argument as above shows that a graded injective module is injective.

Example 1.2.15. \underline{Graded rings associated to filter rings}

A ring A with identity is called a filtered ring if there is an ascending family {A_{i} | i ∈ ℤ} of additive subgroup of A such that 1 ∈ A_{0} and A_{i}A_{j} ⊆ A_{i+j}, for all i, j ∈ ℤ. Let M be a right A-module where A is a filtered ring. M is called a filtered module if there is an ascending family {M_{i} | i ∈ ℤ} of additive subgroup of M such that M_{i}A_{j} ⊆ M_{i+j}, for all i, j ∈ ℤ. An Amodule homomorphism f : M → N of filtered modules M and N is called a filtered homomorphism if f(M_{i}) ⊆ N_{i} for i ∈ ℤ. A category consisting of filtered A-modules for objects and filtered homomorphisms as morphisms is denoted by Filt- A. If M is a filtered A-module then gr M := ⨁_{i∈ℤ} M_{i}/M_{i−1} is a ℤ-graded gr A := ⨁_{i∈ℤ} A_{i}/A_{i−1}-module. The operations here are defined naturally. This gives a functor gr : Filt- A → Gr-A. In Example 1.4.7 we will use a variation of this construction to associate to a valued division algebra a graded division algebra.

In the theory of filtered rings, one defines the concepts of filtered free and projective modules and under certain conditions the functor gr sends these objects to the corresponding objects in the category Gr-A. For a comprehensive study of filtered rings see [71].

1.2.10 Homogeneous idempotents calculus

The idempotents naturally arise in relation with decomposition of rings and modules. The following facts about idempotents are well known in a non-graded setting and one can check that they translate in the graded setting with the similar proofs (cf. [61, §21]). Let P_{i}, 1 ≤ i ≤ l, be graded right ideals of A such that A = P_{1} ⊕ ⋯ ⊕ P_{l}. Then there are homogeneous orthogonal idempotents e_{i} (hence of degree zero) such that 1 = e_{1} + ⋯ + e_{l} and e_{i}A = P_{i}.

Let e and f be homogeneous idempotent elements in the graded ring A. (Note that, in general, there are non-homogeneous idempotents in a graded ring.) Let θ : eA → fA be a (not necessarily) right A-module homomorphism. Then θ(e) = θ(e^{2}) = θ(e)e = fae for some a ∈ A and for b ∈ eA, θ(b) = θ(eb) = θ(e)b. This shows that there is a map

Hom_{A}(eA, fA) → fAe,  (1.28)
θ ↦ θ(e)

and one can easily check this is a group isomorphism. We have fAe = ⨁_{γ∈Γ} fA_{γ}e and by Theorem 1.2.5,

Hom_{A}(eA, fA) ≅ ⨁_{γ∈Γ} Hom_{A}(eA, fA)_{γ}.

Then one can see that the homomorphism (1.28) respects the graded decomposition.

Now if θ : eA → fA(α), where α ∈ Γ, is a graded A-isomorphism, then x = θ(e) ∈ fA_{α}e and y = θ^{−1}(f) ∈ eA_{−α}f, where x and y are homogeneous of degrees α and −α, respectively, such that yx = e and xy = f. Finally, for f = e, the map (1.28) gives a graded ring isomorphism End_{A}(eA) ≅_{gr} eAe. These facts will be used later in Theorem 5.1.3.

1.3 Grading on matrices

Starting from any ring A and any group Γ, one can consider a Γ-grading on the matrix ring 𝕄_{n}(A). At the first glance, this grading looks somewhat artificial. However, this type of grading on matrices appears quite naturally in the graded rings arising from graphs. In this section we study the grading on matrices. We then include a section on graph algebras (including path algebras and Leavitt path algebras, §1.6). These algebras give us a wealth of examples of graded rings and graded matrix rings.

For a free right A-module V of dimension n, there is a ring isomorphism End_{A}(V) ≅ 𝕄_{n}(A). When A is a Γ-graded ring and V is a graded free module of finite rank, by Theorem 1.2.5, End_{A}(V) has a natural Γ-grading. This induces a graded structure on the matrix ring 𝕄_{n}(A). In this section we study this grading on matrices. For an n-tuple (δ_{1}, …, δ_{n}), δ_{i} ∈ Γ, we construct a grading on the matrix ring 𝕄_{n}(A), denoted by 𝕄_{n}(A)(δ_{1}, …, δ_{n}), and we show that

End_{A} (A(−δ_{1}) ⊕ A(−δ_{2}) ⊕ ⋯ ⊕ A(−δ_{n})) ≅_{gr} 𝕄_{n}(A)(δ_{1}, …, δ_{n}).

We will see that these graded structures on matrices appear very naturally when studying the graded structure of path algebras in §1.6.

1.3.1

Let A be a Γ-graded ring and let M = M_{1} ⊕ ⋯ ⊕ M_{n}, where M_{i} are graded finitely generated right A-modules. Then M is also a graded A-module (see §1.2.6). Let

( Hom(M_{j}, M_{i}))_{1≤i,j≤n} = (\begin{matrix}Hom_{A}(M_{1}, M_{1}) & Hom_{A}(M_{2}, M_{1}) & ⋯ & Hom_{A}(M_{n}, M_{1}) \\ Hom_{A}(M_{1}, M_{2}) & Hom_{A}(M_{2}, M_{2}) & ⋯ & Hom_{A}(M_{n}, M_{2}) \\ ⋮ & ⋮ & ... & ⋮ \\ Hom_{A}(M_{1}, M_{n}) & Hom_{A}(M_{2}, M_{n}) & ⋯ & Hom_{A}(M_{n}, M_{n}) \end{matrix}) . (1.29)

It is easy to observe that ( Hom(M_{j}, M_{i}))_{1≤i,j≤n} forms a ring with the component-wise addition and the matrix multiplication. Furthermore, for λ ∈ Γ, assigning the additive subgroup

(\begin{matrix}Hom_{A}(M_{1}, M_{1})_{λ} & Hom_{A}(M_{2}, M_{1})_{λ} & ⋯ & Hom_{A}(M_{n}, M_{1})_{λ} \\ Hom_{A}(M_{1}, M_{2})_{λ} & Hom_{A}(M_{2}, M_{2})_{λ} & ⋯ & Hom_{A}(M_{n}, M_{2})_{λ} \\ ⋮ & ⋮ & ... & ⋮ \\ Hom_{A}(M_{1}, M_{n})_{λ} & Hom_{A}(M_{2}, M_{n})_{λ} & ⋯ & Hom_{A}(M_{n}, M_{n})_{λ} \end{matrix})  (1.30)

as a γ-homogeneous component of ( Hom(M_{j}, M_{i}))_{1≤i,j≤n}, using Theorem 1.2.5, one can show that ( Hom(M_{j}, M_{i}))_{1≤i,j≤n} is a Γ-graded ring.

Let π_{j} : M → M_{j} and κ_{j} : M_{j} → M be the (graded) projection and injection homomorphisms. For the next theorem, we need the following identities

∑_{i=1}^{n} κ_{i}π_{i} = id_{M} and π_{i}κ_{j} = δ_{ij} id_{M_{j}},  (1.31)

where δ_{ij} is the Kronecker delta.

Theorem 1.3.1. Let A be a Γ-graded ring and M = M_{1} ⊕ ⋯ ⊕ M_{n}, where M_{i} are graded finitely generated right A-modules. Then there is a graded ring isomorphism

Φ : End_{A}(M) ⟶ ( Hom(M_{j}, M_{i}))_{1≤i,j≤n}

defined by f ↦ (π_{i}fκ_{j}), 1 ≤ i, j ≤ n.

Proof. The map Φ is clearly well-defined. Since for f, g ∈ End_{A}(M),

Φ(f + g) = (π_{i}(f + g)κ_{j})_{1≤i,j≤n} = (π_{i}fκ_{j} + π_{i}gκ_{j})_{1≤i,j≤n} =

(π_{i}fκ_{j})_{1≤i,j≤n} + (π_{i}gκ_{j})_{1≤i,j≤n} = Φ(f) + Φ(g)

and

Φ(fg) = (π_{i}fgκ_{j})_{1≤i,j≤n} = (π_{i}f(∑_{l=1}^{n} κ_{l}π_{l})gκ_{j})_{1≤i,j≤n} = ( ∑_{l=1}^{n}(π_{i}fκ_{l})(π_{l}gκ_{j}))_{1≤i,j≤n} = Φ(f)Φ(g),

Φ is a ring homomorphism. Furthermore, if f ∈ End_{A}(M)_{λ}, λ ∈ Γ, then π_{i}fκ_{j} ∈ Hom_{A}(M_{j}, M_{i})_{λ}, for 1 ≤ i, j ≤ n. This (see (1.30)) shows that Φ is a graded ring homomorphism. Define the map

Ψ : ( Hom(M_{j}, M_{i}))_{1≤i,j≤n} ⟶ End_{A}(M)
(g_{ij})_{1≤i,j≤n} ⟼ ∑_{1≤i,j≤n} κ_{i}g_{ij}π_{j}

Using the identities 1.31, one can observe that the compositions ΨΦ and ΦΨ give the identity maps of the corresponding rings. Thus Φ is an isomorphism.

For a graded ring A, observe that

Φ_{δ_{j},δ_{i}} : Hom_{A} (A(δ_{i}), A(δ_{j})) ≅_{gr} A(δ_{j} − δ_{i}),  (1.32)

such that Φ_{δ_{k},δ_{i}}(gf) = Φ_{δ_{k},δ_{j}}(g)Φ_{δ_{j},δ_{i}}(f), where f ∈ Hom(A(δ_{i}), A(δ_{j})) and g ∈ Hom(A(δ_{j}), A(δ_{k})) (see 1.10). If

V = A(−δ_{1}) ⊕ A(−δ_{2}) ⊕ ⋯ ⊕ A(−δ_{n}),

then by Theorem 1.3.1,

End_{A}(V) ≅_{gr} ( Hom (A(−δ_{j}), A(−δ_{i})))_{1≤i,j≤n}.

Applying Φ defined in (1.32) to each entry, we have

End_{A}(V) ≅_{gr} ( Hom (A(−δ_{j}), A(−δ_{i})))_{1≤i,j≤n} ≅_{gr} (A(δ_{j} − δ_{i}))_{1≤i,j≤n}.

Denoting the graded matrix ring (A(δ_{j} − δ_{i}))_{1≤i,j≤n} by 𝕄_{n}(A)(δ_{1}, …, δ_{n}), we have

𝕄_{n}(A)(δ_{1}, …, δ_{n}) = (\begin{matrix}A(δ_{1} − δ_{1}) & A(δ_{2} − δ_{1}) & ⋯ & A(δ_{n} − δ_{1}) \\ A(δ_{1} − δ_{2}) & A(δ_{2} − δ_{2}) & ⋯ & A(δ_{n} − δ_{2}) \\ ⋮ & ⋮ & ... & ⋮ \\ A(δ_{1} − δ_{n}) & A(δ_{2} − δ_{n}) & ⋯ & A(δ_{n} − δ_{n}) \end{matrix}) .  (1.33)

By (1.30), 𝕄_{n}(A)(δ_{1}, …, δ_{n})_{λ}, the λ-homogeneous elements, are the n × n-matrices over A with the degree shifted (suspended) as follows:

𝕄_{n}(A)(δ_{1}, …, δ_{n})_{λ} = (\begin{matrix}A_{λ+δ_{1}−δ_{1}} & A_{λ+δ_{2}−δ_{1}} & ⋯ & A_{λ+δ_{n}−δ_{1}} \\ A_{λ+δ_{1}−δ_{2}} & A_{λ+δ_{2}−δ_{2}} & ⋯ & A_{λ+δ_{n}−δ_{2}} \\ ⋮ & ⋮ & ... & ⋮ \\ A_{λ+δ_{1}−δ_{n}} & A_{λ+δ_{2}−δ_{n}} & ⋯ & A_{λ+δ_{n}−δ_{n}} \end{matrix} ) .  (1.34)

This also shows that

deg(e_{ij}(x)) = deg(x) + δ_{i} − δ_{j}.  (1.35)

In particular the zero homogeneous component (which is a ring) is of the form

𝕄_{n}(A)(δ_{1}, …, δ_{n})_{0} = (\begin{matrix}A_{0} & A_{δ_{2}−δ_{1}} & ⋯ & A_{δ_{n}−δ_{1}} \\ A_{δ_{1}−δ_{2}} & A_{0} & ⋯ & A_{δ_{n}−δ_{2}} \\ ⋮ & ⋮ & ... & ⋮ \\ A_{δ_{1}−δ_{n}} & A_{δ_{2}−δ_{n}} & ⋯ & A_{0} \end{matrix}) .  (1.36)

Setting \overline{δ} = (δ_{1}, …, δ_{n}) ∈ Γ^{n}, one denotes the graded matrix ring (1.33) by 𝕄_{n}(A)(\overline{δ}). To summarise, we have shown that there is a graded ring isomorphism

End_{A} (A(−δ_{1}) ⊕ A(−δ_{2}) ⊕ ⋯ ⊕ A(−δ_{n})) ≅_{gr} 𝕄_{n}(A)(δ_{1}, …, δ_{n}).  (1.37)

Consider the graded A-bimodule A^{n}(\overline{δ}) = A(δ_{1}) ⊕ ⋯ ⊕ A(δ_{n}). Then one can check that A^{n}(\overline{δ}) is a graded right 𝕄_{n}(A)(\overline{δ})-module and A^{n}(−\overline{δ}) is a graded left 𝕄_{n}(A)(\overline{δ})-module, where −\overline{δ} = (−δ_{1}, …, −δ_{n}). These will be used in the Morita theory (see Proposition 2.1.1).

One can easily check the graded ring R = 𝕄_{n}(A)(\overline{δ}), where \overline{δ} = (δ_{1}, … δ_{n}), δ_{i} ∈ Γ, has the support

Γ_{R} = ⋃_{1≤i,j≤n} Γ_{A} + δ_{i} − δ_{j}.  (1.38)

One can rearrange the shifting, without changing the graded matrix ring as the following theorem shows (see also [73, pp. 60-61]).

Theorem 1.3.2. Let A be a Γ-graded ring and δ_{i} ∈ Γ, 1 ≤ i ≤ n.

(1) If α ∈ Γ, and π ∈ S_{n} is a permutation then

𝕄_{n}(A)(δ_{1}, …, δ_{n}) ≅_{gr} 𝕄_{n}(A)(δ_{π(1)} + α, …, δ_{π(n)} + α).  (1.39)

(2) If τ_{1}, …, τ_{n} ∈ Γ_{A}^{∗}, then

𝕄_{n}(A)(δ_{1}, …, δ_{n}) ≅_{gr} 𝕄_{n}(A)(δ_{1} + τ_{1}, …, δ_{n} + τ_{n}).  (1.40)

Proof. (1) Let V be a graded free module over A with a homogeneous basis v_{1}, …, v_{n} of degree λ_{1}, …, λ_{n}, respectively. It is easy to see that

V ≅_{gr} A(−λ_{1}) ⊕ ⋯ ⊕ A(−λ_{n}),

and thus End_{A}(V) ≅_{gr} 𝕄_{n}(A)(λ_{1}, …, λ_{n}). Now let π ∈ S_{n}. Rearranging the homogeneous basis as v_{π(1)}, …, v_{π(n)} and defining the A-graded isomorphism ϕ : V → V by ϕ(v_{i}) = v_{π^{−1}(i)}, we get a graded isomorphism in the level of endomorphism rings

𝕄_{n}(A)(λ_{1}, …, λ_{n}) ≅_{gr} End_{A}(V) ⟶^{ϕ} End_{A}(V) ≅_{gr} 𝕄_{n}(A)(λ_{π(1)}, …, λ_{π(n)}).

Moreover, (1.34) shows that it does not make any difference adding a fixed α ∈ Γ to each of the entries in the shifting. This gives us (1.39).

(2) For (1.40), let τ_{i} ∈ Γ_{A}^{∗}, 1 ≤ i ≤ n, that is, τ_{i} = deg(u_{i}) for some units u_{i} ∈ A^{h}. Consider the basis v_{i}u_{i}, 1 ≤ i ≤ n for V. With this basis, End_{A}(V) ≅_{gr} 𝕄_{n}(A)(δ_{1} + τ_{1}, …, δ_{n} + τ_{n}). Consider the A-graded isomorphism id : V → V, by id(v_{i}) = (v_{i}u_{i})u_{i}^{−1}. A similar argument as Part (1) now gives (1.40).

Note that if A has a trivial Γ-grading, i.e., A = ⨁_{γ∈Γ} A_{γ}, where A_{0} = A and A_{γ} = 0, for 0 ≠ γ ∈ Γ, this construction induces a good grading on 𝕄_{n}(A). By definition, this is a grading on 𝕄_{n}(A) such that e_{ij}, the matrix with 1 in the ij-position and zero everywhere else, is homogeneous, for 1 ≤ i, j ≤ n. This particular group gradings on matrix rings have been studied by Dăscălescu et al. [32] (see Remark 1.3.8). Therefore for x ∈ A,

deg(e_{ij}(x)) = δ_{i} − δ_{j}.  (1.41)

One can easily check that for a ring A with trivial Γ-grading, the graded ring 𝕄_{n}(A)(\overline{δ}), where \overline{δ} = (δ_{1}, … δ_{n}), δ_{i} ∈ Γ, has the support \{δ_{i} − δ_{j} | 1 ≤ i, j ≤ n\}. (This follows also immediately from (1.38).)

The grading on matrices appears quite naturally in the graded rings arising from graphs. We will show that the graded structure Leavitt path algebras of acyclic and comet graphs are in effect the graded matrix rings as constructed above (see §1.6, in particular, Theorems 1.6.15 and 1.6.17).

Example 1.3.3. Let A be a ring, Γ be a group and A be graded trivially by Γ, i.e., A is concentrated in degree zero (see §1.1.1). Consider the Γ-graded matrix ring R = 𝕄_{n}(A)(0, −1, …, −n+1), where n ∈ ℕ. By (1.38) the support of R is the set {−n + 1, −n + 2, …, n − 2, n − 1} and by (1.34) we have the following arrangements for the homogeneous elements of R. For k ∈ ℤ,

R_{k} = ( \begin{matrix}A_{k} & A_{k−1} & … & A_{k+1−n} \\ A_{k+1} & A_{k} & … & A_{k+2−n} \\ ⋮ & ⋮ & ... & ⋮ \\ A_{k+n−1} & A_{k+n−2} & … & A_{k} \end{matrix} ) .

Thus

R_{0} = ( \begin{matrix}A & 0 & … & 0 \\ 0 & A & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & A \end{matrix} ) , R_{−1} = ( \begin{matrix}0 & 0 & … & 0 \\ A & 0 & … & 0 \\ ⋮ & ... & ... & ⋮ \\ 0 & … & A & 0 \end{matrix} ) , …, R_{−n+1} = ( \begin{matrix}0 & 0 & … & 0 \\ 0 & 0 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ A & 0 & … & 0 \end{matrix} ) ,

R_{1} = ( \begin{matrix}0 & A & … & 0 \\ 0 & 0 & ... & ⋮ \\ ⋮ & ⋮ & ... & A \\ 0 & 0 & … & 0 \end{matrix} ) , …, R_{n−1} = ( \begin{matrix}0 & 0 & … & A \\ 0 & 0 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 0 \end{matrix} ) .

In Chapter 2, we will see that R = 𝕄_{n}(A)(0, −1, …, −n + 1) is graded Morita equivalent to the trivially graded ring A.

Example 1.3.4. Let R be a ring, R[x, x^{−1}] the ℤ-graded Laurent polynomial ring and A = R[x^{3}, x^{−3}] the ℤ-graded subring with the support 3ℤ (see Example 1.1.20). Consider the ℤ-graded matrix ring

𝕄_{6}(A)(0, 1, 1, 2, 2, 3).

By (1.34), the homogeneous elements of degree 1 have the form

( \begin{matrix}A_{1} & A_{0} & A_{0} & A_{−1} & A_{−1} & A_{−2} \\ A_{2} & A_{1} & A_{1} & A_{0} & A_{0} & A_{−1} \\ A_{2} & A_{1} & A_{1} & A_{0} & A_{0} & A_{−1} \\ A_{3} & A_{2} & A_{2} & A_{1} & A_{1} & A_{0} \\ A_{3} & A_{2} & A_{2} & A_{1} & A_{1} & A_{0} \\ A_{4} & A_{3} & A_{3} & A_{2} & A_{2} & A_{1} \end{matrix} ) = ( \begin{matrix}0 & R & R & 0 & 0 & 0 \\ 0 & 0 & 0 & R & R & 0 \\ 0 & 0 & 0 & R & R & 0 \\ Rx^{3} & 0 & 0 & 0 & 0 & R \\ Rx^{3} & 0 & 0 & 0 & 0 & R \\ 0 & Rx^{3} & Rx^{3} & 0 & 0 & 0 \end{matrix} ) .

Example 1.3.5. Let K be a field. Consider the ℤ-graded ring A = 𝕄_{5}(K)(0, 1, 2, 2, 3). Then the support of this ring is {0, ±1, ±2} and by (1.36) the zero homogeneous component (which is a ring) is

A_{0} = (\begin{matrix}K & 0 & 0 & 0 & 0 \\ 0 & K & 0 & 0 & 0 \\ 0 & 0 & K & K & 0 \\ 0 & 0 & K & K & 0 \\ 0 & 0 & 0 & 0 & K \end{matrix}) ≅ K ⊕ K ⊕ 𝕄_{2}(K) ⊕ K.

Example 1.3.6. \underline{𝕄_{n}(A)(δ_{1}, …, δ_{n}), for Γ = \{δ_{1}, …, δ_{n}\}, is a skew group ring}

Let A be a Γ-graded ring where Γ = {δ_{1}, …, δ_{n}} is a finite group. Consider 𝕄_{n}(A)(δ_{1}, …, δ_{n}), which is a Γ-graded ring with its homogeneous components described by (1.34). We will show that this graded ring is a skew group ring 𝕄_{n}(A)_{0} ⋆ Γ. In particular, by Proposition 1.1.16(3), it is a strongly graded ring. Consider the matrix u_{α} ∈ 𝕄_{n}(A)(δ_{1}, …, δ_{n})_{α}, where in each row i, we have 1 in (i, j) position, where δ_{j} − δ_{i} + α = 0 and zero everywhere else. One can easily see that u_{α} is a permutation matrix with exactly one 1 in each row and column. Furthermore, for α, β ∈ Γ, u_{α}u_{β} = u_{α+β}. Indeed, consider the ith row of u_{α}, with the only 1 in jth column where δ_{j} − δ_{i} + α = 0. Now, consider the jth row of u_{β} with a kth column such that δ_{k} − δ_{j} + β = 0 and so with 1 in (j, k) row. Thus multiplying u_{α} with u_{β}, we have zero everywhere in ith row except in (i, k)th position. On the other hand since δ_{k} − δ_{i} + α + β = 0, in ith row of u_{α+β} we have zero except in (i, k)th position. Repeating this argument for each row of u_{α} shows that u_{α}u_{β} = u_{α+β}.

Now defining ϕ : Γ → Aut(𝕄_{n}(A)_{0}) by ϕ(α)(a) = u_{α}au_{α}^{−1}, and setting the 2-cocycle ψ trivial, by §1.1.3, R = 𝕄_{n}(A)_{0} ⋆_{ϕ} Γ.

This was observed in [74], where it was proved that 𝕄_{n}(A)(δ_{1}, …, δ_{n})_{0} is isomorphic to the smash product of Cohen and Montgomery [27] (see Remark 2.3.11).

Example 1.3.7. The following examples (from [32, Example 1.3]) provide two instances of ℤ_{2}- grading on 𝕄_{2}(K), where K is a field. The first grading is a good grading, whereas the second one is not a good grading.

1. Let R = 𝕄_{2}(K) with the ℤ_{2}-grading defined by

R_{0} = \{(\begin{matrix}a & 0 \\ 0 & b \end{matrix}) | a, b ∈ K\} and R_{1} = \{(\begin{matrix}0 & c \\ d & 0 \end{matrix}) | c, d ∈ K\} .

Since e_{11}, e_{22} ∈ R_{0} and e_{12}, e_{21} ∈ R_{1}, by definition, this is a good grading. Note that R = 𝕄_{2}(K)(0, 1).

2. Let S = 𝕄_{2}(K) with the ℤ_{2}-grading defined by

S_{0} = \{(\begin{matrix}a & b − a \\ 0 & b \end{matrix}) | a, b ∈ K\} and S_{1} = \{(\begin{matrix}d & c \\ d & −d \end{matrix}) | c, d ∈ K\} .

Then S is a graded ring, such that the ℤ_{2}-grading is not a good grading, since e_{11} is not homogeneous. Furthermore, comparing S_{0} with (1.36), shows that the grading on S does not come from the construction given by (1.33). Consider the map

f : R ⟶ S; (\begin{matrix}a & b \\ c & d \end{matrix}) ⟼ (\begin{matrix}a + c & b + d − a − c \\ c & d − c \end{matrix}) .

This map is in fact a graded ring isomorphism, and so R ≅_{gr} S. This shows that the good grading is not preserved under graded isomorphisms.

Remark 1.3.8. \underline{Good gradings on matrix algebras}

Let K be a field and Γ be an abelian group. One can put a Γ-grading on the ring 𝕄_{n}(K), by assigning a degree (an element of the group Γ) to each matrix e_{i,j}, 1 ≤ i, j ≤ n. This is called a good grading. This grading has been studied in [32]. In particular it has been shown that a grading on 𝕄_{n}(K) is good if and only if it can be described as 𝕄_{n}(K)(δ_{1}, …, δ_{n}) for some δ_{i} ∈ Γ. Furthermore, any grading on 𝕄_{n}(K) is a good grading if Γ is torsion free. It has also been shown that if R = 𝕄_{n}(K) has a Γ-grading such that e_{ij} is a homogeneous for some 1 ≤ i, j ≤ n, then there exists a good grading on S = 𝕄_{n}(K) with a graded isomorphism R ≅ S. It is shown that if Γ is finite, then the number of good gradings on 𝕄_{n}(K) is |Γ|^{n−1}. Furthermore, (for a finite Γ) the class of strongly graded and crossed-product good gradings of 𝕄_{n}(K) have been classified.

1.3.2 Mixed shifting

For a Γ-graded ring A, \overline{α} = (α_{1}, … α_{m}) ∈ Γ^{m} and \overline{δ} = (δ_{1}, …, δ_{n}) ∈ Γ^{n}, set

𝕄_{m×n}(A)[\overline{α}][\overline{δ}] := (\begin{matrix}A_{α_{1}−δ_{1}} & A_{α_{1}−δ_{2}} & ⋯ & A_{α_{1}−δ_{n}} \\ A_{α_{2}−δ_{1}} & A_{α_{2}−δ_{2}} & ⋯ & A_{α_{2}−δ_{n}} \\ ⋮ & ⋮ & ... & ⋮ \\ A_{α_{m}−δ_{1}} & A_{α_{m}−δ_{2}} & ⋯ & A_{α_{m}−δ_{n}} \end{matrix} ) .

So 𝕄_{m×n}(A)[\overline{α}][\overline{δ}] consists of matrices with the ij-entry in A_{α_{i}−δ_{j}}. If a ∈ 𝕄_{m×n}(A)[\overline{α}][\overline{δ}], then one can easily check that multiplying a from the left induces a graded right A-module homomorphism

A^{n}(\overline{δ}) ⟶ A^{m}(\overline{α}),

(a_{1} ; a_{2} ; ⋮ ; a_{n}) ⟼ a (a_{1} ; a_{2} ; ⋮ ; a_{n}) .  (1.42)

Conversely, suppose ϕ : A^{n}(\overline{δ}) → A^{m}(\overline{α}) is graded right A-module homomorphism. Let e_{j} denote the standard basis element of A^{n}(\overline{δ}) with 1 in the j-th entry and zeros elsewhere. Let ϕ(e_{j}) = (a_{1j}, a_{2j}, …, a_{mj}), 1 ≤ j ≤ n. Since ϕ is a graded map, comparing the grading of both sides, one can observe that deg(a_{ij}) = α_{i} − δ_{j}. So that the map ϕ is represented by the left multiplication with the matrix a = (a_{ij})_{m×n} ∈ 𝕄_{m×n}(A)[\overline{α}][\overline{δ}].

The following simple Lemma comes in handy.

Lemma 1.3.9. Let a ∈ 𝕄_{m×n}(A)[\overline{α}][\overline{δ}] and b ∈ 𝕄_{n×k}(A)[\overline{δ}][\overline{β}]. Then ab ∈ 𝕄_{m×k}(A)[\overline{α}][\overline{β}].

Proof. Let ϕ_{a} : A^{n}(\overline{δ}) → A^{m}(\overline{α}) and ϕ_{b} : A^{k}(\overline{β}) → A^{n}(\overline{δ}) be graded A-module homomorphisms induced by multiplications with a and b, respectively. Then

ϕ_{ab} = ϕ_{a}ϕ_{b} : A^{k}(\overline{β}) ⟶ A^{m}(\overline{α}).

This shows that ab ∈ 𝕄_{m×k}(A)[\overline{α}][\overline{β}]. (This can also be checked directly, by multiplying the matrices a and b and taking into account the shifting arrangements.)

In particular 𝕄_{m×m}(A)[\overline{α}][\overline{α}] represents End(A^{m}(α), A^{m}(α))_{0}. Combining this with (1.37), we get

𝕄_{m×m}(A)[\overline{α}][\overline{α}] = 𝕄_{m}(A)(−\overline{α})_{0}.  (1.43)

Proposition 1.3.10. Let A be a Γ-graded ring and let \overline{α} = (α_{1}, …, α_{m}) ∈ Γ^{m}, \overline{δ} = (δ_{1}, …, δ_{n}) ∈ Γ^{n}. Then the following are equivalent:

(1) A^{m}(\overline{α}) ≅_{gr} A^{n}(\overline{δ}) as graded right A-modules;

(2) A^{m}(−\overline{α}) ≅_{gr} A^{n}(−\overline{δ}) as graded left A-modules;

(3) There exist a = (a_{ij}) ∈ 𝕄_{n×m}(A)[\overline{δ}][\overline{α}] and b = (b_{ij}) ∈ 𝕄_{m×n}(A)[\overline{α}][\overline{δ}] such that ab = 𝕀_{n} and ba = 𝕀_{m}.

Proof. (1) ⇒ (3) Let ϕ : A^{m}(\overline{α}) → A^{n}(\overline{δ}) and ψ : A^{n}(\overline{δ}) → A^{m}(\overline{α}) be graded right A-module isomorphisms such that ϕψ = 1 and ψϕ = 1. The paragraph prior to the Proposition shows that the map ϕ is represented by the left multiplication with a matrix a = (a_{ij})_{n×m} ∈ 𝕄_{n×m}(A)[\overline{δ}][\overline{α}]. In the same way one can construct b ∈ 𝕄_{m×n}(A)[\overline{α}][\overline{δ}] which induces ψ. Now ϕψ = 1 and ψϕ = 1 translate to ab = 𝕀_{n} and ba = 𝕀_{m}.

(3) ⇒ (1) If a ∈ 𝕄_{n×m}(A)[\overline{δ}][\overline{α}], then multiplication from the left, induces a graded right Amodule homomorphism ϕ_{a} : A^{m}(\overline{α}) ⟶ A^{n}(\overline{δ}). Similarly b induces ψ_{b} : A^{n}(\overline{δ}) ⟶ A^{m}(\overline{α}). Now ab = 𝕀_{n} and ba = 𝕀_{m} translate to ϕ_{a}ψ_{b} = 1 and ψ_{b}ϕ_{a} = 1.

(2) ⟺ (3) This part is similar to the previous cases by considering the matrix multiplication from the right. Namely, the graded left A-module homomorphism ϕ : A^{m}(−\overline{α}) → A^{n}(−\overline{δ}) represented by a matrix multiplication from the right of the form 𝕄_{m×n}(A)[\overline{α}][\overline{δ}] and similarly ψ gives a matrix in 𝕄_{n×m}(A)[\overline{δ}][\overline{α}]. The rest follows easily.

The following corollary shows that A(α) ≅_{gr} A as graded right A-modules if and only if α ∈ Γ_{A}^{∗}. In fact, replacing m = n = 1 in Proposition 1.3.10 we obtain the following. Corollary 1.3.11. Let A be a Γ-graded ring and α ∈ Γ. Then the following are equivalent:

(1) A(α) ≅_{gr} A as graded right A-modules;

(2) A(−α) ≅_{gr} A as graded right A-modules;

(3) A(α) ≅_{gr} A as graded left A-modules;

(4) A(−α) ≅_{gr} A as graded left A-modules;

(5) There is an invertible homogeneous element of degree α. Proof. This follows from Proposition 1.3.10. Corollary 1.3.12. Let A be a Γ-graded ring. Then the following are equivalent:

(1) A is a crossed-product;

(2) A(α) ≅_{gr} A, as graded right A-modules, for all α ∈ Γ;

(3) A(α) ≅_{gr} A, as graded left A-modules, for all α ∈ Γ;

(4) The shift functor 𝒯_{α} : Gr-A → Gr-A is isomorphic to identity functor, for all α ∈ Γ. Proof. This follows from Corollary 1.3.11, (1.13) and the definition of the crossed-product rings (§1.1.2).

The Corollary above will be used to show that the action of Γ on the graded Grothendieck group of a crossed-product algebra is trivial (see Example 3.1.9). Example 1.3.13. \underline{The Leavitt algebra ℒ(1, n)}

In [63] Leavitt considered the free associative ring A with coefficient in ℤ generated by symbols {x_{i}, y_{i} | 1 ≤ i ≤ n} subject to relations

x_{i}y_{j} = δ_{ij}, for all 1 ≤ i, j ≤ n,  (1.44)

∑_{i=1}^{n} y_{i}x_{i} = 1,

where n ≥ 2 and δ_{ij} is the Kronecker delta. The relations guarantee the right A-module homomorphism

ϕ : A ⟶ A^{n}  (1.45)
a ↦ (x_{1}a, x_{2}a, …, x_{n}a)

has an inverse

ψ : A^{n} ⟶ A  (1.46)
(a_{1}, …, a_{n}) ↦ y_{1}a_{1} + ⋯ + y_{n}a_{n},

so A ≅ A^{n} as a right A-module. He showed that A is universal with respect to this property, of type (1, n − 1) (see §1.7) and it is a simple ring.

Leavitt’s algebra constructed in (1.44) has a natural grading; assigning 1 to y_{i} and −1 to x_{i}, 1 ≤ i ≤ n, since the relations are homogeneous (of degree zero), the ring A is a ℤ-graded ring (see §1.6.1 for a general construction of graded rings from free algebras). The isomorphism (1.45) induces a graded isomorphism

ϕ : A ⟶ A(−1)^{n}  (1.47)
a ↦ (x_{1}a, x_{2}a, …, x_{n}a),

where A(−1) is the suspension of A by −1. In fact letting

y = (y_{1}, …, y_{n}), and

x = (x_{1} ; x_{2} ; ⋮ ; x_{n}) ,

we have y ∈ 𝕄_{1×n}(A)[\overline{α}][\overline{δ}] and x ∈ 𝕄_{n×1}(A)[\overline{δ}][\overline{α}], where \overline{α} = (0) and \overline{δ} = (−1, …, −1). Thus by Proposition 1.3.10, A ≅_{gr} A(−1)^{n}.

Motivated by this algebra, the Leavitt path algebras were introduced in [2, 6], which associate to a direct graph a certain algebra. When the graph has one vertex and n loops, the algebra corresponds to this graph is the Leavitt algebra constructed in (1.44) and is denoted by ℒ(1, n) or ℒ_{n}. The Leavitt path algebras will provide a vast array of examples of graded algebras. We will study these algebras in §1.6.3.

1.4 Graded division rings

Graded fields and its noncommutative version, graded division rings, are among the simplest graded rings. With a little effort, we can completely compute the invariants of these algebras which we are interested in, namely, the graded Grothendieck groups (§3.6) and the graded Picard groups (§4).

Recall from §1.1.4 that a Γ-graded ring A = ⨁_{γ∈Γ} A_{γ} is called a graded division ring if every nonzero homogeneous element has a multiplicative inverse. Throughout this section we are considering graded (right) modules over graded division rings. Note that we work with the abelian graded group, however all the results are valid for non-abelian grading as well. We first show that for graded modules over a graded division ring, there is well-defined notion of dimension. The proofs follow the standard proofs in the non-graded setting (see [52, §IV, Thms. 2.4, 2.7, 2.13]), or the graded setting (see [73, Prop. 4.6.1], [51, p. 79]).

Proposition 1.4.1. Let A be a Γ-graded division ring and M be a graded A-module. Then M is a graded free A-module. More generally, any linearly independent subset of M consisting of homogeneous elements can be extended to a homogeneous basis of M.

Proof. Since A is a graded division ring (i.e., all homogeneous elements are invertible), for any m ∈ M^{h}, {m} is a linearly independent subset of M. This immediately gives the first statement of the theorem as a consequence of the second.

Fix a linearly independent subset X of M consisting of homogeneous elements. Let

F = {Q ⊆ M^{h} | X ⊆ Q and Q is A-linearly independent}.

This is a non-empty partially ordered set with inclusion, and every chain Q_{1} ⊆ Q_{2} ⊆ … in F has an upper bound ⋃ Q_{i} ∈ F. By Zorn’s Lemma, F has a maximal element, which we denote by P. If ⟨P ⟩ ≠ M, then there is a homogeneous element m ∈ M^{h} \ ⟨P ⟩. We will show that P ∪ {m} is a linearly independent set containing X, contradicting the maximality of P.

Suppose ma + ∑ p_{i}a_{i} = 0, where a, a_{i} ∈ A, p_{i} ∈ P with a ≠ 0. Then there is a homogeneous component of a, say a_{λ}, which is also non-zero. Considering the λ deg(m)-homogeneous component of this sum, we have m = ma_{λ}a_{λ}^{−1} = − ∑ p_{i}a′_{i}a_{λ}^{−1} for a′_{i} homogeneous, which contradicts the assumption m ∈ M^{h} \ ⟨P ⟩. Hence a = 0, which implies each a_{i} = 0. This gives the required contradiction, so M = ⟨P ⟩, completing the proof.

The following proposition shows in particular that a graded division ring has graded invariant basis number (we will discuss this type of rings in §1.7).

Proposition 1.4.2. Let A be a Γ-graded division ring and M be a graded A-module. Then any two homogeneous bases of M over A have the same cardinality.

Proof. By [52, Thm. IV.2.6], if a module M has an infinite basis over a ring, then any other basis of M has the same cardinality. This proves the Proposition in the case the the homogeneous basis is infinite.

Now suppose that M has two finite homogeneous bases X and Y. Then X = {x_{1}, …, x_{n}} and Y = {y_{1}, …, y_{m}}, for x_{i}, y_{i} ∈ M^{h} \ 0. As X is a basis for M, we can write

y_{m} = x_{1}a_{1} + ⋯ + x_{n}a_{n},

for some a_{i} ∈ A^{h}, where deg(y_{m}) = deg(a_{i}) deg(x_{i}) for each 1 ≤ i ≤ n. Since y_{m} ≠ 0, we have at least one a_{i} ≠ 0. Let a_{k} be the first non-zero a_{i}, and we note that a_{k} is invertible as it is non-zero and homogeneous in A. Then

x_{k} = y_{m}a_{k}^{−1} − x_{k+1}a_{k+1}a_{k}^{−1} − ⋯ − x_{n}a_{n}a_{k}^{−1},

and the set X′ = {y_{m}, x_{1}, …, x_{k−1}, x_{k+1}, …, x_{n}} spans M since X spans M. So

y_{m−1} = y_{m}b_{m} + x_{1}c_{1} + ⋯ + x_{k−1}c_{k−1} + x_{k+1}c_{k+1} + ⋯ + x_{n}c_{n},

for b_{m}, c_{i} ∈ A^{h}. There is at least one non-zero c_{i}, since if all the c_{i} are zero, then either y_{m} and y_{m−1} are linearly dependent or y_{m−1} is zero which are not the case. Let c_{j} denote the first non-zero c_{i}. Then x_{j} can be written as a linear combination of y_{m−1}, y_{m} and those x_{i} with i ≠ j, k. Therefore the set X′′ = {y_{m−1}, y_{m}} ∪ {x_{i} : i ≠ j, k} spans M since X′ spans M.

Continuing this process of adding a y and removing an x gives, after the k-th step, a set which spans M consisting of y_{m}, y_{m−1}, …, y_{m−k+1} and n − k of the x_{i}. If n < m, then after the n-th step, we would have that the set {y_{m}, …, y_{m−n+1}} spans M. But if n < m, then m − n + 1 ≥ 2, so this set does not contain y_{1}, and therefore y_{1} can be written as a linear combination of the elements of this set. This contradicts the linear independence of Y, so we must have m ≤ n. Repeating a similar argument with X and Y interchanged gives n ≤ m, so n = m.

The Propositions 1.4.1 and 1.4.2 above show that for a graded module M over a graded division ring A, M has a homogeneous basis and any two homogeneous bases of M have the same cardinality. The cardinal number of any homogeneous basis of M is called the dimension of M over A, and it is denoted by dim_{A}(M) or [M : A].

Proposition 1.4.3. Let A be a Γ-graded division ring and M be a graded A-module. If N is a graded submodule of M, then

dim_{A}(N) + dim_{A}(M/N) = dim_{A}(M).

Proof. By Proposition 1.4.1, the submodule N is a graded free A-module with a homogeneous basis Y which can be extended to a homogeneous basis X of M. We will show that U = {x + N | x ∈ X \ Y } is a homogeneous basis of M/N. Note that by (1.8), U consists of homogeneous elements. Let t ∈ (M/N)^{h}. Again by (1.8), t = m + N, where m ∈ M^{h} and m = ∑ x_{i}a_{i} + ∑ y_{j}b_{j} where a_{i}, b_{j} ∈ A, y_{j} ∈ Y and x_{i} ∈ X \ Y. So m + N = ∑(x_{i} + N)a_{i}, which shows that U spans M/N. If ∑(x_{i} + N)a_{i} = 0, for a_{i} ∈ A, x_{i} ∈ X \ Y, then ∑ x_{i}a_{i} ∈ N which implies that ∑ x_{i}a_{i} = ∑ y_{k}b_{k} for b_{k} ∈ A and y_{k} ∈ Y, which implies that a_{i} = 0 and b_{k} = 0 for all i, k. Therefore U is a homogeneous basis for M/N and as we can construct a bijective map X \ Y → U, we have |U| = |X \ Y |. Then dim_{A} M = |X| = |Y | + |X \ Y | = |Y | + |U| = dim_{A} N + dim_{A}(M/N).

The following statement is the graded version of a similar statement on simple rings (see [52, §IX.1]). This is required for the proof of Theorem 1.4.5.

Proposition 1.4.4. Let A and B be Γ-graded division rings. If

𝕄_{n}(A)(λ_{1}, …, λ_{n}) ≅_{gr} 𝕄_{m}(B)(γ_{1}, …, γ_{m})

as graded rings, where λ_{i}, γ_{j} ∈ Γ, 1 ≤ i ≤ n, 1 ≤ j ≤ m, then n = m and A ≅_{gr} B.

Proof. The proof follows the ungraded case (see [52, §IX.1]) with an extra attention given to the grading. We refer the reader to [70, §4.3] for a detailed proof.

We can further determine the relations between the shifting (λ_{1}, …, λ_{n}) and (γ_{1}, …, γ_{m}) in the above proposition. For this we need to extend [25, Theorem 2.1] (see also [73, Theorem 9.2.18]) from fields (with trivial grading) to graded division algebras. The following theorem states that two graded matrix algebras over a graded division ring with two shiftings are isomorphic if and only if one can obtain one shifting from the other by applying (1.39) and (1.40).

Theorem 1.4.5. Let A be a Γ-graded division ring. Then for λ_{i}, γ_{j} ∈ Γ, 1 ≤ i ≤ n, 1 ≤ j ≤ m,

𝕄_{n}(A)(λ_{1}, …, λ_{n}) ≅_{gr} 𝕄_{m}(A)(γ_{1}, …, γ_{m})  (1.48)

if and only if n = m and for a suitable permutation π ∈ S_{n}, we have λ_{i} = γ_{π(i)} + τ_{i} + σ, 1 ≤ i ≤ n, where τ_{i} ∈ Γ_{A} and a fixed σ ∈ Γ, i.e., (λ_{1}, …, λ_{n}) is obtained from (γ_{1}, …, γ_{m}) by applying (1.39) and (1.40).

Proof. One direction is Theorem 1.3.2, noting that since A is a graded division ring, Γ_{A} = Γ_{A}^{∗}.

We now prove the converse. That n = m follows from Proposition 1.4.4. By (1.4.1) one can find ϵ = (ε_{1}, …, ε_{1}, ε_{2}, …, ε_{2}, …, ε_{k}, …, ε_{k}) in Γ such that 𝕄_{n}(A)(λ_{1}, …, λ_{n}) ≅_{gr} 𝕄_{n}(A)(ϵ) as in (1.52). Now set

V = A(−ε_{1}) × ⋯ × A(−ε_{1}) × ⋯ × A(−ε_{k}) × ⋯ × A(−ε_{k})

and pick the (standard) homogeneous basis e_{i}, 1 ≤ i ≤ n and define E_{ij} ∈ End_{A}(V) by E_{ij}(e_{t}) = δ_{j,t}e_{i}, 1 ≤ i, j, t ≤ n. One can easily see that E_{ij} is a A-graded homomorphism of degree ε_{s_{i}} − ε_{s_{j}} where ε_{s_{i}} and ε_{s_{j}} are i-th and j-th elements in ϵ. Moreover, End_{A}(V) ≅_{gr} 𝕄_{n}(A)(ϵ) and E_{ij} corresponds to the matrix e_{ij} in 𝕄_{n}(A)(ϵ). In a similar manner, one can find ϵ′ = (ε′_{1}, …, ε′_{1}, ε′_{2}, …, ε′_{2}, …, ε′_{k′}, …, ε′_{k′}) and a graded A-vector space W such that

𝕄_{n}(A)(γ_{1}, …, γ_{n}) ≅_{gr} 𝕄_{n}(A)(ϵ′),

and End_{A}(W) ≅_{gr} 𝕄_{n}(A)(ϵ′). Therefore (1.48) provides a graded ring isomorphism θ : End_{A}(V) → End_{A}(W). Define E′_{ij} := θ(E_{ij}) and E′_{ii}(W) = Q_{i}, for 1 ≤ i, j ≤ n. Since {E_{ii} | 1 ≤ i ≤ n} is a complete system of orthogonal idempotents, so is {E′_{ii} | 1 ≤ i ≤ n}. It follows that

W ≅_{gr} _{⨁1≤j≤n}Q_{j}.

Furthermore, E′_{ij}E′_{tr} = δ_{j,t}E′_{ir} and E′_{ii} acts as identity on Q_{i}. These relations show that restricting E′_{ij} on Q_{j} induces an A-graded isomorphism E′_{ij} : Q_{j} → Q_{i} of degree ε_{s_{i}} −ε_{s_{j}} (same degree as E_{ij}). So Q_{j} ≅_{gr} Q_{1}(ε_{s_{1}} − ε_{s_{j}}) for any 1 ≤ j ≤ n. Therefore

W ≅_{gr} _{⨁1≤j≤n}Q_{1}(ε_{s_{1}} − ε_{s_{j}}).

By dimension count (see Proposition 1.4.3), it follows that dim_{A} Q_{1} = 1.

A similar argument for the identity map id : End_{A}(V) → End_{A}(V) produces

V ≅_{gr} _{⨁1≤j≤n}P_{1}(ε_{s_{1}} − ε_{s_{j}}),

where P_{1} = E_{11}(V), with dim_{A} P_{1} = 1.

Since P_{1} and Q_{1} are A-graded vector spaces of dimension 1, there is σ ∈ Γ, such that Q_{1} ≅_{gr} P_{1}(σ). Using the fact that for an A-graded module P and α, β ∈ Γ, P (α)(β) = P (α+β) = P (β)(α), we have

W ≅_{gr} _{⨁1≤j≤n}Q_{1}(ε_{s_{1}} − ε_{s_{j}}) ≅_{gr} _{⨁1≤j≤n}P_{1}(σ)(ε_{s_{1}} − ε_{s_{j}}) ≅_{gr}

⨁1≤j≤nP_{1}(ε_{s_{1}} − ε_{s_{j}})(σ) ≅_{gr} V (σ). (1.49)

We denote this A-graded isomorphism with ϕ : W → V (σ). Let e′_{i}, 1 ≤ i ≤ n be a (standard) homogeneous basis of degree ε′_{s_{i}} in W. Then ϕ(e′_{i}) = ∑_{1≤j≤n} e_{j}a_{j}, where a_{j} ∈ A^{h} and e_{j} are homogeneous of degree ε_{s_{j}} − σ in V (σ). Since deg(ϕ(e′_{i})) = ε′_{s_{i}}, all e_{j}’s with non-zero a_{j} in the sum have the same degree. For if ε_{s_{j}} − σ = deg(e_{j}) ≠ deg(e_{l}) = ε_{s_{l}} − σ, then since deg(e_{j}a_{j}) = deg(e_{l}a_{l}) = ε′_{s_{i}} it follows that ε_{s_{j}} − ε_{s_{l}} ∈ Γ_{A} which is a contradiction as Γ_{A} + ε_{s_{j}} and Γ_{A} + ε_{s_{l}} are distinct. Thus ε′_{s_{i}} = ε_{s_{j}} + τ_{j} − σ where τ_{j} = deg(a_{j}) ∈ Γ_{A}. In the same manner one can show that, ε′_{s_{i}} = ε′_{s_{i′}} in ϵ′ if and only if ε_{s_{j}} and ε_{s_{j′}} assigned to them by the previous argument are also equal. This shows that ϵ′ can be obtained from ϵ by applying (1.39) and (1.40). Since ϵ′ and ϵ are also obtained from γ_{1}, …, γ_{n} and λ_{1}, …, λ_{n}, respectively, by applying (1.39) and (1.40), putting these together shows that λ_{1}, …, λ_{n} and γ_{1}, …, γ_{n} have the similar relations, i.e., λ_{i} = γ_{π(i)} + τ_{i} + σ, 1 ≤ i ≤ n, where τ_{i} ∈ Γ_{A} and a fixed σ ∈ Γ.

A graded division algebra A is defined to be a graded division ring with centre R such that [A : R] < ∞. Note that since R is a graded field, by Propositions 1.4.1 and 1.4.2, A has a welldefined dimension over R. A graded division algebra A over its centre R is said to be unramified if Γ_{A} = Γ_{R} and totally ramified if A_{0} = R_{0}.

Let A be a graded division ring and let R be a graded subfield of A which is contained in the centre of A. It is clear that R_{0} = R ∩ A_{0} is a field and A_{0} is a division ring. The group of invertible homogeneous elements of A is denoted by A^{h∗}, which is equal to A^{h} \ 0. Considering A as a graded R-module, since R is a graded field, there is a uniquely defined dimension [A : R] by Theorem 1.4.1. The proposition below has been proven by Hwang, Wadsworth [50, Prop. 2.2] for two graded fields R ⊆ S with a torsion-free abelian grade group.

Proposition 1.4.6. Let A be a graded division ring and let R be a graded subfield of A which is contained in the centre of A. Then

[A : R] = [A_{0} : R_{0}]|Γ_{A} : Γ_{R}|.

Proof. Since A is a graded division ring, A_{0} is a division ring. Furthermore, R_{0} is a field. Let {x_{i}}_{i∈I} be a basis for A_{0} over R_{0}. Consider the cosets of Γ_{A} over Γ_{R} and let {δ_{j}}_{j∈J} be a coset representative, where δ_{j} ∈ Γ_{A}. Take {y_{j}}_{j∈J} ⊆ A^{h∗} such that deg(y_{j}) = δ_{j} for each j. We will show that {x_{i}y_{j}} is a basis for A over F.

Consider the map

ψ : A^{h∗} ⟶ Γ_{A}/Γ_{R},
a ⟼ deg(a) + Γ_{R}.

This is a group homomorphism with kernel A_{0}R^{h∗}, since for any a ∈ ker(ψ) there is some r ∈ R^{h∗} with ar^{−1} ∈ A_{0}. For a ∈ A, a = ∑_{γ∈Γ} a_{γ} where a_{γ} ∈ A_{γ} and ψ(a_{γ}) = γ + Γ_{R} = δ_{j} + Γ_{F} for some δ_{j} in the coset representative of Γ_{A} over Γ_{R}. Then there is some y_{j} with deg(y_{j}) = δ_{j} and a_{γ}y_{j}^{−1} ∈ ker(ψ) = A_{0}R^{h∗}. So

a_{γ}y_{j}^{−1} = (∑_{i} x_{i}r_{i})g

for g ∈ R^{h∗} and r_{i} ∈ R_{0}. Since R is in the centre of A, It follows a_{γ} = ∑_{i} x_{i}y_{j}r_{i}g. So a can be written as an R-linear combination of the elements of {x_{i}y_{j}}.

To show linear independence, suppose

∑_{i=1}^{n} x_{i}y_{i}r_{i} = 0,  (1.50)

for r_{i} ∈ R. Write r_{i} as the sum of its homogeneous components, and then consider a homogeneous component of the sum (1.50), say ∑_{k=1}^{m} x_{k}y_{k}r′_{k}, where deg(x_{k}y_{k}r′_{k}) = α. Since x_{k} ∈ A_{0}, deg(r′_{k}) + deg(y_{k}) = α for all k, so all of the y_{k} are the same. This implies that ∑_{k} x_{k}r′_{k} = 0, where all of the r′_{k} have the same degree. If r′_{k} = 0 for all k then r_{i} = 0 for all i we are done. Otherwise, for some r′_{l} ≠ 0, we have ∑_{k} x_{k}(r′_{k}r′_{l}^{−1}) = 0. Since {x_{i}} forms a basis for A_{0} over R_{0}, this implies r′_{k} = 0 for all k and thus r_{i} = 0 for all 1 ≤ i ≤ n.

Example 1.4.7. \underline{A graded division algebra associated to a valued division algebra}

Let D be a division algebra with a valuation. To this one associates a graded division algebra gr(D) = ⨁_{γ∈Γ_{D}} gr(D)_{γ}, where Γ_{D} is the value group of D and the summands gr(D)_{γ} arise from the filtration on D induced by the valuation (see details below and also Example 1.2.15). As it is illustrated in [51, 45], even though computations in the graded setting are often easier than working directly with D, it seems that not much is lost in passage from D to its corresponding graded division algebra gr(D). This has provided motivation to systematically study this correspondence, notably by Hwang, Tignol and Wadsworth [50, 51], and to compare certain functors defined on these objects, notably the Brauer group and the reduced Whitehead group SK_{1} [45, 89]. We introduce this correspondence here and in §3 we calculate their graded Grothendieck groups (Example 3.6.5).

Let D be a division algebra finite dimensional over its centre F, with a valuation v : D^{∗} → Γ. So Γ is a totally ordered abelian group, and for any a, b ∈ D^{∗}, v satisfies the following conditions

(i) v(ab) = v(a) + v(b);

(ii) v(a + b) ≥ min{v(a), v(b)} (b ≠ −a).

Let

V_{D} = {a ∈ D^{∗} : v(a) ≥ 0} ∪ {0}, the valuation ring of v;
M_{D} = {a ∈ D^{∗} : v(a) > 0} ∪ {0}, the unique maximal left (and right) ideal of V_{D};
\overline{D} = V_{D}/M_{D}, the residue division ring of v on D; and
Γ_{D} = im(v), the value group of the valuation.

For background on valued division algebras, see [54] or the survey paper [91]. One associates to D a graded division algebra as follows: For each γ ∈ Γ_{D}, let

D^{≥γ} = {d ∈ D^{∗} : v(d) ≥ γ} ∪ {0}, an additive subgroup of D;
D^{>γ} = {d ∈ D^{∗} : v(d) > γ} ∪ {0}, a subgroup of D^{≥γ}; and
gr(D)_{γ} = D^{≥γ}/D^{>γ}.

Then define

gr(D) = ⨁_{γ∈Γ_{D}} gr(D)_{γ}.

Because D^{>γ}D^{≥δ} + D^{≥γ}D^{>δ} ⊆ D^{>(γ+δ)} for all γ, δ ∈ Γ_{D}, the multiplication on gr(D) induced by multiplication on D is well-defined, giving that gr(D) is a Γ-graded ring, called the associated graded ring of D. The multiplicative property (i) of the valuation v implies that gr(D) is a graded division ring. Clearly, we have gr(D)_{0} = \overline{D}, and Γ_{gr(D)} = Γ_{D}. For d ∈ D^{∗}, we write d̃for the image d + D^{>v(d)} of d in gr(D)_{v(d)}. Thus, the map given by d ↦ d̃ is a group epimorphism D^{∗} → gr(D)^{∗} with kernel 1 + M_{D}.

The restriction v|_{F} of the valuation on D to its centre F is a valuation on F, which induces a corresponding graded field gr(F). Then it is clear that gr(D) is a graded gr(F)-algebra, and one can prove that for

[gr(D) : gr(F)] = [\overline{D} : \overline{F}] |Γ_{D} : Γ_{F} | ≤ [D : F] < ∞.

Now let F be a field with a henselian valuation v. Then the valuation of F extends uniquely to D (see [91, Theorem 2.1]), and with respect to this valuation, D is said to be tame if Z(\overline{D}) is separable over \overline{F} and char(\overline{F}) ∤ ind(D)/( ind(\overline{D})[Z(\overline{D}) : \overline{F}]). It is known ([51, Proposition 4.3]) that D is tame if and only if [gr(D) : gr(F)] = [D : F] and Z(gr(D)) = gr(F).

We will compute the graded Grothendieck group and the graded Picard group of these division algebras in Examples 3.6.5 and 4.2.6.

1.4.1 Ring of zero homogeneous elements of a graded central simple ring

Let A be a Γ-graded division ring and 𝕄_{n}(A)(λ_{1}, …, λ_{n}) be a graded simple ring, where λ_{i} ∈ Γ, 1 ≤ i ≤ n. Since A is a division ring, Γ_{A} is a subgroup of Γ. Consider the quotient group Γ/Γ_{A} and let Γ_{A} + ε_{1}, …, Γ_{A} + ε_{k} be the distinct elements in Γ/Γ_{A} representing the cosets Γ_{A} + λ_{i}, 1 ≤ i ≤ n, and for each ε_{l}, let r_{l} be the number of i with Γ_{A} + λ_{i} = Γ_{A} + ε_{l}. It was observed in [51, Proposition 1.4] that

𝕄_{n}(A)_{0} ≅ 𝕄_{r_{1}}(A_{0}) × ⋯ × 𝕄_{r_{k}}(A_{0})  (1.51)

and in particular 𝕄_{n}(A)_{0} is a simple ring if and only if k = 1. Indeed, using (1.39) and (1.40) we get

𝕄_{n}(A)(λ_{1}, …, λ_{n}) ≅_{gr} 𝕄_{n}(A)(ε_{1}, …, ε_{1}, ε_{2}, …, ε_{2}, …, ε_{k}, …, ε_{k}),  (1.52)

with each ε_{l} occurring r_{l} times. Now (1.34) for λ = 0 and

(δ_{1}, …, δ_{n}) = (ε_{1}, … ε_{1}, ε_{2}, …, ε_{2}, …, ε_{k}, …, ε_{k})

immediately gives (1.51).

Remark 1.4.8. \underline{The graded Artin-Wedderburn structure theorem}

The Artin-Wedderburn theorem shows that division rings are basic “building blocks” of ring theory. A graded version of Artin-Wedderburn structure theorem holds. We state the statement here without proof. We refer the reader to [73, 51] for proofs of these statements.

A Γ-graded ring B is isomorphic to 𝕄_{n}(A)(λ_{1}, …, λ_{n}), where A is a Γ-graded division ring and λ_{i} ∈ Γ, 1 ≤ i ≤ n, if and only if B is graded right Artinian (i.e., a decreasing chain of graded right ideals becomes stationary) and graded simple.

A Γ-graded ring B is isomorphic to a finite product of matrix rings overs graded division rings (with suitable shiftings) if and only if B is graded right Artinian and graded primitive (i.e., J^{gr}(B) = 0).

1.5 Strongly graded rings and Dade’s theorem

Let A be a Γ-graded ring and Ω be a subgroup of Γ. Recall from Example 1.1.8 the Ω-graded ring A_{Ω} = ⨁_{γ∈Ω} A_{γ}. If A is a Γ-strongly graded ring, then it is immediate that A_{Ω} is a Ωstrongly graded ring. In this case one can show that the category of Γ-graded A-modules, Gr^{Γ}-A, is equivalent to the category of Ω-graded A_{Ω}-modules, Gr^{Ω}-A_{Ω}. In fact, the equivalence

Gr^{Γ}-A ≈ Gr^{Ω}-A_{Ω},

under the given natural functors (see Theorem 1.5.7) implies that A is Γ-strongly graded ring. This was first proved by Dade [30] in the case of Ω = 0. We prove Dade’s theorem (Theorem 1.5.1) and then state this more general case in Theorem 1.5.7.

Let A be a Γ-graded ring. For any right A_{0}-module N and any γ ∈ Γ, we identify the right A_{0}-module N ⊗_{A_{0}} A_{γ} with its image in N ⊗_{A_{0}} A. Since A = ⨁_{γ∈Γ} A_{γ} and A_{γ} are A_{0}-bimodules, N ⊗_{A_{0}} A is a Γ-graded right A-module, with

N ⊗_{A_{0}} A = ⨁_{γ∈Γ} (N ⊗_{A_{0}} A_{γ}).  (1.53)

Consider the restriction functor

\begin{matrix}𝒢 := (−)_{0} : Gr-A & ⟶ Mod- A_{0} \\ M & ⟼ M_{0} \\ ψ & ⟼ ψ|_{M_{0}}, \end{matrix}

and the induction functor defined by

\begin{matrix}ℐ := − ⊗_{A_{0}} A : Mod- A_{0} & ⟶ Gr-A \\ N & ⟼ N ⊗_{A_{0}} A \\ ϕ & ⟼ ϕ ⊗ id_{A} . \end{matrix}

One can easily check that 𝒢 ∘ ℐ ≅ id_{A_{0}} with the natural transformation,

𝒢 ℐ(N) = 𝒢(N ⊗_{A_{0}} A) = N ⊗_{A_{0}} A_{0} ⟶ N,  (1.54)
n ⊗ a ↦ na.

On the other hand, there is a natural transformation,

ℐ 𝒢(M) = ℐ(M_{0}) = M_{0} ⊗_{A_{0}} A ⟶ M,  (1.55)
m ⊗ a ↦ ma.

The theorem below shows that ℐ ∘𝒢 ≅ id_{A} (under (1.55)), if and only if A is a strongly graded ring. Theorem 1.5.1 was proved by Dade [30, Theorem 2.8] (see also [73, Thm. 3.1.1]).

Theorem 1.5.1 (Dade’s Theorem). Let A be a Γ-graded ring. Then A is strongly graded if and only if the functors (−)_{0} : Gr-A → Mod- A_{0} and −⊗_{A_{0}} A : Mod- A_{0} → Gr-A form mutually inverse equivalences of categories.

Proof. Suppose A is strongly graded. One can easily check (without using the assumption that A is strongy graded) that 𝒢 ∘ ℐ ≅ id_{A_{0}}. We show that ℐ ∘𝒢 ≅ id_{A}.

For a graded A-module M we have ℐ ∘𝒢(M) = M_{0} ⊗_{A_{0}} A. We will show that the natural homomorphism ϕ : M_{0} ⊗_{A_{0}} A → M; m ⊗ a ↦ ma is a graded A-module isomorphism. The map ϕ is clearly graded (see (1.53)). Since A is strongly graded, it follows that for γ, δ ∈ Γ,

M_{γ+δ} = M_{γ+δ}A_{0} = M_{γ+δ}A_{−γ}A_{γ} ⊆ M_{δ}A_{γ} ⊆ M_{γ+δ}.  (1.56)

Thus M_{δ}A_{γ} = M_{γ+δ}. Therefore, ϕ(M_{0} ⊗_{A_{0}} A_{γ}) = M_{0}A_{γ} = M_{γ}, which implies that ϕ is surjective.

Let N = ker(ϕ), which is a graded A-submodule of M_{0} ⊗_{A_{0}} A, so N_{0} = N ∩ (M_{0} ⊗_{A_{0}} A_{0}). However the restriction of ϕ to M_{0} ⊗_{A_{0}} A_{0} → M_{0} is the canonical isomorphism, so N_{0} = 0. Since N is a graded A-module, a similar argument as (1.56) shows N_{γ} = N_{0}A_{γ} = 0 for all γ ∈ Γ. It follows that ϕ is injective. Thus ℐ ∘𝒢(M) = M_{0} ⊗_{A_{0}} A ≅ M. Since all the homomorphisms involved are natural, this shows that ℐ ∘𝒢 ≅ id_{A}.

For the converse suppose ℐ and 𝒢 are mutually inverse (under (1.55) and (1.54)). For any graded A-module M, since ℐ 𝒢(M) ≅_{gr} M, we have, for any α ∈ Γ,

M_{0} ⊗_{A_{0}} A_{α} ⟶ M_{α},
m ⊗ a ↦ ma

is an bijective map. This immediately implies

M_{0}A_{α} = M_{α}.  (1.57)

Now for any β ∈ Γ, consider the graded A-module A(β). Replacing M by A(β) in (1.57), we get A(β)_{0}A_{α} = A(β)_{α}, i.e., A_{β}A_{α} = A_{β+α}. This shows that A is strongly graded.

Corollary 1.5.2. Let A be a strongly Γ-graded ring. Then the functors

(−)_{0} : Gr^{Γ/Ω}-A → Mod- A_{Ω},

and

− ⊗_{A_{Ω}} A : Mod- A_{Ω} → Gr^{Γ/Ω}-A

form mutually inverse equivalences of categories.

Proof. Since A is a strongly Γ-graded ring, it follows that A is a Γ/Ω-graded ring (Example 1.1.18). The result now follows from Theorem 1.5.1.

Remark 1.5.3. Recall that gr- A denotes the category of graded finitely generated right A-modules and Pgr-A denotes the category of graded finitely generated projective right A-modules. Note that in general the restriction functor (−)_{0} : Gr-A → Mod- A_{0} does not induce a functor (−)_{0} : Pgr-A → Pr- A_{0}. In fact, one can easily produce a graded finitely generated projective A-module P such that P_{0} is not projective A_{0}-module. As an example, consider the ℤ-graded ring T of Example 1.1.5. Then T (1) is clearly a graded finitely generated projective T -module. However T (1)_{0} = M is not T_{0} = R-module.

Remark 1.5.4. The proof of Theorem 1.5.1 also shows that A is strongly graded if and only if gr- A ≅ mod- A_{0}, if and only if Pgr-A ≅ Pr- A_{0}, (see Remark 1.5.3) via the same functors (−)_{0} and − ⊗_{A_{0}} A of the Theorem 1.5.1.

Remark 1.5.5. \underline{Strongly graded modules}

Let A be a Γ-graded ring and M be a graded A-module. Then M is called strongly graded A-module if

M_{α}A_{β} = M_{α+β},  (1.58)

for any α, β ∈ Γ. The proof of Theorem 1.5.1 shows that A is strongly graded if and only if any graded A-module is strongly graded. Indeed if A is strongly graded then (1.56) shows that any graded A-module is strongly graded. Conversely, if any graded module is strongly graded, then considering A as a graded A-module, (1.58) for M = A, shows that A_{α}A_{β} = A_{α+β} for any α, β ∈ Γ.

Remark 1.5.6. \underline{Ideal correspondence between A_{0} and A for a strongly graded ring}

The proof of Theorem 1.5.1 shows that there is a one-to-one correspondence between the right ideals of A_{0} and the graded right ideals of A (similarly for the left ideals). However, this correspondence does not hold between two sided ideals. As an example, A = 𝕄_{2}(K[x^{2}, x^{−2}])(0, 1), where K is a field, is a strongly ℤ-graded simple ring, whereas A_{0} ≅ K ⊗ K is not a simple ring (see §1.4.1).

In the same way, the equivalence Gr-A ≈ Mod- A_{0} of Theorem 1.5.1 gives a correspondence between several (one-sided) properties of graded objects in A with objects over A_{0}. For example, one can easily show that A is graded right (left) Noetherian if and only if A_{0} is right (left) Noetherian (see also Corollary 1.5.9).

Using Theorem 1.5.1, we will see that the graded Grothendieck group of a strongly graded ring coincides with the (classical) Grothendieck group of the ring of zero homogeneous part (see §3.1.3).

Theorem 1.5.7. Let A be a Γ-graded ring and Ω be a subgroup of Γ. Then A is Γ-strongly graded ring if and only if

(−)_{Ω} : Gr^{Γ}-A ⟶ Gr^{Ω}-A_{Ω}
M ⟼ M_{Ω}
ψ ⟼ ψ|_{M_{Ω}},

and

− ⊗_{A_{0}} A : Gr^{Ω}-A_{Ω} ⟶ Gr^{Γ}-A
N ⟼ N_{0} ⊗_{A_{0}} A
ϕ ⟼ ϕ_{0} ⊗ id_{A},

form mutually inverse equivalences of categories.

Proof. The proof is similar to the proof of Theorem 1.5.1 and it is omitted.

Example 1.5.8. Let A be a Γ × Ω-strongly graded ring. Then by Theorem 1.5.7

Gr^{Γ×Ω}-A ≈ Gr^{Ω}-A_{(0,−)},

where A_{(0,−)} = ⨁_{ω∈Ω} A_{(0,ω)}. This example will be used in §6.3. Compare this also with Corollary 1.2.10.

Another application of Theorem 1.5.1 is to provide a condition when a strongly graded ring is a graded von Neumann ring (§1.1.6). This will be used later in Coroallry 1.6.13 to show that the Leavitt path algebras are von Neumann regular rings.

Corollary 1.5.9. Let A be a strongly graded ring. Then A a is graded von Neumann regular ring if and only if A_{0} is a von Neumann regular ring.

Sketch of proof. Since any (graded) flat module is a direct limit of (graded) projective modules, from the equivalence of categories Gr-A ≈_{gr} Mod- A_{0} (Theorem 1.5.1), it follows that A is graded von Neumann regular if and only if A_{0} is von Neumann regular.

Remark 1.5.10. An element-wise proof of Corollary 1.5.9 can also be found in [95, Theorem 3].

For a Γ-graded ring A, and α, β ∈ Γ, one has a A_{0}-bimodule homomorphism

ϕ_{α,β} : A_{α} ⊗_{A_{0}} A_{β} ⟶ A_{α+β}  (1.59)
a ⊗ b ⟼ ab.

The following theorem gives another characterisation for strongly graded rings.

Theorem 1.5.11. Let A be a Γ-graded ring. Then A is a strongly graded ring if and only if for any γ ∈ Γ, the homomorphism

ϕ_{γ,−γ} : A_{γ} ⊗ A_{−γ} ⟶ A_{0},
a ⊗ b ⟼ ab,

is an isomorphism.

Proof. Suppose that for any γ ∈ Γ, the map ϕ_{γ,−γ} : A_{γ} ⊗ A_{−γ} → A_{0} is an isomorphism. Thus there are a_{i} ∈ A_{γ}, b_{i} ∈ A_{−γ} such that

∑_{i} a_{i}b_{i} = ϕ_{γ,−γ}( ∑_{i} a_{i} ⊗ b_{i}) = 1.

So 1 ∈ A_{γ}A_{−γ}. Now by Proposition 1.1.16(1) A is strongly graded.

Conversely, suppose A is a strongly graded ring. We prove that the homomorphism (1.59) is an isomorphism. The definition of strongly graded implies that ϕ_{α,β} is surjective. Suppose

ϕ_{α,β}(∑_{i} a_{i} ⊗ b_{i}) = ∑_{i} a_{i}b_{i} = 0.  (1.60)

Using Proposition 1.1.16(1), write 1 = ∑_{j} x_{j}y_{j}, where x_{j} ∈ A_{−β} and y_{j} ∈ A_{β}. Then

∑_{i} a_{i} ⊗ b_{i} = (∑_{i} a_{i} ⊗ b_{i})(∑_{j} x_{j}y_{j}) = ∑_{i} (a_{i} ⊗ ∑_{j} b_{i}x_{j}y_{j}) =

∑_{i}(∑_{j}(a_{i}b_{i}x_{j} ⊗ y_{j})) = ∑_{j} ∑_{i}(a_{i}b_{i}x_{j} ⊗ y_{j}) = ∑_{j} ( ∑_{i}(a_{i}b_{i})x_{j} ⊗ y_{j})) = 0.

This shows that ϕ_{α,β} is injective. Now setting α = γ and β = −γ finishes the proof.

1.5.1

Let A and B be rings and P be a A − B-bimodule. Then P is called an invertible A − B-bimodule, if there is a B − A-bimodule Q such that P ⊗_{B} Q ≅ A as A − A-bimodules and Q ⊗_{A} P ≅ B as B − B-bimodules and the following diagrams are commutative.

\xymatrix{P ⊗_{B} Q ⊗_{A} P \ar[r] \ar[d] & A ⊗_{A} P \ar[d] & Q ⊗_{A} P ⊗_{B} Q \ar[r] \ar[d] & B ⊗_{B} Q \ar[d] \\ P ⊗_{B} B \ar[r] & P & Q ⊗_{A} A \ar[r] & Q}

One can prove that P is a finitely generated projective A and B-modules.

Now Theorem 1.5.11 shows that for a strongly Γ-graded ring A, the A_{0}-bimodules A_{γ}, γ ∈ Γ, is an invertible module and thus is a finitely generated projective A_{0}-module. This in return implies that A is a projective A_{0}-module. Note that in general, one can easily construct a graded ring A where A is not projective over A_{0} (see Example 1.1.5) and A_{γ} is not finitely generated A_{0}-module, such as the ℤ-graded ring ℤ[x_{i} | i ∈ ℕ] of Example 1.1.10.

Remark 1.5.12. \underline{Other terminologies for strongly graded rings}

The term “strongly graded” for such rings was coined by E. Dade in [30] which is now commonly in use. Other terms for these rings are fully graded and generalised crossed products. See [31] for a history of development of such rings in literature.

1.6 Grading on graph algebras

1.6.1 Grading on free rings

Let X be a nonempty set of symbols and Γ be a group. (As always we assume the groups are abelian, although the entire theory can be written for an arbitrary group.) Let d : X → Γ be a map. One can extend d in a natural way to a map from the set of finite words on X to Γ, which is called d again. For example if x, y, z ∈ X and xyz is a word, then d(xyz) = d(x) + d(y) + d(z). One can easily see that if w_{1}, w_{2} are two words, then d(w_{1}w_{2}) = d(w_{1}) + d(w_{2}). If we allow an empty word, which will be the identity element in the free ring, then we assign the identity of Γ to this word.

Let R be a ring and R(X) be the free ring (with or without identity) on a set X with coefficients in R. The elements of R(X) are of the form ∑_{w} r_{w}w, where r_{w} ∈ R and w stands for a word on X. The multiplication is defined by

( ∑_{w} r_{w}w)( ∑_{v} r_{v}v) = ∑_{z} ( ∑_{{w,v|z=wv, r_{w},r_{v}≠0}} r_{w}r_{v})z.

Define R(X)_{γ} = {∑_{w} r_{w}w | d(w) = γ). One can check that R(X) = ⨁_{γ∈Γ} R(X)_{γ}. Thus R(X) is a Γ-graded ring. Note that if we don’t allow the empty word in the construction, then R(X) is a graded ring without identity (see Remark 1.1.15). It is easy to see that R(X) is never a strongly graded ring.

Example 1.6.1. Let R be a ring and R(X) be the free ring on a set X with a graded structure induced by a map d : X → Γ. Let Ω be a subgroup of Γ and consider the map

\overline{d} : X ⟶ Γ/Ω,
x ⟼ Ω + d(x).

The map \overline{d} induces a Γ/Ω-graded structure on R(X) which coincides with the general construction of quotient grading given in Example 1.1.8.

Example 1.6.2. Let X = {x} be a set of symbols with one element and ℤ_{n} be the cyclic group with n elements. Assign 1 ∈ ℤ_{n} to x and generate the free ring with identity on X with coefficients in a field F. This ring is the usual polynomial ring F [x] which, by the above construction, is equipped by a ℤ_{n}-grading. Namely,

F [x] = ⨁_{k∈ℤ_{n}} ( ∑_{\begin{matrix}l∈ℕ, \\ \overline{l}=k \end{matrix}} F x^{l}),

where \overline{l} is the image of l in the group ℤ_{n}. For a ∈ F, since the polynomial x^{n} − a is a homogeneous element of degree zero, the ideal ⟨x^{n} − a⟩ is a graded ideal and thus the quotient ring F [x]/⟨x^{n} − a⟩ is also a ℤ_{n}-graded ring (see §1.1.5). In particular if x^{n} − a is an irreducible polynomial in F [x], then the field F [x]/⟨x^{n} − a⟩ is a ℤ_{n}-graded field as well.

Example 1.6.3. Let {x, y} be a set of symbols. Assign 1 ∈ ℤ_{2} to x and y and consider the graded free ring ℝ(x, y). The ideal generated by homogeneous elements {x^{2} + 1, y^{2} + 1, xy + yx} is graded and thus we retrieve the ℤ_{2}-graded Hamiltonian quaternion algebra of Example 1.1.21 as follows:

ℍ ≅ ℝ(x, y)/⟨x^{2} + 1, y^{2} + 1, xy + yx⟩.

Furthermore, assigning (1, 0) ∈ ℤ_{2}×ℤ_{2} to x and (0, 1) ∈ ℤ_{2}×ℤ_{2} to y we obtained the ℤ_{2}×ℤ_{2}-graded quaternion algebra of Example 1.1.21.

Example 1.6.4. \underline{The Weyl algebra}

For a (commutative) ring R, the Weyl algebra R(x, y)/⟨xy − yx − 1⟩ can be considered as a ℤ-graded ring by assigning 1 to x and −1 to y.

Example 1.6.5. \underline{The Leavitt algebra ℒ(n, k + 1)}

Let K be a field, n and k positive integers and A be the free associative K-algebra with identity generated by symbols {x_{ij}, y_{ji} | 1 ≤ i ≤ n + k, 1 ≤ j ≤ n} subject to relations (coming from)

Y · X = I_{n,n}  and  X · Y = I_{n+k,n+k},

where

Y = (\begin{matrix}y_{11} & y_{12} & … & y_{1,n+k} \\ y_{21} & y_{22} & … & y_{2,n+k} \\ ⋮ & ⋮ & ... & ⋮ \\ y_{n,1} & y_{n,2} & … & y_{n,n+k} \end{matrix}) , X = (\begin{matrix}x_{11} & x_{12} & … & x_{1,n} \\ x_{21} & x_{22} & … & x_{2,n} \\ ⋮ & ⋮ & ... & ⋮ \\ x_{n+k,1} & x_{n+k,2} & … & x_{n+k,n} \end{matrix}) .  (1.61)

In Example 1.3.13 we studied a special case of this algebra when n = 1 and k = n − 1. This algebra was studied by Leavitt in relation with its type in [63, p.130] which is shown that for arbitrary n and k the algebra is of type (n, k) (see §1.7) and when n ≥ 2 they are domains. We denote this algebra by ℒ(n, k + 1). (Cohn’s notation in [26] for this algebra is V_{n,n+k}.)

Assigning deg(y_{ji}) = (0, …, 0, 1, 0 …, 0) and deg(x_{ij}) = (0, …, 0, −1, 0 …, 0), 1 ≤ i ≤ n + k, 1 ≤ j ≤ n, in ⨁_{n} ℤ, where 1 and −1 are in j-th entries respectively, makes the free algebra generated by x_{ij} and y_{ji} a graded ring. Furthermore, one can easily observe that the relations coming from (1.61) are all homogeneous with respect to this grading, so that the Leavitt algebra ℒ(n, k + 1) is a ⨁_{n} ℤ-graded ring. In particular, ℒ(1, k) is a ℤ-graded ring (Example 1.3.13).

1.6.2 Corner skew Laurent polynomial rings

Let R be a ring with identity and p an idempotent of R. Let ϕ : R → pRp be a corner isomorphism, i.e, a ring isomorphism such that ϕ(1) = p. A corner skew Laurent polynomial ring with coefficients in R, denoted by R[t_{+}, t_{−}, ϕ], is a unital ring which is constructed as follows: The elements of R[t_{+}, t_{−}, ϕ] are formal expressions

t_{−}^{j}r_{−j} + t_{−}^{j−1}r_{−j+1} + ⋯ + t_{−}r_{−1} + r_{0} + r_{1}t_{+} + ⋯ + r_{i}t_{+}^{i},

where r_{−n} ∈ p_{n}R and r_{n} ∈ Rp_{n}, for all n ≥ 0, where p_{0} = 1 and p_{n} = ϕ^{n}(p_{0}). The addition is component-wise, and the multiplication is determined by the distribution law and the following rules:

t_{−}t_{+} = 1,  t_{+}t_{−} = p,  rt_{−} = t_{−}ϕ(r),  t_{+}r = ϕ(r)t_{+}.  (1.62)

The corner skew Laurent polynomial rings are studied in [7], where their K_{1}-groups are calculated. This construction is a special case of a so called fractional skew monoid rings constructed in [8]. Assigning −1 to t_{−} and 1 to t_{+} makes A := R[t_{+}, t_{−}, ϕ] a ℤ-graded ring with A = ⨁_{i∈ℤ} A_{i} (see [8, Proposition 1.6])

A_{i} = Rp_{i}t_{+}^{i}, for i > 0,
A_{i} = t_{−}^{i}p_{−i}R, for i < 0,
A_{0} = R.

Clearly, when p = 1 and ϕ is the identity map, then R[t_{+}, t_{−}, ϕ] reduces to the familiar ring R[t, t^{−1}].

In the next two propositions we will characterise those corner skew Laurent polynomials which are strongly graded rings (§1.1.2) and graded von Neumann regular rings (§1.1.6).

Recall that an idempotent element p of the ring R is called a full idempotent if RpR = R.

Proposition 1.6.6. Let R be a ring with identity and A = R[t_{+}, t_{−}, ϕ] be a corner skew Laurent polynomial ring. Then A is strongly graded if and only if ϕ(1) is a full idempotent.

Proof. First note that A_{1} = Rϕ(1)t_{+} and A_{−1} = t_{−}ϕ(1)R. Furthermore, since ϕ(1) = p, we have

r_{1}ϕ(1)t_{+}t_{−}ϕ(1)r_{2} = r_{1}ϕ(1)pϕ(1)r_{2} = r_{1}pppr_{2} = r_{1}ϕ(1)r_{2}.

Suppose A is strongly graded. Then 1 ∈ A_{1}A_{−1}. That is

1 = ∑_{i} (r_{i}ϕ(1)t_{+})(t_{−}ϕ(1)r′_{i}) = ∑_{i} r_{i}ϕ(1)r′_{i},  (1.63)

where r_{i}, r′_{i} ∈ R. So Rϕ(1)R = R, that is ϕ(1) is a full idempotent.

On the other hand suppose ϕ(1) is a full idempotent. Since ℤ is generated by 1, in order to prove that A is strongly graded, it is enough to show that 1 ∈ A_{1}A_{−1} and 1 ∈ A_{−1}A_{1} (see §1.1.2). But

t_{−}ϕ(1)ϕ(1)t_{+} = t_{−1}ϕ(1)t_{+} = 1t_{−}t_{+} = 1,

shows that 1 ∈ A_{−1}A_{1}. Since ϕ(1) is a full idempotent, there are r_{i}, r′_{i} ∈ R, i ∈ I such that ∑ r_{i}ϕ(1)r′_{i} = 1. Then Equation 1.63 shows that 1 ∈ A_{1}A_{−1}.

Proposition 1.6.7. Let R be a ring with identity and A = R[t_{+}, t_{−}, ϕ] be a corner skew Laurent polynomial ring. Then A is a graded von Neumann regular ring if and only if R is a von Neumann regular ring.

Proof. If a graded ring is graded von Neumann regular, then it is easy to see that its zero component ring is von Neumann regular. This proves one direction of the theorem. For the converse, suppose R is regular. Let x ∈ A_{i}, where i > 0. So x = rp_{i}t_{+}^{i}, for some r ∈ R, where p_{i} = ϕ^{i}(1). By relations (1.62) and induction, we have t_{+}^{i}t_{−}^{i} = ϕ^{i}(p_{0}) = p_{i}. Since R is regular, there is an s ∈ R such that rp_{i}srp_{i} = rp_{i}. Then choosing y = t_{−}^{i}p_{i}s, we have

xyx = (rp_{i}t_{+}^{i})(t_{−}^{i}p_{i}s)(rp_{i}t_{+}^{i}) = (rp_{i}t_{+}^{i}t_{−}^{i}p_{i}s)(rp_{i}t_{+}^{i}) = rp_{i}p_{i}p_{i}srp_{i}t_{+}^{i} = rp_{i}t_{+}^{i} = x.

A similar argument shows that for x ∈ A_{i}, where i < 0, there is a y such that xyx = x. This shows that A is a graded von Neumann regular ring.

Note that in a corner skew Laurent polynomial ring R[t_{+}, t_{−}, ϕ], t_{+} is a left invertible element with a right inverse t_{−} (see the relations (1.62)). In fact this property characterises such rings. Namely, a graded ring A = ⨁_{i∈ℤ} A_{i} such that A_{1} has a left invertible element is a corner skew Laurent polynomial ring as the following theorem shows. The following theorem (first established in [8]) will be used to realise Leavitt path algebras (§1.6.3) as corner skew Laurent polynomial rings.

Theorem 1.6.8. Let A be a ℤ-graded ring which has a left invertible element t_{+} ∈ A_{1}. Then its right inverse t_{−} ∈ A_{−1}, and A = A_{0}[t_{+}, t_{−}, ϕ], where ϕ : A_{0} → t_{+}t_{−}A_{0}t_{+}t_{−}, ϕ(a) = t_{+}at_{−}.

Proof. Since t_{−}t_{+} = 1 and t_{+} ∈ A_{1}, it easily follows that t_{−} ∈ A_{−1}. Furthermore t_{+}t_{−} = t_{+}t_{−}t_{+}t_{−} is a homogeneous idempotent of degree zero. Observe that

ϕ : A_{0} ⟶ t_{+}t_{−}A_{0}t_{+}t_{−},
a ⟼ t_{+}at_{−}

is a (unital) ring isomorphism. Consider the corner skew Laurent polynomial ring Ã = A_{0}[t̃_{+}, t̃_{−}, ϕ]. Since ϕ(a) = t_{+}at_{−}, it follows that t_{−}ϕ(a) = at_{−} and ϕ(a)t_{+} = t_{+}a. Thus t_{+} and t_{−} satisfy all the relations in (1.62). Therefore there is a well-defined map ψ : Ã → A, such that ψ(t̃_{±}) = t_{±} and the restriction of ψ on A_{0} is the identity and

ψ( ∑_{k=1}^{j} t̃_{−}^{k}a_{−k} + a_{0} + ∑_{k=1}^{i} a_{i}t̃_{+}^{i}) = ∑_{k=1}^{j} t_{−}^{k}a_{−k} + a_{0} + ∑_{k=1}^{i} a_{i}t_{+}^{i}.

This also shows that ψ is a graded homomorphism. In order to show that ψ is an isomorphism, it suffices to show that its restriction to each homogeneous component ψ : Ã_{i} → A_{i} is a bijection. Suppose x ∈ Ã_{i}, i > 0 such that ψ(x) = 0. Then x = dt̃_{+}^{i} for some d ∈ A_{0}p_{i} where p_{i} = ϕ^{i}(1) and ψ(x) = dt_{+}^{i}. Note that ϕ^{i}(1) = t_{+}^{i}t_{−}^{i}. Thus dϕ^{i}(1) = dt_{+}^{i}t_{−}^{i} = ψ(x)t_{−}^{i} = 0. It now follows x = dt̃_{+}^{i} = dϕ^{i}(1)t̃_{+}^{i} = 0 in Ã_{i}. This shows ψ is injective. Suppose y ∈ A_{i}. Then yt_{−}^{i} ∈ A_{0} and yt_{−}^{i}t_{+}^{i}t_{−}^{i} = yt_{−}^{i}ϕ^{i}(1) ∈ A_{0}ϕ^{i}(1) = A_{0}p_{i}. This shows yt_{−}^{i}t_{+}^{i}t_{−}^{i}t̃_{+}^{i} ∈ Ã_{i}. But ψ(yt_{−}^{i}t_{+}^{i}t_{−}^{i}t̃_{+}^{i}) = yt_{−}^{i}t_{+}^{i}t_{−}^{i}t_{+}^{i} = y. This shows that ψ : Ã_{i} → A_{i}, i > 0 is a bijection. A similar argument can be written for the case of i < 0. The case i = 0 is obvious. This completes the proof.

1.6.3 Graphs and Leavitt path algebras

In this subsection we gather some graph-theoretic definitions and recall the basics on path and Leavitt path algebras. A directed graph E = (E^{0}, E^{1}, r, s) consists of two countable sets E^{0}, E^{1} and maps r, s : E^{1} → E^{0}. The elements of E^{0} are called vertices and the elements of E^{1} edges. If s^{−1}(v) is a finite set for every v ∈ E^{0}, then the graph is called row-finite. In this note we will consider only row-finite graphs. In this setting, if the number of vertices, i.e., |E^{0}|, is finite, then the number of edges, i.e., |E^{1}|, is finite as well and we call E a finite graph.

For a graph E = (E^{0}, E^{1}, r, s), a vertex v for which s^{−1}(v) is empty is called a sink, while a vertex w for which r^{−1}(w) is empty is called a source. An edge with the same source and range is called a loop. A path μ in a graph E is a sequence of edges μ = μ_{1} … μ_{k}, such that r(μ_{i}) = s(μ_{i+1}), 1 ≤ i ≤ k − 1. In this case, s(μ) := s(μ_{1}) is the source of μ, r(μ) := r(μ_{k}) is the range of μ, and k is the length of μ which is denoted by |μ|. We consider a vertex v ∈ E^{0} as a trivial path of length zero with s(v) = r(v) = v. If μ is a nontrivial path in E, and if v = s(μ) = r(μ), then μ is called a closed path based at v. If μ = μ_{1} … μ_{k} is a closed path based at v = s(μ) and s(μ_{i}) ≠ s(μ_{j}) for every i ≠ j, then μ is called a cycle.

For two vertices v and w, the existence of a path with the source v and the range w is denoted by v ≥ w. Here we allow paths of length zero. By v ≥_{n} w, we mean there is a path of length n connecting these vertices. Therefore v ≥_{0} v represents the vertex v. Also, by v > w, we mean a path from v to w where v ≠ w. In this note, by v ≥ w′ ≥ w, it is understood that there is a path connecting v to w and going through w′ (i.e., w′ is on the path connecting v to w). For n ≥ 2, we define E^{n} to be the set of paths of length n and E^{∗} = ⋃_{n≥0} E^{n}, the set of all paths.

For a graph E, let n_{v,w} be the number of edges with the source v and range w. Then the adjacency matrix of the graph E is A_{E} = (n_{v,w}). Usually one orders the vertices and then writes A_{E} based on this ordering. Two different ordering of vertices give different adjacency matrices. However if A_{E} and A′_{E} are two adjacency matrices of E, then there is a permutation matrix P such that A′_{E} = P A_{E}P^{−1}.

A graph E is called essential if E does not have sinks and sources. Furthermore, a graph is called irreducible if for every ordered pair vertices v and w, there is a path from v to w.

A path algebra, with coefficients in the field K, is constructed as follows: consider a K-vector space with finite paths as the basis and define the multiplication by concatenation of paths. A path algebra has a natural graded structure by assigning paths as homogeneous elements of degree equal to their lengths. A formal definition of path algebras with coefficients in a ring R is given below.

Definition 1.6.9. Path algebras. For a graph E and a ring R with identity, we define the path algebra of E, denoted by 𝒫_{R}(E), to be the algebra generated by the sets {v | v ∈ E^{0}}, {α | α ∈ E^{1}} with the coefficients in R, subject to the relations

1. v_{i}v_{j} = δ_{ij}v_{i} for every v_{i}, v_{j} ∈ E^{0}.

2. s(α)α = αr(α) = α for all α ∈ E^{1}.

Here the ring R commutes with the generators {v, α | v ∈ E^{0}, α ∈ E^{1}}. When the coefficient ring R is clear from the context, we simply write 𝒫(E) instead of 𝒫_{R}(E). When R is not commutative, then we consider 𝒫_{R}(E) as a left R-module. Using the above two relations, it is easy to see that when the number of vertices is finite, then 𝒫_{R}(E) is a ring with identity ∑_{v∈E^{0}} v.

When the graph has one vertex and n-loops, the path algebra associated to this graph is isomorphic to R⟨x_{1}, …, x_{n}⟩, i.e., a free associative unital algebra over R with n non-commuting variables.

Setting deg(v) = 0 for v ∈ E^{0} and deg(α) = 1 for α ∈ E^{1}, we obtain a natural ℤ-grading on the free R-ring generated by {v, α | v ∈ E^{0}, α ∈ E^{1}} (§1.6.1). Since the relations in Definition 1.6.9 are all homogeneous, the ideal generated by these relations is homogeneous and thus we have a natural ℤ-grading on 𝒫_{R}(E). Note that 𝒫(E) is positively graded, and for any m, n ∈ ℕ,

𝒫(E)_{m} 𝒫(E)_{n} = 𝒫(E)_{m+n}.

However by Proposition 1.1.16(2), 𝒫(E) is not an strongly ℤ-graded ring.

The theory of Leavitt path algebras were introduced in [2, 6] which associate to directed graphs certain type of algebras. These algebras were motivated by Leavitt’s construction of universal non-IBN rings [63]. Leavitt path algebras are quotients of path algebras by relations resembling those in construction of algebras studied by Leavitt (see Example 1.3.13).

Definition 1.6.10. Leavitt path algebras. For a row-finite graph E and a ring R with identity, we define the Leavitt path algebra of E, denoted by ℒ_{R}(E), to be the algebra generated by the sets {v | v ∈ E^{0}}, {α | α ∈ E^{1}} and {α^{∗} | α ∈ E^{1}} with the coefficients in R, subject to the relations

1. v_{i}v_{j} = δ_{ij}v_{i} for every v_{i}, v_{j} ∈ E^{0}.

2. s(α)α = αr(α) = α and r(α)α^{∗} = α^{∗}s(α) = α^{∗} for all α ∈ E^{1}.

3. α^{∗}α′ = δ_{αα′}r(α), for all α, α′ ∈ E^{1}.

4. ∑_{{α∈E^{1},s(α)=v}} αα^{∗} = v for every v ∈ E^{0} for which s^{−1}(v) is non-empty.

Here the ring R commutes with the generators {v, α, α^{∗} | v ∈ E^{0}, α ∈ E^{1}}. When the coefficient ring R is clear from the context, we simply write ℒ(E) instead of ℒ_{R}(E). When R is not commutative, then we consider ℒ_{R}(E) as a left R-module. The elements α^{∗} for α ∈ E^{1} are called ghost edges. One can show that ℒ_{R}(E) is a ring with identity if and only if the graph E is finite (otherwise, ℒ_{R}(E) is a ring with local identities).

Setting deg(v) = 0, for v ∈ E^{0}, deg(α) = 1 and deg(α^{∗}) = −1 for α ∈ E^{1}, we obtain a natural ℤ-grading on the free R-ring generated by {v, α, α^{∗} | v ∈ E^{0}, α ∈ E^{1}}. Since the relations in Definition 1.6.10 are all homogeneous, the ideal generated by these relations is homogeneous and thus we have a natural ℤ-grading on ℒ_{R}(E).

If μ = μ_{1} … μ_{k}, where μ_{i} ∈ E^{1}, is an element of ℒ(E), then we denote by μ^{∗} the element μ_{k}^{∗} … μ_{1}^{∗} ∈ ℒ(E). Further we define v^{∗} = v for any v ∈ E^{0}. Since α^{∗}α′ = δ_{αα′}r(α), for all α, α′ ∈ E^{1}, any word in the generators {v, α, α^{∗} | v ∈ E^{0}, α ∈ E^{1}} in ℒ(E) can be written as μγ^{∗} where μ and γ are paths in E (recall that vertices were considered paths of length zero). The elements of the form μγ^{∗} are called monomials.

If the graph E is infinite, ℒ_{R}(E) is a graded ring without identity (see Remark 1.1.15).

Taking the grading into account, one can write ℒ_{R}(E) = _{⨁k∈ℤ} ℒ_{R}(E)_{k} where,

ℒ_{R}(E)_{k} = { ∑_{i} r_{i}α_{i}β_{i}^{∗} | α_{i}, β_{i} are paths, r_{i} ∈ R, and |α_{i}| − |β_{i}| = k for all i}.

For simplicity we denote ℒ_{R}(E)_{k}, the homogeneous elements of degree k, by ℒ_{k}. The following theorem was proved in [46] which determines the finite graphs so that their associated Leavitt path algebras are strongly graded.

Theorem 1.6.11. Let E be a finite graph. Then L(E) is strongly graded if and only if E does not have sinks.

The proof of this theorem is quite long and does not fit for purpose of this note. However, we can realise the Leavitt path algebras of finite graphs with no source in terms of corner skew Laurent polynomial rings (see §1.6.2). This way we can provide a short proof for the above theorem when the graph has no sources.

Let E be a finite graph with no source and E^{0} = {v_{1}, …, v_{n}} be the set of all vertices of E. For each 1 ≤ i ≤ n, we choose an edge e_{i} such that r(e_{i}) = v_{i} and consider t_{+} = e_{1} + ⋯ + e_{n} ∈ ℒ(E)_{1}. Then t_{−} = e_{1}^{∗} + ⋯ + e_{n}^{∗} is its right inverse. Thus by Theorem 1.6.8, ℒ(E) = ℒ(E)_{0}[t_{+}, t_{−}, ϕ], where ϕ : ℒ(E)_{0} → t_{+}t_{−} ℒ(E)_{0}t_{+}t_{−}, ϕ(a) = t_{+}at_{−}.

Using this interpretation of Leavitt path algebras we are able to prove the following theorem.

Theorem 1.6.12. Let E be a finite graph with no source. Then L(E) is strongly graded if and only if E does not have sinks.

Proof. Write ℒ(E) = ℒ(E)_{0}[t_{+}, t_{−}, ϕ], where ϕ(1) = t_{+}t_{−}. The theorem now follows from an easy to prove observation that t_{+}t_{−} is a full idempotent if and only if E does not have sinks along with Proposition 1.6.6, that ϕ(1) is a full idempotent if and only if ℒ(E)_{0}[t_{+}, t_{−}, ϕ] is strongly graded.

As a consequence of Theorem 1.6.11, we can show that Leavitt path algebras associated to finite graphs with no sinks are graded regular von Neumann rings (§1.1.6).

Corollary 1.6.13. Let E be a finite graph with no sinks. Then ℒ(E) is a graded von Neumann regular ring.

Proof. Since ℒ(E) is strongly graded (Theorem 1.6.11), by Corollary 1.5.9, ℒ(E) is von Neumann regular if ℒ(E)_{0} is a von Neumann regular ring. But we know that the zero component ring ℒ(E)_{0} is an ultramatricial algebra which is von Neumann regular (see the proof of [6, Theorem 5.3]). This finishes the proof.

Example 1.6.14. \underline{Leavitt path algebras are not graded unit regular rings}

Recall that a graded ring is graded von Neumann unit regular (or graded unit regular for short) if for any homogeneous element x, there is an invertible homogeneous element y such that xyx = x. Clearly any graded unit regular ring is von Neumann regular. However the converse is not the case. For example, Leavitt path algebras are not in general unit regular as the following example shows. Consider the graph:

[figure: E :; •; y_{1}; y_{2}]

Then it is easy to see that there is no homogeneous invertible element x such that y_{1}xy_{1} = y_{1} in ℒ(E).

The following theorem determines the graded structure of Leavitt path algebras associated to acyclic graphs.

Theorem 1.6.15. Let K be a field and E a finite acyclic graph with sinks {v_{1}, …, v_{t}}. For any sink v_{s}, let R(v_{s}) = {p_{1}^{v_{s}}, …, p_{n}^{v_{s}}} denote the set of all paths ending at v_{s}. Then there is a ℤ-graded isomorphism

L_{K}(E) ≅_{gr} ⨁_{s=1}^{t} M_{n(v_{s})}(K)(|p_{1}^{v_{s}}|, …, |p_{n(v_{s})}^{v_{s}}|).  (1.64)

Sketch of proof. Fix a sink v_{s} and denote R(v_{s}) = {p_{1}, …, p_{n}}. The set

I_{v_{s}} = { ∑ kp_{i}p_{j}^{∗}|k ∈ K, p_{i}, p_{j} ∈ R(v_{s})}

is an ideal of L_{K}(E), and we have an isomorphism ϕ : I_{v_{s}} → M_{n(v_{s})}(K), kp_{i}p_{j}^{∗} ↦ k(e_{ij}), where k ∈ K, p_{i}, p_{j} ∈ R(v_{s}) and e_{ij} is the standard matrix unit. Now, considering the grading on M_{n(v_{s})}(K)(|p_{1}^{v_{s}}|, …, |p_{n(v_{s})}^{v_{s}}|), we show that ϕ is a graded isomorphism. Let p_{i}p_{j}^{∗} ∈ I_{v_{s}}. Then

deg(p_{i}p_{j}^{∗}) = |p_{i}| − |p_{j}| = deg(e_{ij}) = deg(ϕ(p_{i}p_{j}^{∗})).

So ϕ respects the grading. Hence ϕ is a graded isomorphism. One can check that

L_{K}(E) = ⨁_{s=1}^{t} I_{v_{s}} ≅_{gr} ⨁_{s=1}^{t} M_{n(v_{s})}(K)(|p_{1}^{v_{s}}|, …, |p_{n(v_{s})}^{v_{s}}|).

Example 1.6.16. Theorem 1.6.15 shows that the Leavitt path algebras of the graphs E_{1} and E_{2} below with coefficients from the field K are graded isomorphic to 𝕄_{5}(K)(0, 1, 1, 2, 2) and thus ℒ(E_{1}) ≅_{gr} ℒ(E_{2}). However ℒ(E_{3}) ≅_{gr} 𝕄_{5}(K)(0, 1, 2, 2, 3).

\xymatrix{ &  & • \ar[d] \\  &  & • \ar[d] &  & • \ar[d] & • \ar[d] &  & • \ar[d] & • \ar[d] \\ E_{1} : • \ar[r] & • \ar[r] & • & E_{2} : • \ar[r] & • \ar[r] & • & E_{3} : & • \ar[r] & • \ar[r] & •}

Theorem 1.6.17. Let K be a field and E a C_{n}-comet with the cycle C of length n ≥ 1. Let v be a vertex on the cycle C with e be the edge in the cycle with s(e) = v. Eliminate the edge e and consider the set {p_{i}|1 ≤ i ≤ m} of all paths with end in v. Then

L_{K}(E) ≅_{gr} 𝕄_{m}(K[x^{n}, x^{−n}])(|p_{1}|, …, |p_{m}|).  (1.65)

Sketch of proof. One can show that the set of monomials {p_{i}C^{k}p_{j}^{∗}|1 ≤ i, j ≤ n, k ∈ ℤ} are a basis of L_{K}(E) as a K-vector space. Define a map

ϕ : L_{K}(E) → 𝕄_{m}(K[x^{n}, x^{−n}])(|p_{1}|, …, |p_{m}|), by ϕ(p_{i}C^{k}p_{j}^{∗}) = e_{ij}(x^{kn}),

where e_{ij}(x^{kn}) is a matrix with x^{kn} in the ij-position and zero elsewhere. Extend this linearly to L_{K}(E). We have

ϕ((p_{i}C^{k}p_{j}^{∗})(p_{r}C^{t}p_{s}^{∗})) = ϕ(δ_{jr}p_{i}C^{k+t}p_{s}^{∗})
= δ_{jr}e_{is}x^{(k+t)n}
= (e_{ij}x^{kn})(e_{rs}x^{tn})
= ϕ(p_{i}C^{k}p_{j}^{∗})ϕ(p_{r}C^{t}p_{s}^{∗}).

Thus ϕ is a homomorphism. Also, ϕ sends the basis to the basis, so ϕ is an isomorphism. We now need to show that ϕ is graded. We have

deg(p_{i}C^{k}p_{j}^{∗}) = |p_{i}C^{k}p_{j}^{∗}| = nk + |p_{i}| − |p_{j}|

and

deg(ϕ(p_{i}C^{k}p_{j}^{∗})) = deg(e_{ij}(x^{kn})) = nk + |p_{i}| − |p_{j}|.

Therefore ϕ respects the grading. This finishes the proof.

Example 1.6.18. By Theorem 1.6.11, the Leavitt path algebra of the following graph with coefficients in a field K,

\xymatrix{ & • \ar[dr] \\ E : &  & • \ar[r] & • \ar[l] \\  & • \ar[ur]}

is strongly graded. By Theorem 1.6.17,

ℒ_{K}(E) ≅_{gr} 𝕄_{4}(K[x^{2}, x^{−2}])(0, 1, 1, 1).  (1.66)

However this algebra is not a crossed-product. Set B = K[x, x^{−1}] with the grading B = _{⨁n∈ℤ}Kx^{n} and consider A = K[x^{2}, x^{−2}] as a graded subring of B with A_{n} = Kx^{n} if n ≡ 0 (mod 2), and A_{n} = 0 otherwise. Using the graded isomorphism of (1.66), by (1.34) a homogeneous element of degree 1 in ℒ_{K}(E) has the form

(\begin{matrix}A_{1} & A_{2} & A_{2} & A_{2} \\ A_{0} & A_{1} & A_{1} & A_{1} \\ A_{0} & A_{1} & A_{1} & A_{1} \\ A_{0} & A_{1} & A_{1} & A_{1} \end{matrix}) .

Since A_{1} = 0, the determinants of these matrices are zero, and thus no homogeneous element of degree 1 is invertible. Thus ℒ_{K}(E) is not crossed-product (see §1.1.2).

Now consider the following graph,

\xymatrix{E : & • \ar[r]^{f} & • \ar[r]^{g} & • \ar[l]^{h} & • \ar[l]^{e}}

By Theorem 1.6.17,

ℒ_{K}(E) ≅_{gr} 𝕄_{4}(K[x^{2}, x^{−2}])(0, 1, 1, 2).  (1.67)

Using the graded isomorphism of (1.67), by (1.36) homogeneous elements of degree 0 in ℒ_{K}(E) have the form

ℒ_{K}(E)_{0} = (\begin{matrix}A_{0} & A_{1} & A_{1} & A_{2} \\ A_{−1} & A_{0} & A_{0} & A_{1} \\ A_{−1} & A_{0} & A_{0} & A_{1} \\ A_{−2} & A_{−1} & A_{−1} & A_{0} \end{matrix} ) = (\begin{matrix}K & 0 & 0 & Kx^{2} \\ 0 & K & K & 0 \\ 0 & K & K & 0 \\ Kx^{−2} & 0 & 0 & K \end{matrix}) .

In the same manner, homogeneous elements of degree one have the form,

ℒ_{K}(E)_{1} = (\begin{matrix}0 & Kx^{2} & Kx^{2} & 0 \\ K & 0 & 0 & Kx^{2} \\ K & 0 & 0 & Kx^{2} \\ 0 & K & K & 0 \end{matrix}) .

Choose

u = (\begin{matrix}0 & 0 & x^{2} & 0 \\ 0 & 0 & 0 & x^{2} \\ 1 & 0 & 0 & x^{2} \\ 0 & 1 & 0 & 0 \end{matrix}) ∈ ℒ(E)_{1}

and observe that u is invertible (this matrix corresponds to the element g + h + fge^{∗} + ehf^{∗} ∈ ℒ_{K}(E)_{1}).

Thus ℒ_{K}(E) is crossed-product (and therefore a skew group ring as the grading is cyclic), i.e.,

ℒ_{K}(E) ≅_{gr} ⨁_{i∈ℤ} ℒ_{K}(E)_{0}u^{i}

and a simple calculation shows that one can describe this algebra as follows:

ℒ_{K}(E)_{0} ≅ 𝕄_{2}(K) × 𝕄_{2}(K),

and

ℒ_{K}(E) ≅_{gr} ( 𝕄_{2}(K) × 𝕄_{2}(K)) ⋆_{τ} ℤ,  (1.68)

where,

τ((\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}) , (\begin{matrix}b_{11} & b_{12} \\ b_{21} & b_{22} \end{matrix})) = ((\begin{matrix}b_{22} & b_{21} \\ b_{12} & b_{11} \end{matrix}) , (\begin{matrix}a_{22} & a_{21} \\ a_{12} & a_{11} \end{matrix})).

Remark 1.6.19. For a graph E, the Leavitt path algebra ℒ_{K}(E) has a ℤ-graded structure. This grading was obtained by assigning 0 to vertices, 1 to edges and −1 to ghost edges. However, one can equip ℒ_{K}(E) with other graded structures as well. Let Γ be an arbitrary group with the identity element e. Let w : E^{1} → Γ be a weight map and further define w(α^{∗}) = w(α)^{−1}, for any edge α ∈ E^{1} and w(v) = e for v ∈ E^{0}. The free K-algebra generated by the vertices, edges and ghost edges is a Γ-graded K-algebra (see §1.6.1). Furthermore, the Leavitt path algebra is the quotient of this algebra by relations in Definition 1.6.10 which are all homogeneous. Thus ℒ_{K}(E) is a Γ-graded K-algebra.

As an example, consider the graphs

[figure: E :; •; f; •; e; F :; •; g; •; h]

and assign 1 for the degree of f and 2 for the degree of e in E and 1 for the degrees of g and h in F. Then the proof of Theorem 1.6.17 shows that ℒ_{K}(E) ≅ 𝕄_{2}(K[x^{2}, x^{−2}])(0, 1) and ℒ_{K}(F) ≅ 𝕄_{2}(K[x^{2}, x^{−2}])(0, 1). So with these gradings, ℒ_{K}(E) ≅_{gr} ℒ_{K}(F).

1.7 The graded IBN and graded type

A ring A with identity has invariant basis number (IBN) or invariant basis property if any two bases of a free (right) A-module have the same cardinality, i.e., if A^{n} ≅ A^{m} as A-modules, then n = m. When A does not have IBN, the type of A is defined as a pair of positive integers (n, k) such that A^{n} ≅ A^{n+k} as A-modules and these are the smallest number with this property, that is, (n, k) is the minimum under the usual lexicographic order. This means any two bases of a free A-module have the unique cardinality if one of the basis has the cardinality less than n and further if a free module has rank n, then a free module with the smallest cardinality (other than n) isomorphic to this module is of rank n + k. Another way to describe a type (n, k) is that, A^{n} ≅ A^{n+k} is the first repetition in the list A, A^{2}, A^{3}, . . ..

It was shown that if A has type (n, k), then A^{m} ≅ A^{m′} if and only if m = m′ or m, m′ ≥ n and m ≡ m′ (mod k) (see [26, p. 225], [63, Theorem 1]).

One can show that a (right) noetherian ring has IBN. Furthermore, if there is a ring homomorphism A → B, (which preserves 1), and B has IBN then A has IBN as well. Indeed, if A^{m} ≅ A^{n}, then B^{m} ≅ A^{m} ⊗_{A} B ≅ A^{n} ⊗_{A} B ≅ B^{n}, so n = m. One can describe the type of a ring by using the monoid of isomorphism classes of finitely generated projective modules (see Example 3.1.4). For a nice discussion about these rings see [20, 26, 66].

A graded ring A has a graded invariant basis number (gr-IBN) if any two homogeneous bases of a graded free (right) A-module have the same cardinality, i.e., if A^{m}(\overline{α}) ≅_{gr} A^{n}(\overline{δ}), where \overline{α} = (α_{1}, …, α_{m}) and \overline{δ} = (δ_{1}, …, δ_{n}), then m = n. Note that contrary to the non-graded case, this does not imply that two graded free modules with bases of the same cardinality are graded isomorphic (see Proposition 1.3.10). A graded ring A has IBN in gr-A, if A^{m} ≅_{gr} A^{n} then m = n. If A has IBN in gr- A, then A_{0} has IBN. Indeed, if A_{0}^{m} ≅ A_{0}^{n} as A_{0}-modules, then A^{m} ≅_{gr} A_{0}^{m} ⊗_{A_{0}} A ≅ A_{0}^{n} ⊗_{A_{0}} A ≅_{gr} A^{n}, so n = m (see [73, p. 215]).

When the graded ring A does not have gr-IBN, the graded type of A is defined as a pair of positive integers (n, k) such that A^{n}(\overline{δ}) ≅_{gr} A^{n+k}(\overline{α}) as A-modules, for some \overline{δ} = (δ_{1}, …, δ_{n}) and \overline{α} = (α_{1}, …, α_{n+k}) and these are the smallest number with this property. In Proposition 1.7.1 we show that the Leavitt algebra ℒ(n, k + 1) (see Example 1.6.5) has graded type (n, k).

Parallel to the non-graded setting, one can show that a graded (right) noetherian ring has gr-IBN. Furthermore, if there is a graded ring homomorphism A → B, (which preserves 1), and B has gr-IBN then A has gr-IBN as well. Indeed, if A^{m}(\overline{α}) ≅_{gr} A^{n}(\overline{δ}), where \overline{α} = (α_{1}, …, α_{m}) and \overline{δ} = (δ_{1}, …, δ_{n}), then

B^{m}(\overline{α}) ≅_{gr} A^{m}(\overline{α}) ⊗_{A} B ≅ A^{n}(\overline{δ}) ⊗_{A} B ≅_{gr} B^{n}(\overline{δ}),

which implies n = m. Using this, one can show that any graded commutative ring has gr-IBN. For, there exists a graded maximal ideal and its quotient ring is a graded field which has gr-IBN (see §1.1.5 and Proposition 1.4.2).

Let A be a Γ-graded ring such that A^{m}(\overline{α}) ≅_{gr} A^{n}(\overline{δ}), where \overline{α} = (α_{1}, …, α_{m}) and \overline{δ} = (δ_{1}, …, δ_{n}). Then there is a universal Γ-graded ring R such that R^{m}(\overline{α}) ≅_{gr} R^{n}(\overline{δ}), and a graded ring homomorphism R → A which induces the graded isomorphism A^{m}(\overline{α}) ≅_{gr} R^{m}(\overline{α}) ⊗_{R} A ≅_{gr} R^{n}(\overline{δ})⊗_{R}A ≅_{gr} A^{n}(\overline{δ}). Indeed, by Proposition 1.3.10, there are matrices a = (a_{ij}) ∈ 𝕄_{n×m}(A)[\overline{δ}][\overline{α}] and b = (b_{ij}) ∈ 𝕄_{m×n}(A)[\overline{α}][\overline{δ}] such that ab = 𝕀_{n} and ba = 𝕀_{m}. The free ring generated by symbols in place of a_{ij} and b_{ij} subject to relations imposed by ab = 𝕀_{n} and ba = 𝕀_{m} is the desired universal graded ring. In detail, let F be a free ring generated by x_{ij}, 1 ≤ i ≤ n, 1 ≤ j ≤ m and y_{ij}, 1 ≤ i ≤ m, 1 ≤ j ≤ n. Assign the degrees deg(x_{ij}) = δ_{i}−α_{j} and deg(y_{ij}) = α_{i}−δ_{j} (see §1.6.1). This makes F a Γ-graded ring. Let R be a ring F modulo the relations ∑_{s=1}^{m} x_{is}y_{sk} = δ_{ik}, 1 ≤ i, k ≤ n and ∑_{t=1}^{n} y_{it}x_{tk} = δ_{ik}, 1 ≤ i, k ≤ m, where δ_{ik} is the Kronecker delta. Since all the relations are homogeneous, R is a Γ-graded ring. Clearly the map sending x_{ij} to a_{ij} and y_{ij} to b_{ij} induces a graded ring homomorphism R → A. Again Proposition 1.3.10 shows that R^{m}(\overline{α}) ≅_{gr} R^{n}(\overline{δ}).

Proposition 1.7.1. Let R = ℒ(n, k + 1) be the Leavitt algebra of type (n, k). Then

(1) R is a universal ⨁_{n} ℤ-graded ring which does not have gr-IBN;

(2) R has graded type (n, k);

(3) For n = 1, R has IBN in gr- R.

Proof. (1) Consider the Leavitt algebra ℒ(n, k + 1) constructed in Example 1.6.5, which is a ⨁_{n} ℤgraded ring and is universal. Furthermore (1.61) combined by Proposition 1.3.10(3) shows that R^{n} ≅_{gr} R^{n+k}(\overline{α}). Here \overline{α} = (α_{1}, …, α_{n+k}), where α_{i} = (0, …, 0, 1, 0 …, 0) and 1 is in i-th entry. This shows that R = ℒ(n, k + 1) does not have gr-IBN.

(2) By [26, Theorem 6.1], R is of type (n, k). This immediately implies the graded type of R is also (n, k).

(3) Suppose R^{n} ≅_{gr} R^{m} as graded R-modules. Then R_{0}^{n} ≅ R_{0}^{m} as R_{0}-modules. But R_{0} is an ultramatricial algebra, i.e., direct limit of matrices over a field. Since IBN respects direct limits ([26, Theorem 2.3]), R_{0} has IBN. Therefore, n = m.

Remark 1.7.2. Assignment of deg(y_{ij}) = 1 and deg(x_{ij}) = −1, for all i, j, makes R = ℒ(n, k + 1) a ℤ-graded algebra of graded type (n, k) with R^{n} ≅_{gr} R^{n+k}(1).

Remark 1.7.3. Let A be a Γ-graded ring. In [75, Proposition 4.4], it was shown that if Γ is finite then A has gr-IBN if and only if A has IBN.

1.8 The graded stable rank

The notion of stable rank was defined by H. Bass [12] in order to study K_{1}-group of rings with finite Krull dimension. For a concise introduction of the stable rank, we refer the reader to [60, 61], and for its applications to K-theory to [11, 12]. It seems the natural notion of the graded stable rank in the context of graded ring theory, has not yet been investigated in the literature. In this section we define the graded stable rank and study the important case of graded rings with graded stable rank 1. This will be used later in §3 in relation with the graded Grothendieck groups.

A row (a_{1}, …, a_{n}) of homogeneous elements of a Γ-graded ring A is called a graded left unimodular row if the graded ideal generated by a_{i}, 1 ≤ i ≤ n, is A. When n ≥ 2, a graded left unimodular row is called stable if there exists homogeneous elements r_{1}, …, r_{n−1} of A such that the graded left ideal generated by homogeneous elements a_{i} + r_{i}a_{n}, 1 ≤ i ≤ n − 1, is A.

The graded left stable rank of a ring A is defined to be n, denoted sr^{gr}(A) = n, if any graded unimodular row of length n + 1 is stable, but there exists a non-stable unimodular row of length n. If such n does not exist (i.e. there are non-stable unimodular rows of arbitrary length) we say that the graded stable rank of A is infinite.

In order that this definition would be well-defined, one needs to show that if any graded unimodular row of fixed length n is stable, so is any unimodular row of a greater length. This can be proved similar to the non-graded case and we omit the proof (see for example [60, Proposition 1.3]).

When the graded group Γ is a trivial group, the above definitions reduce to the standard definitions of unimodular rows and stable ranks.

The case of graded stable rank 1 is of special importance. Suppose A is a Γ-graded with sr^{gr}(A) = 1. Then from the definition it follows that, if a, b ∈ A^{h} such that Aa + Ab = A, then there is a homogeneous element c such that the homogeneous element a+cb is left invertible. When sr^{gr}(A) = 1, any left invertible homogeneous element is in fact invertible. For, suppose c ∈ A^{h} is left invertible, i.e., there is a ∈ A^{h} such that ac = 1. Then the row (a, 1−ca) is graded left unimodular. Thus, there is a s ∈ A^{h} such that u := a + s(1 − ca) is left invertible. But uc = 1. Thus u is (left and right) invertible and consequently, c is an invertible homogeneous element.

The graded stable rank 1 is quite a strong condition. In fact if sr^{gr}(A) = 1 then Γ_{A} = Γ_{A}^{∗}. For, if a ∈ A_{γ} is a non-zero element, then since (a, 1) is unimodular and sr^{gr}(A) = 1, there is b ∈ A^{h}, such that a + b is an invertible homogeneous element, necessarily of degree γ. Thus γ ∈ Γ_{A}^{∗}.

Example 1.8.1. \underline{Graded division rings have graded stable rank 1}

Since any nonzero homogeneous element of a graded division ring is invertible, one can easily show that its graded stable rank is 1. Thus for a field K, sr^{gr}(K[x, x^{−1}]) = 1, whereas sr(K[x, x^{−1}]) = 2.

Example 1.8.2. \underline{For a strongly graded ring A, sr^{gr}(A) ≠ sr(A_{0})}

Let A = ℒ(1, 2) be the Leavitt algebra generated by x_{1}, x_{2}, y_{1}, y_{2} (see Example 1.3.13). Then relations (1.44) show that y_{1} is left invertible but it is not invertible. This shows that sr^{gr}(A) ≠ 1. On the other hand, since A_{0} is an ultramatricial algebra, sr(A_{0}) = 1 (see §3.8.3, [60, Corollary 5.5] and [38]).

We have the following theorem which is a graded version of the Cancellation Theorem with a similar proof (see [61, Theorem 20.11]).

Theorem 1.8.3. Graded Cancellation Theorem Let A be a Γ-graded ring and M, N, P be graded right A-modules, with P being finitely generated. If the graded ring E := End_{A}(P) has graded left stable rank 1, then P ⊕ M ≅_{gr} P ⊕ N as A-modules implies M ≅_{gr} N as A-modules.

Proof. Let h : P ⊕ M → P ⊕ N be a graded A-module isomorphism. Then the composition of the maps

P ⟶^{i_{1}} P ⊕ M ⟶^{h} P ⊕ N ⟶^{π_{1}} P,

M ⟶^{i_{2}} P ⊕ M ⟶^{h} P ⊕ N ⟶^{π_{1}} P

induces a graded split epimorphism of degree zero, denoted by (f, g) : P ⊕ M → P. Here (f, g)(p, m) = f(p) + g(m), where f = π_{1}hi_{1} and g = π_{1}hi_{2}. It is clear that ker(f, g) ≅_{gr} N.

Let (\begin{matrix}f′ \\ g′ \end{matrix}) : P → P ⊕ M be the split homomorphism. Thus

1 = (f, g) (\begin{matrix}f′ \\ g′ \end{matrix}) = ff′ + gg′.

This shows that the left ideal generated by f′ and gg′ is E. Since E has graded stable rank 1, it follows there is e ∈ E of degree 0 such that u := f′ + e(gg′) is an invertible element of E. Writing

u = (1, eg) (\begin{matrix}f′ \\ g′ \end{matrix}) implies that both ker(f, g) and ker(1, eg) are graded isomorphic to

P ⊕ M/ Im (\begin{matrix}f′ \\ g′ \end{matrix}) .

Thus ker(f, g) ≅_{gr} ker(1, eg). But ker(f, g) ≅_{gr} N and ker(1, eg) ≅_{gr} M. Thus M ≅_{gr} N.

The following corollary will be used in §3 to show that for a graded ring with graded stable rank 1, the monoid of graded finitely generated projective modules injects into the graded Grothendieck group (see Corollary 3.1.8).

Corollary 1.8.4. Let A be a Γ-graded ring with graded left stable rank 1 and M, N, P be graded right A-modules. If P is a graded finitely generated projective A-module, then P ⊕ M ≅_{gr} P ⊕ N as A-modules implies M ≅_{gr} N as A-modules.

Proof. Suppose P ⊕ M ≅_{gr} P ⊕ N as A-modules. Since P is a graded finitely generated A-module, there is a graded A-module Q such that P ⊕ Q ≅_{gr} A^{n}(\overline{α}) (see (1.27)). It follows that

A^{n}(\overline{α}) ⊕ M ≅_{gr} A^{n}(\overline{α}) ⊕ N.

We prove that if

A(α) ⊕ M ≅_{gr} A(α) ⊕ N,  (1.69)

then M ≅_{gr} N. The corollary then follows by an easy induction.

By (1.37) there is a graded ring isomorphism End_{A}(A(α)) ≅_{gr} A. Since A has graded stable rank 1, so does End_{A}(A(α)). Now by Theorem 1.8.3, from (1.69) it follows that M ≅_{gr} N. This finishes the proof.

Graded Morita Theory

Starting from a right A-module P, one can construct a 6-tuple, (A, P, P^{∗}, B; ϕ, ψ), where P^{∗} = Hom_{A}(P, A), B = Hom_{A}(P, P), ϕ : P^{∗} ⊗_{B} P → A and ψ : P ⊗_{R} P^{∗} → B, which have appropriate bimodule structures. This is called the Morita context associated with P. When P is a progenerator (i.e., finitely generated projective and a generator), then one can show that Mod- A ≈ Mod- B, i.e, the category of (right) A-modules is equivalent to the category of (right) B-modules. Conversely, if for two rings A and B, Mod- A ≈ Mod- B, then one can construct a A-progenerator P such that B ≅ End_{A}(P). The feature of this theory, called the Morita theory, is that the whole process involves working with Hom and tensor functors, both of which respect the grading. Thus starting from a graded ring A and a graded progenerator P, and carrying out the Morita theory, one can naturally extending the equivalence from Gr-A ≈ Gr-B to Mod- A ≈ Mod- B and vice versa.

Extending the equivalence from the (sub)categories of graded modules to the categories of modules is not at the first glance obvious. Recall that two categories 𝒞 and 𝒟 are equivalent if and only if there is a functor ϕ : 𝒞 → 𝒟 which is fully faithful and for any D ∈ 𝒟, there is C ∈ 𝒞 such that ϕ(C) = D.

Suppose ϕ : Gr-A → Gr-B is a graded equivalent (see Definition 2.3.2). Then for two graded A-modules, M and N, (where M is finitely generated) we have

Hom_{A}(M, N)_{0} = Hom_{Gr-A}(M, N) ≅ Hom_{Gr-B}(ϕ(M), ϕ(N)) = Hom_{B}(ϕ(M), ϕ(N))_{0}.

The fact that, this can be extended to

Hom_{A}(M, N) ≅ Hom_{B}(ϕ(M), ϕ(N)),

is not immediate. We will show that this is indeed the case.

In this chapter we study the equivalences between the categories of graded modules and their relations with the categories of modules. The main theorem of this chapter shows that for two graded rings A and B, if Gr-A ≈_{gr} Gr-B then Mod- A ≈ Mod- B (Theorem 2.3.5). This was first studied by Gordon and Green [40] in the setting of ℤ-graded rings.

Throughout this chapter, A is a Γ-graded ring unless otherwise stated. Furthermore, all functors are additive functors. For the theory of (non-graded) Morita theory we refer the reader to [62] and [4]

2.1 First instance of the graded Morita equivalence

Before describing the general graded Morita equivalence, we treat a special case of matrix algebras, which, in the word of T.Y. Lam, is the “first instance” of equivalence between module categories [62, §17B]. This special case is quite explicit and will help us in calculating the graded Grothendieck group of matrix algebras. Recall from §1.3 that for a Γ-graded ring A, A^{n}(\overline{δ}) = A(δ_{1}) ⊕ ⋯ ⊕ A(δ_{n}) is a graded free A-bimodule. Furthermore, A^{n}(\overline{δ}) is a graded right 𝕄_{n}(A)(\overline{δ})-module and A^{n}(−\overline{δ}) is a graded left 𝕄_{n}(A)(\overline{δ})-module, where −\overline{δ} = (−δ_{1}, …, −δ_{n}).

Proposition 2.1.1. Let A be a Γ-graded ring and let \overline{δ} = (δ_{1}, …, δ_{n}), where δ_{i} ∈ Γ, 1 ≤ i ≤ n. Then the functors

ψ : Gr- 𝕄_{n}(A)(\overline{δ}) ⟶ Gr-A,
P ⟼ P ⊗_{𝕄_{n}(A)(\overline{δ})} A^{n}(−\overline{δ})

and

φ : Gr-A ⟶ Gr- 𝕄_{n}(A)(\overline{δ}),
Q ⟼ Q ⊗_{A} A^{n}(\overline{δ})

form equivalences of categories and commute with suspensions, i.e, ψ𝒯_{α} = 𝒯_{α}ψ, α ∈ Γ.

Proof. One can check that there is a Γ-graded A-bimodule isomorphism

f : A^{n}(\overline{δ}) ⊗_{𝕄_{n}(A)(\overline{δ})} A^{n}(−\overline{δ}) ⟶ A,  (2.1)

(a_{1}, …, a_{n}) ⊗ (b_{1} ; ⋮ ; b_{n}) ⟼ a_{1}b_{1} + ⋯ + a_{n}b_{n}

with

f^{−1} : A ⟶ A^{n}(\overline{δ}) ⊗_{𝕄_{n}(A)(\overline{δ})} A^{n}(−\overline{δ}),

a ⟼ (a, 0, …, 0) ⊗ (1 ⋮ 0) .

Furthermore, there is a Γ-graded 𝕄_{n}(A)(\overline{δ})-bimodule isomorphism

g : A^{n}(−\overline{δ}) ⊗_{A} A^{n}(\overline{δ}) ⟶ 𝕄_{n}(A)(\overline{δ}),  (2.2)

(a_{1} ; ⋮ ; a_{n}) ⊗ (b_{1}, …, b_{n}) ⟼ (\begin{matrix}a_{1}b_{1} & ⋯ & a_{1}b_{n} \\ ⋮ & & ⋮ \\ a_{n}b_{1} & ⋯ & a_{n}b_{n} \end{matrix})

with

g^{−1} : 𝕄_{n}(A)(\overline{δ}) ⟶ A^{n}(−\overline{δ}) ⊗_{A} A^{n}(−\overline{δ}),

(a_{i,j}) ⟼ (a_{1,1} ; a_{2,1} ; ⋮ ; a_{n,1}) ⊗ (1, 0, …, 0) + ⋯ + ( a_{1,n} ; ⋮ ; a_{n−1,n} ; a_{n,n}) ⊗ (1, 0, …, 0).

Now using (2.1) and (2.2), it easily follows that φψ and ψφ are equivalent to identity functors. The general fact that for α ∈ Γ,

(P ⊗ Q)(α) = P (α) ⊗ Q = P ⊗ Q(α),

(see §1.2.6) shows that the suspension functor commutes with ψ and ϕ.

Since the functors ϕ and ψ, being tensor functors, preserve the projectivity, and send finitely generated modules to finitely generated modules, we immediately get the following corollary which we will use it to compute graded K_{0} of matrix algebras.

Corollary 2.1.2. Let A be a Γ-graded ring and let \overline{δ} = (δ_{1}, …, δ_{n}), where δ_{i} ∈ Γ, 1 ≤ i ≤ n. Then the functors

ψ : Pgr- 𝕄_{n}(A)(\overline{δ}) ⟶ Pgr-A,
P ⟼ P ⊗_{𝕄_{n}(A)(\overline{δ})} A^{n}(−\overline{δ})

and

φ : Pgr-A ⟶ Pgr- 𝕄_{n}(A)(\overline{δ}),
Q ⟼ Q ⊗_{A} A^{n}(\overline{δ})

form equivalences of categories and commute with suspensions, i.e, ψ𝒯_{α} = 𝒯_{α}ψ, α ∈ Γ.

Example 2.1.3. \underline{Strongly graded is not a Morita invariant property}

Let K be a field and consider the ℤ-graded ring K[x^{2}, x^{−2}] which has the support 2ℤ. By Proposition 2.1.1, A = 𝕄_{2}(K[x^{2}, x^{−2}])(0, 1) is graded Morita equivalent to B = K[x^{2}, x^{−2}]. However, one can easily see that A is strongly graded whereas B is not.

One can also observe that although A and B are graded Morita equivalent, the support set of A is ℤ, whereas the support of B is 2ℤ (see (1.38)).

Remark 2.1.4. Let A be a Γ-graded ring and let \overline{δ} = (δ_{1}, …, δ_{n}), where δ_{i} ∈ Γ, 1 ≤ i ≤ n. Suppose Ω is a subgroup of Γ such that Γ_{A} ⊆ Ω and \{δ_{1}, …, δ_{n}\} ⊆ Ω. Since the support of 𝕄_{n}(A)(\overline{δ}) is a subset of Ω (see (1.38)), we can naturally consider A and 𝕄_{n}(A)(\overline{δ}) as Ω-graded rings (see Remark 1.2.4). Since the Γ-isomorphisms (2.1) and (2.2) in the proof of Proposition 2.1.1 can be considered as Ω-isomorphism, we can easily see that Gr^{Ω}- 𝕄_{n}(A)(\overline{δ}) is equivalent to Gr^{Ω}-A as well. The converse of this statement is valid for the general graded Morita Theory, see Remark 2.3.9.

2.2 Graded generators

We start with a categorical definition of graded generators. In this section, as usual, the modules are (graded) right A-modules.

Definition 2.2.1. Let A be a Γ-graded ring. A graded A-module P is called a graded generator if f : M → N is a nonzero graded A-module homomorphism, then there exists α ∈ Γ and a graded homomorphism g : P (α) → M such that fg : P (α) → M → N is not zero map.

Let A be a Γ-graded ring and P be a graded right A-module. Define the graded trace ideal of P as follows

Tr^{gr}(P) := { ∑_{i} f_{i}(p_{i}) | f_{i} ∈ Hom(P, A)_{α}, α ∈ Γ, p_{i} ∈ P^{h}}.

One can check that Tr^{gr}(P) is a graded two sided ideal of A. The following theorem provides a set theoretic way to define a graded generator.

Theorem 2.2.2. For any graded A-module P, the following are equivalent:

(1) P is a graded generator;

(2) Tr^{gr}(P) = A;

(3) A is a direct summand of a finite direct sum ⨁_{i} P (α_{i}), where α_{i} ∈ Γ;

(4) A is a direct summand of a direct sum ⨁_{i} P (α_{i}), where α_{i} ∈ Γ;

(5) Every graded A-module M is a homomorphic image of ⨁_{i} P (α_{i}), where α_{i} ∈ Γ.

Proof. (1) ⇒ (2) First note that

Tr^{gr}(P) = { ∑_{i} f_{i}(P (α)) | f_{i} ∈ Hom_{A -Gr} (P (α), A), α ∈ Γ}.

Suppose that Tr^{gr}(P) ≠ A. Then the graded canonical projection f : A → A/ Tr^{gr}(P) is not a zero map. Since P is graded generator, there is g ∈ Hom_{A -Gr}(P (α), A) such that fg is not zero. But this implies g(P (α)) ⊈ Tr^{gr}(P) which is a contradiction.

(2) ⇒ (3) Since Tr^{gr}(P) = A, one can find g_{i} ∈ Hom_{A -Gr}(P (α_{i}), A), α_{i} ∈ Γ, 1 ≤ i ≤ n such that ∑_{i=1}^{n} g_{i}(P (α_{i})) = A. Consider the graded A-module epimorphism ⨁_{i=1}^{n} P (α_{i}) → A, (p_{1}, …, p_{n}) ↦ ∑_{i=1}^{n} g_{i}(p_{i}). Since A is graded projective, this map splits. Thus ⨁_{i=1}^{n} P (α_{i}) ≅_{gr} A ⊕ Q, for some graded A-module Q. This gives (3).

(3) ⇒ (4) This is immediate.

(4) ⇒ (5) Since any module is a homomorphic image of a graded free A-module, there is a graded epimorphism ⨁_{j} A(α′_{j}) → M. By (3) there is a graded epimorphism ⨁_{i} P (α_{i}) → A, so an epimorphism, ⨁_{i} P (α_{i} + α′_{j}) → A(α′_{j}). Thus ⨁_{j} ⨁_{i} P (α_{i} + α′_{j}) → ⊕_{j}A(α′_{j}) → M is a graded epimorphism.

(5) ⇒ (1) Let f : M → N be a nonzero graded homomorphism. By (4) there is an epimorphism ⨁_{i} P (α_{i}) → M. So the composition ⨁_{i} P (α_{i}) → M → N is not zero. This immediately implies there is an i such that the composition P (α_{i}) → M → N is not zero. This gives (1).

Example 2.2.3. Let A be a graded simple ring and P be a graded projective A-module. Since Tr^{gr}(P) is a graded two sided ideal, one can easily show, using Proposition 2.2.2(2), that P is a graded generator.

Recall that in the category of A-right modules, Mod- A, a generator is defined as in Definition 2.2.1 by dropping all the graded adjective. Furthermore a similar theorem as in 2.2.2 can be written in the non-graded case, by considering Γ to be the trivial group (see [62, Theorem 18.8]). In particular P is a generator if and only if Tr(P) = A, where

Tr(P) := { ∑_{i} f_{i}(p_{i}) | f_{i} ∈ Hom_{A}(P, A), p_{i} ∈ P }.

Remark 2.2.4. Considering Gr-A as a subcategory of Mod- A (which is clearly not a full subcategory), one can define generators for Gr-A. In this case one can easily see that ⨁_{γ∈Γ} A(γ) is a generator for the category Gr-A. Recall that, in comparison, A is a generator for Mod- A and A is a graded generator for Gr-A.

An A-module P is called a progenerator if it is both a finitely generated projective and a generator, i.e., there is an n ∈ ℕ such that A^{n} ≅ P ⊕ K and P^{n} ≅ A ⊕ L, where K and L are A-modules. Similarly, a graded A-module is called a graded progenerator if it is both a graded finitely generated projective and a graded generator.

Reminiscent of the case of graded projective modules (Proposition 1.2.12), we have the following relation between generators and graded generators.

Theorem 2.2.5. Let P be a finitely generated A-module. Then P is a graded generator if and only if P is graded and is a generator.

Proof. Suppose P is a graded generator. By Theorem 2.2.2, Tr^{gr}(P) = A which implies that there are graded homomorphisms f_{i} (of possibly different degrees) in Hom_{A -Gr}(P (α_{i}), A) and p_{i} ∈ P^{h}, such that ∑_{i} f_{i}(p_{i}) = 1. This immediately implies Tr(P), being an ideal, is A. Thus P is a generator.

Conversely, suppose P is graded and a generator. Thus there are homomorphisms f_{i} in Hom(P, A) and p_{i} ∈ P such that ∑_{i} f_{i}(p_{i}) = 1. Since P is finitely generated, by Theorem 1.2.5, f_{i} can be written as a sum of graded homomorphisms, and p_{i} as sum of homogeneous elements in P. This shows 1 ∈ Tr^{gr}(P). Since Tr^{gr}(P) is an ideal, Tr^{gr}(P) = A and so P is a graded generator by Theorem 2.2.2.

2.3 General graded Morita equivalence

Let P be a right A-module. Consider the ring B = Hom_{A}(P, P). Then P has a natural B − Abimodule structure. The actions of A and B on P are defined by p.a = pa and g.p = g(p), respectively, where g ∈ B, p ∈ P and a ∈ A. Consider the dual P^{∗} = Hom_{A}(P, A). Then P^{∗} has a natural A − B-bimodule structure. The actions of A and B on P^{∗} defined by (a.q)(p) = aq(p) and q.g = q ∘ g, respectively, where g ∈ B, q ∈ P^{∗}, p ∈ P and a ∈ A. Furthermore one defines

ϕ : P^{∗} ⊗_{B} P ⟶ A,
q ⊗ p ⟼ q(p),

and

ψ : P ⊗_{A} P^{∗} ⟶ B
p ⊗ q ⟼ pq,

where pq(p′) = p(q(p′)). One can check that ϕ is a A − A-bimodule homomorphism and ψ is a B −B-bimodule homomorphism. We leave it to the reader to check these and that with the actions introduced above, P has a B − A-bimodule structure and P^{∗} has a A − B-bimodule structure (see [62, §18C]). These compatibility conditions amount to the fact that the Morita ring

M = (\begin{matrix}A & P^{∗} \\ P & B \end{matrix}) ,  (2.3)

with the matrix multiplication has the associativity property. In fact M is a formal matrix ring as defined in Example 1.1.4.

As part of the Morita theory, one proves that when P is a generator, then ϕ is an isomorphism. Similarly, if P is finitely generated and projective, then ψ is an isomorphism (see [62, §18C]).

Putting these facts together, it is an easy observation that

− ⊗_{A}P^{∗} : Mod- A → Mod- B,  and  − ⊗_{B}P : Mod- B → Mod- A,  (2.4)

are inverse of each other and so these two categories are (Morita) equivalent.

If P is a graded finitely generated right A-module, then by Theorem 1.2.5, B = End_{A}(P, P) is also a graded ring and P^{∗} a graded left A-module. In fact, one can easily check that with the actions defined above, P is a graded B − A-bimodule, P^{∗} is a graded A − B-module and similarly ϕ and ψ are graded A − A and B − B-module homomorphisms, respectively. The Morita ring M of (2.3) is a graded formal matrix ring (see Example 1.2.6), with

M_{α} = (\begin{matrix}A_{α} & P_{α}^{∗} \\ P_{α} & B_{α} \end{matrix}) = (\begin{matrix}A_{α} & Hom_{Gr-A}(P, A(α)) \\ P_{α} & Hom_{Gr-A}(P, P (α)) \end{matrix}) .

We demonstrate here that P^{∗} is a graded A−B-bimodule and leave the others which are similar and routine to the reader. Recall that P^{∗} = ⨁_{α∈Γ} Hom_{A}(P, A)_{α}. Let a ∈ A_{α} and q ∈ P_{β}^{∗} = Hom_{A}(P, A)_{β}. Then a.q ∈ P^{∗}, where (a.q)(p) = aq(p). If p ∈ P_{γ}, then one easily sees that (a.q)(p) ∈ A_{α+β+γ}. This shows that aq ∈ Hom_{A}(P, A)_{α+β} = P_{α+β}^{∗}. On the other hand, if q ∈ P_{α}^{∗} and g ∈ Hom(P, P)_{β}, then q.g ∈ P^{∗}, where q.g(p) = qg(p). So if p ∈ P_{γ} then (q.g)(p) = qg(p) ∈ A_{α+β+γ}. So q.g ∈ P_{α+β}^{∗} as well.

By Theorem 2.2.5 and Proposition 1.2.12, if P is a graded finitely generated projective and graded generator, then P is finitely generated projective and a generator. Since the grading is preserved under the tensor products, the restriction of the functors − ⊗_{A} P^{∗} and − ⊗_{B} P of (2.4) induce an equivalence

− ⊗_{A}P^{∗} : Gr-A → Gr-B,  and  − ⊗_{B}P : Gr-B → Gr-A.  (2.5)

Furthermore, these functors commute with suspensions. Thus we get a commutative diagram where the vertical maps are forgetful functor (see §1.2.7),

\xymatrix{Gr-A \ar[r]^{−⊗_{A}P^{∗}} \ar[d]_{U} & Gr-B \ar[d]^{U} \\ Mod- A \ar[r]^{−⊗_{A}P^{∗}} & Mod- B.}  (2.6)

We call − ⊗_{A} P^{∗} a graded equivalence functor (see Definition 2.3.2).

Example 2.3.1. Let A be a graded ring and e be a full homogeneous idempotent of A, i.e., e is a homogeneous element, e^{2} = e and AeA = A. Clearly e has degree zero. Consider P = eA. One can readily see that P is a right graded progenerator. Then P^{∗} = Hom_{A}(eA, A) ≅_{gr} Ae and B = End_{A}(eA, eA) ≅_{gr} eAe (see §1.2.10). The maps ϕ and ψ described above as a part of Morita context, takes the form ϕ : Ae ⊗_{eAe} eA → A and ψ : eA ⊗_{A} Ae → eAe which are graded isomorphisms. Thus we get an (graded) equivalence between Gr-A and Gr-eAe which lifts to an (graded) equivalence between Mod- A and Mod- eAe, as it is shown in the diagram below.

\xymatrix{Gr-A \ar[r]^{−⊗_{A}Ae} \ar[d]_{U} & Gr-eAe \ar[d]^{U} \\ Mod- A \ar[r]^{−⊗_{A}Ae} & Mod- eAe.}  (2.7)

Before stating the general graded Morita equivalence, we need to make some definitions. Recall from §1.1 that for α ∈ Γ, the α-suspension functor 𝒯_{α} : Gr-A → Gr-A, M ↦ M(α) is an isomorphism with the property 𝒯_{α}𝒯_{β} = 𝒯_{α+β}, α, β ∈ Γ.

Definition 2.3.2. Let A and B be Γ-graded rings.

1. A functor ϕ : Gr-A → Gr-B is called a graded functor if ϕ𝒯_{α} = 𝒯_{α}ϕ.

2. A graded functor ϕ : Gr-A → Gr-B is called a graded equivalence if there is a graded functor ψ : Gr-B → Gr-A such that ψϕ ≅ 1_{Gr-A} and ϕψ ≅ 1_{Gr-B}.

3. If there is a graded equivalence between Gr-A and Gr-B, we say A and B are graded equivalent or graded Morita equivalence and we write Gr-A ≈_{gr} Gr-B, or Gr^{Γ}-A ≈_{gr} Gr^{Γ}-B to emphasis the categories are Γ-graded.

4. A functor ϕ′ : Mod- A → Mod- B is called a graded functor if there is a graded functor ϕ : Gr-A → Gr-B such that the following diagram, where the vertical functors are forgetful functors (see §1.2.7), is commutative.

\xymatrix{Gr-A \ar[r]^{ϕ} \ar[d]_{U} & Gr-B \ar[d]^{U} \\ Mod- A \ar[r]^{ϕ′} & Mod- B.}  (2.8)

The functor ϕ is called an associated graded functor of ϕ′.

5. A functor ϕ : Mod- A → Mod- B is called a graded equivalence if it is graded and an equivalence.

Definition 2.3.2 of graded functors is formulated for the category of (graded) right modules. A similar definition can be written for the category of (graded) left modules. We will see that the notion of graded equivalence is a left-right symmetric (see Remark 2.3.8).

Example 2.3.3. The equivalence between Mod- A and Mod- eAe in Example 2.3.1 is a graded equivalence as it is demonstrated in Diagram 2.7.

If Q is an object in Gr-A, then we denote U(Q) ∈ Mod- A also by Q, forgetting its graded structure (see §1.2.7). Also when working with graded matrix rings, say 𝕄_{n}(A)(\overline{δ}), we write simply Mod- 𝕄_{n}(A) when considering the category of (non-graded) modules over this matrix algebra. These should not cause any confusion in the text.

Example 2.3.4. Proposition 2.1.1 can be written for the the category of A-modules, which gives us a non-graded version of Morita equivalence. We have the following commutative diagram which shows that the functor − ⊗_{A} A^{n} : Mod- A → Mod- 𝕄_{n}(A) is a graded equivalence.

\xymatrix{Gr-A \ar[r]^{−⊗_{A}A^{n}(\overline{δ})} \ar[d]_{U} & Gr- 𝕄_{n}(A)(\overline{δ}) \ar[d]^{U} \\ Mod- A \ar[r]^{−⊗_{A}A^{n}} & Mod- 𝕄_{n}(A).}

We are in a position to state the main Theorem of this chapter.

Theorem 2.3.5. Let A and B be two Γ-graded rings. Let ϕ : Gr-A ⟶ Gr-B be a graded equivalence. Then there is a graded equivalence ϕ′ : Mod- A ⟶ Mod- B with an associated graded functor isomorphic to ϕ. Indeed, there is a graded A − B-bimodule Q such that ϕ ≅ − ⊗_{A} Q and consequently the following diagram commutes.

\xymatrix{Gr-A \ar[r]^{−⊗_{A}Q} \ar[d]_{U} & Gr-B \ar[d]^{U} \\ Mod- A \ar[r]^{−⊗_{A}Q} & Mod- B.}

Proof. Let the graded functor ψ : Gr-B → Gr-A be an inverse of the functor ϕ with

f : ψϕ ≅ 1_{Gr-A}.

Since B is a graded finitely generated projective and a graded generator in Gr-B, it follows that P = ψ(B) is a graded finitely generated projective and a graded generator in Gr-A. Thus by Theorem 2.2.5 and Proposition 1.2.12, P is a finitely generated projective and a generator in Mod- A as well. This shows that Hom_{A}(P, −) : Mod- A → Mod- B is a graded equivalence. We will show that ϕ ≅ Hom_{A}(P, −) on the category of Gr-A.

By Theorem 1.2.5, Hom_{B}(B, B) = ⨁_{α∈Γ} Hom_{B}(B, B)_{α}, and by (1.9) we can write Hom(B, B)_{α} = Hom_{Gr-B}(B, B(α)). Applying ψ on each of these components, since ψ is a graded functor, we get

Hom_{B}(B, B) = ⨁_{α∈Γ} Hom_{B}(B, B)_{α} ⟶^{ψ} ⨁_{α∈Γ} Hom_{A}(P, P)_{α} = Hom_{A}(P, P).  (2.9)

One can immediately see that this gives a graded isomorphism of rings between Hom_{B}(B, B) and Hom_{A}(P, P).

For any b ∈ B consider the right B-module homomorphism

η_{b} : B → B
x ↦ bx.

Then the regular representation map η : B → Hom_{B}(B, B), η(b) = η_{b}, is a graded isomorphism of rings. Thus we have a graded isomorphism of rings

B ⟶^{η} Hom_{B}(B, B) ⟶^{ψ} Hom_{A}(P, P),  (2.10)

where P is a graded A-progenerator.

Now since for any graded A-module X, Hom_{A}(P, X) is a graded right Hom_{A}(P, P)-module, the isomorphisms 2.10 induces a graded B-module structure on Hom_{A}(P, X). Namely, for homogeneous elements b ∈ B and t ∈ Hom_{A}(P, X)

t.b = tψ(η_{b})  (2.11)

which extends linearly to all elements.

We show that ϕ ≅ Hom_{A}(P, −). Let X be a graded A-module. Then

ϕ(X) ≅_{gr} Hom_{B}(B, ϕ(X)) = ⨁_{α∈Γ} Hom_{B}(B, ϕ(X))_{α} ⟶^{ψ} ⨁_{α∈Γ} Hom_{A}(P, ψϕ(X))_{α}

⟶^{Hom(1,f)} ⨁_{α∈Γ} Hom_{A}(P, X)_{α} = Hom_{A}(P, X). (2.12)

This shows that ϕ(X) is isomorphic to Hom_{A}(P, X) as a graded abelian groups. We need to show that this isomorphism, call it Θ, is a B-module isomorphism. Let z ∈ ϕ(X) and b ∈ B be homogeneous elements. Since ϕ(X) ≅_{gr} Hom_{B}(B, ϕ(X)), for z ∈ ϕ(X), we denote the corresponding homomorphism in Hom_{B}(B, ϕ(X)) also by z. Then

Θ(z.b) = Θ(zη_{b}) = Hom(1, f)ψ(zη_{b}) = fψ(z)ψ(η_{b}).

But

Θ(z).b = ( Hom(1, f)ψ(z)).b = fψ(z)ψ(η_{b}) (by (2.11)).

Thus ϕ ≅ Hom_{A}(P, −) on Gr-A. Now considering ϕ′ = Hom_{A}(P, −) : Mod- A → Mod- B. This gives the first part of the Theorem.

Setting P^{∗} = Hom_{A}(P, A), one can easily check that for any graded right A-module M, the map

M ⊗_{A} P^{∗} ⟶ Hom_{A}(P, M),
m ⊗ q ⟼ (m ⊗ q)(p) = mq(p),

where m ∈ M, q ∈ P^{∗} and p ∈ P, is a graded B-homomorphism (recall that by (2.10), B ≅_{gr} End_{A}(P)). Since P is a graded progenerator, by Theorem 2.2.5, it a progenerator, which in return gives that the above homomorphism is in fact an isomorphism (see [62, Remark 18.25]). Thus ϕ ≅ Hom_{A}(P, −) ≅ − ⊗_{A} P^{∗}. This gives the second part of the Theorem.

Theorem 2.3.6. Let A and B be two Γ-graded rings. The following are equivalent:

(1) Mod- A is graded equivalent to Mod- B;

(2) Gr-A is graded equivalent to Gr-B;

(3) B ≅_{gr} End_{A}(P) for a graded A-progenerator P;

(4) B ≅_{gr} e 𝕄_{n}(A)(\overline{δ})e for a full homogeneous idempotent e ∈ 𝕄_{n}(A)(\overline{δ}), where \overline{δ} = (δ_{1}, …, δ_{n}), δ_{i} ∈ Γ.

Proof. (1) ⇒ (2) Let ϕ : Mod- A → Mod- B be a graded equivalence. Using (2.8), it follows that ϕ(A) = P is a graded right B-module. Also a similar argument as in (2.10) shows that P is a graded left A-module. Since ϕ is an equivalence, from the (non-graded) Morita theory it follows that ϕ ≅ − ⊗_{A} P with an inverse − ⊗_{B} P^{∗}. Since the tensor product respects the grading, the same functor ϕ induces a graded equivalence between Gr-A and Gr-B.

(2) ⇒ (3) This is (2.10) in the proof of Theorem 2.3.5.

(3) ⇒ (4) Since P is a graded finitely generated projective A-module, P ⊕ Q ≅_{gr} A^{n}(−\overline{δ}), where \overline{δ} = (δ_{1}, …, δ_{n}) and n ∈ ℕ. Let e ∈ End_{A}(A^{n}(−\overline{δ})) ≅_{gr} 𝕄_{n}(A)(\overline{δ}) be the graded homomorphism which sends Q to zero and acts as identity on P. Thus e ∈ End_{A}(A^{n}(−\overline{δ}))_{0} ≅ 𝕄_{n}(A)(\overline{δ})_{0} and P = eA^{n}(−\overline{δ}). Define the map

θ : End_{A}(P) ⟶ e End_{A}(A^{n}(−\overline{δ}))e

by θ(f)|_{P} = f and θ(f)|_{Q} = 0. Since e is homogeneous of degree zero, it is straightforward to see that this is a graded isomorphism of rings (which preserve the identity). Thus

B ≅_{gr} End_{A}(P) ≅_{gr} e End_{A}(A^{n}(−\overline{δ}))e ≅_{gr} e 𝕄_{n}(A)(\overline{δ})e.

We are left to show that e is full. By [62, Exercise 2.8], 𝕄_{n}(A)e 𝕄_{n}(A) = 𝕄_{n}(Tr(P)), where P = eA^{n}. Since P is graded progenerator, it is a finitely generated projective A-module and a generator (see Theorem 2.2.5), thus Tr(P) = A and therefore e is a (homogeneous) full idempotent.

(4) ⇒ (1) Example 2.3.1 shows that there is a graded equivalence between Mod- 𝕄_{n}(A) and Mod- e 𝕄_{n}(A)e. On the other hand, Example 2.3.4 shows that there is a graded equivalence between Mod- 𝕄_{n}(A) and Mod- A. This finishes the proof.

Example 2.3.7. \underline{Gr-A ≅ Gr-B does not imply Gr-A ≅_{gr} Gr-B}

One can easily construct examples of two Γ-graded rings A and B such that the categories Gr-A and Gr-B are equivalent, but not graded equivalent, i.e., the equivalent functors do not commute with suspensions (see Definition 2.3.2). Let A be a Γ-graded ring and ϕ : Γ → Aut(A) be a group homomorphism. Consider the group ring A[Γ] and the skew group ring A ⋆_{ϕ} Γ (see §1.1.3). These rings are strongly Γ-graded, and thus by Dade’s Theorem 1.5.1, Gr-A[Γ] and Gr-A ⋆_{ϕ} Γ are equivalent to Mod- A and thus Gr-A[Γ] ≅ Gr-A ⋆_{ϕ} Γ. However one can easily show that these two graded rings are not necessarily graded equivalent.

Remark 2.3.8.

1. The graded Morita theory has a left-right symmetry property. Indeed, starting from the category of graded left modules, one can prove a similar statement as in Proposition 2.3.6, which in turn, part (4) is independent of the left-right assumption. This shows that Gr-A is graded equivalent to Gr-B if and only if A -Gr is graded equivalent to B -Gr

2. Proposition 2.3.6(3) shows that if all graded finitely generated projective A-modules are graded free, then Mod- A is graded equivalence to Mod- B if and only if B ≅_{gr} 𝕄_{n}(A)(\overline{δ}) for some n ∈ ℕ and \overline{δ} = (δ_{1}, …, δ_{n}), δ_{i} ∈ Γ.

Remark 2.3.9. \underline{Gr^{Ω}- A ≈_{gr} Gr^{Ω}- B implies Gr^{Γ}- A ≈_{gr} Gr^{Γ}- B}

Recall from Definition 2.3.2 that we write Gr^{Γ}-A ≈_{gr} Gr^{Γ}-B, if there is a graded equivalence between the categories of Γ-graded A-modules Gr^{Γ}-A and Γ-graded B-modules Gr^{Γ}-B.

Let A and B be Γ-graded rings and Ω a subgroup of Γ such that Γ_{A}, Γ_{B} ⊆ Ω ⊆ Γ. Then A and B can be naturally considered as Ω-graded ring. If Gr^{Ω}-A ≈_{gr} Gr^{Ω}-B, then by Theorem 2.3.6 there is a Ω-isomorphism ϕ : B ≅_{gr} e 𝕄_{n}(A)(\overline{δ})e for a full homogeneous idempotent e ∈ 𝕄_{n}(A)(\overline{δ}), where \overline{δ} = (δ_{1}, …, δ_{n}), δ_{i} ∈ Ω. Since Ω ⊆ Γ, another application of Theorem 2.3.6 shows that Gr^{Γ}-A ≈_{gr} Gr^{Γ}-B. One can also use Theorem 1.2.8 to obtain this statement.

Remark 2.3.10. \underline{Gr^{Γ}- A ≈_{gr} Gr^{Γ}- B implies Gr^{Γ/Ω}- A ≈_{gr} Gr^{Γ/Ω}- B}

Let A and B be two Γ-graded rings. Theorem 2.3.6 shows the equivalence Gr-A ≈_{gr} Gr-B induces an equivalence Mod- A ≈ Mod- B. Haefner in [42] observed that Gr-A ≈_{gr} Gr-B induces other equivalences between different “layers” of grading. We briefly recount this result here.

Let A be a Γ-graded ring and let Ω be subgroup of Γ. Recall from Example 1.1.8 and §1.2.8 that A can be considered at Γ/Ω-graded ring. Recall also that the category of Γ/Ω-graded Amodules, denoted by Gr^{Γ/Ω}-A, consists of the Γ/Ω-graded A-modules as objects and A-module homomorphisms ϕ : M → N which are grade-preserving in the sense that ϕ(M_{Ω+α}) ⊆ N_{Ω+α} for all Ω + α ∈ Γ/Ω as morphisms of the category. In two extreme cases Ω = 0 and Ω = Γ we have Gr^{Γ/Ω}-A = Gr-A and Gr^{Γ/Ω}-A = Mod- A, respectively.

In [42] Haefner shows that, for any two Γ-graded equivalent rings A and B and for any subgroup Ω of Γ, there are equivalences between the categories Gr^{Γ/Ω}-A and Gr^{Γ/Ω}-B. In fact, Haefner works with an arbitrary (non-abelian) group Γ and any subgroup Ω. In this case one needs to adjust the definitions as follows.

Let Γ/Ω denote a set of (right) cosets of Ω (we use the multiplication notation here). An Γ/Ω-graded right A-module M is defined as a right A-module M with an internal direct sum decomposition M = ⨁_{Ωα∈Γ/Ω} M_{Ωα}, where each M_{Ωα} is an additive subgroup of M such that M_{Ωα}A_{β} ⊆ M_{Ωαβ} for all Ωα ∈ Γ/Ω and β ∈ Γ. The decomposition is called an Γ/Ω-grading of M. With the abuse of notation we denote the category of Γ/Ω-graded right A-modules with Gr^{Γ/Ω}-A. Then in [42] it was shown that, Gr^{Γ}-A ≈_{gr} Gr^{Γ}-B implies Gr^{Γ/Ω}-A ≈_{gr} Gr^{Γ/Ω}-B.

Consider the truncation of A at Ω, i.e., A_{Ω} = ⊕_{γ∈Ω}A_{γ} (see Example 1.1.8). As usual, let Mod- A_{Ω} denote the category of right A_{Ω}-modules. If A is strongly Γ-graded, one can show that the categories Gr^{Γ/Ω}-A and Mod- A_{Ω} are equivalent via the functors truncation (−)_{Ω} : Gr^{Γ/Ω}-A → Mod- A_{Ω} and induction − ⊗_{A_{Ω}} A : Mod- A_{Ω} → Gr^{Γ/Ω}-A (see [42, Lemma 7.3]). Again, in the case that the subgroup Ω is trivial, this gives the equivalences of Gr-A and Mod- A_{0} (see Theorem 1.5.1).

Remark 2.3.11. \underline{The case of Gr-A ≈ Mod- R}

If a Γ-graded ring A is strongly graded, then by Theorem 1.5.1, Gr-A is equivalent to Mod- A_{0}. In [69], the conditions under which Gr-A is equivalent to Mod- R, for some ring R with identity, are given. It was shown that, among other things, Gr-A ≈ Mod- R, for some ring R with identity, if and only if Gr-A ≈_{gr} Gr-B, for a strongly Γ-graded ring B, if and only if there is a finite subset Ω of Γ such that for any τ ∈ Γ,

A_{0} = ∑_{γ∈Ω} A_{τ−γ}A_{γ−τ} .  (2.13)

Furthermore, it was shown that if Gr-A ≈ Mod- R then A -Gr ≈ R -Mod and A_{0} ≅ e 𝕄_{n}(R)e for some n ∈ ℕ, and some idempotent e ∈ 𝕄_{n}(R).

If the graded group Γ is finite, then (2.13) shows that there is a ring R such that Gr-A is equivalent to Mod- R. In fact, Cohen and Montgomery in [27] construct a so called smash product A#ℤ[Γ]^{∗} and show that Gr-A is equivalent to Mod- A#ℤ[Γ]^{∗}. In [74] it was shown that A#ℤ[Γ]^{∗} is isomorphic to the ring End_{Gr-A} ( ⨁_{α∈Γ} A(α)) (see Example 1.3.6).

Remark 2.3.12. \underline{The case of Gr^{Γ}- A ≈ Gr^{Λ}- B}

One can consider several variations under which two categories are equivalent. For example, for a Γ-graded ring A and Λ-graded ring B, the situation when the categories Gr^{Γ}-A and Gr^{Λ}-B are equivalent is investigated in [80, 81] (See also Remark 1.1.26).

Furthermore, for two Γ-graded rings A and B, when Gr-A is equivalent to Gr-B (not necessarily respect the shifting) has been studied in [85, 98].

Graded Grothendieck Groups

One of the main aim of this note is to study the graded Grothendieck group, K_{0}^{gr}(A), where A is a Γ-graded ring. In fact, K_{0}^{gr}(A) not only is an abelian group but it has also an extra ℤ[Γ]-module structure. As we will see throughout this note, this extra structure carries a substantial information about the graded ring A. In §3.1 we construct, in detail, the graded Grothendieck groups using the concept of group completions. Here we briefly give an overview of different equivalent constructions.

For an abelian monoid V, we denote by V^{+} the group completion of V. This gives a left adjoint functor to the forgetful functor from the category of abelian groups to abelian monoids. When the monoid V has a Γ-module structure, where Γ is a group, then V^{+} inherits a natural Γ-module structure, or equivalently, ℤ[Γ]-module structure (see §3.1).

There is also a more direct way to construct the V^{+} which we recall here. Consider the set of symbols {[m] | m ∈ V } and let V^{+} be the free abelian group generated by this set modulo the relations [m] + [n] − [m + n], m, n ∈ V. There is a natural (monoid) homomorphism V ⟶ V^{+}, m ↦ [m], which is universal. Using the universality, one can show that the group V^{+} obtained here coincides with the one constructed above using the group completion.

Now let Γ be a group which acts on a monoid V. Then Γ acts in a natural way on the free abelian group generated by symbols {[m] | m ∈ V }. Furthermore the subgroup generated by relations [m] + [n] − [m + n], m, n ∈ V is closed under this action. Thus V^{+} has a natural structure of Γ-module, or equivalently, V^{+} is a ℤ[Γ]-module.

For a ring A with identity and a finitely generated projective (right) A-module P, let [P] denote the class of A-modules isomorphic to P. Then the set

𝒱(A) = {[P] | P is a finitely generated projective A-module}  (3.1)

with addition [P]+[Q] = [P ⨁ Q] forms an abelian monoid. The Grothendieck group of A, denoted by K_{0}(A), is by definition 𝒱(A)^{+}.

The graded Grothendieck group of a graded ring is constructed similarly, by using graded finitely generated projective modules everywhere in the above process. Namely, for a Γ-graded ring A with identity and a graded finitely generated projective (right) A-module P, let [P] denote the class of graded A-modules graded isomorphic to P. Then the monoid

𝒱^{gr}(A) = {[P] | P is graded finitely generated projective A-module}  (3.2)

has a Γ-module structure defined as follows: for γ ∈ Γ and [P] ∈ 𝒱^{gr}(A), γ.[P] = [P (γ)]. The group 𝒱^{gr}(A)^{+} is called the graded Grothendieck group and is denoted by K_{0}^{gr}(A), which as the above discussion shows is a ℤ[Γ]-module.

The above construction of the graded Grothendieck groups is carried over the category of graded right A-modules. Since for a graded finitely generated right A-module P, the dual module P^{∗} = Hom_{A}(P, A) is a graded left A-module, and furthermore, if P is projective, then P^{∗∗} ≅_{gr} P, taking duals gives an equivalence between the category of graded finitely generated projective right Amodules and the category of graded finitely generated projective left A-modules, i.e.,

A -Pgr ≈_{gr} Pgr-A.

This implies that constructing a graded Grothendieck group, using the graded left-modules, call it K_{0}^{gr}(A)_{l}, is isomorphic K_{0}^{gr}(A)_{r}, the group constructed using graded right A-modules. Furthermore, defining the action of Γ on generators of K_{0}^{gr}(A)_{l}, by α[P] = [P (−α)] and extend it to the whole group, and defining the action of Γ on K_{0}^{gr}(A)_{r}, in the usual way, by α[P] = [P (α)], then (1.12) shows that these two groups are ℤ[Γ]-module isomorphic.

As emphasised above, one of the main differences of a graded Grothendieck group versus the non-graded group, is that the former has an extra module structure. For instance, in Example 3.6.3, we will see that for the ℤ-graded ring A = K[x^{n}, x^{−n}], where K is a field and n ∈ ℕ,

K_{0}^{gr}(A) ≅ ⨁_{n} ℤ,

which is a ℤ[x, x^{−1}]-module, with the action of x on (a_{1}, …, a_{n}) ∈ ⨁_{n} ℤ is as follows:

x(a_{1}, …, a_{n}) = (a_{n}, a_{1}, …, a_{n−1}).

In this chapter we study the graded Grothendieck groups in detail and calculate them for certain types of graded rings, including the graded division rings (§3.6), graded local rings (§3.7) and path algebras (§3.8). In §5 we will use the graded Grothendieck groups to classify the so called graded ultramatricial algebras. We will show that certain information decoded in the graded Grothendieck group could be used to give a complete invariant for such algebras. In §6 we compare the graded versus non-graded Grothendieck groups.

For a comprehensive study of non-graded Grothendieck groups see [59, 66, 82].

3.1 The graded Grothendieck group K_{0}^{gr}

3.1.1 Group completions

Let V be a monoid and Γ be a group which acts on V. The group completion of V (this is also called the Grothendieck group of V) has a natural Γ-module structure. We recall the construction here. Define a relation ∼ on V × V as follows: (x_{1}, y_{1}) ∼ (x_{2}, y_{2}) if there is a z ∈ V such that

x_{1} + y_{2} + z = y_{1} + x_{2} + z.  (3.3)

This is an equivalence relation and we denote the equivalence classes of V × V/ ∼ by V^{+} := {[(x, y)] | (x, y) ∈ V × V }. Define

[(x_{1}, y_{1})] + [(x_{2}, y_{2})] = [(x_{1} + x_{2}, y_{1} + y_{2})],
α[(x_{1}, y_{1})] = [(αx_{1}, αy_{1})].

One can easily check that these operations are well-defined, V^{+} is an abelian group and further it is a Γ-module. The map

ϕ : V ⟶ V^{+}
x ⟼ [(x, 0)]  (3.4)

is a Γ-module homomorphism and ϕ is universal, i.e., if there is a Γ-module G and a Γ-module homomorphism ψ : V → G, then there exists a unique Γ-module homomorphism f : V^{+} → G such that fϕ = ψ.

We record the following properties of V^{+} whose proofs are easy and are left to the reader.

Lemma 3.1.1. Let V be a monoid, Γ a group which acts on V, and let ϕ : V → V^{+}, x ↦ [(x, 0)], be the universal homomorphism.

(1) For x, y ∈ V, if ϕ(x) = ϕ(y) then there is a z ∈ V such that x + z = y + z in V.

(2) Each element of V^{+} is of the form ϕ(x) − ϕ(y) for some x, y ∈ V.

(3) V^{+} is generated by V as a group.

Example 3.1.2. Let G be a group and ℕ be the monoid of positive integers. Then ℕ[G] is a monoid equipped by the natural action of G. Its group completion is ℤ[G] which has a natural G-module structure. This will be used in Proposition 3.6.1 to calculate the graded Grothendieck group of graded fields.

Example 3.1.3. \underline{A nonzero monoid whose group completion is zero}

Let V be a monoid with the trivial module structure (i.e., Γ is trivial). Suppose that V \{0} is an abelian group. Then one can check that V^{+} ≅ V \{0}.

Now let V = {0, v}, with v + v = v and 0 as the trivial element. One can check that with this operation V is a monoid and V \{0} is an abelian group which is a trivial group.

Example 3.1.4. \underline{The type of a ring, IBN and the monoid of projective modules}

Let A be a ring (with trivial grading) and 𝒱(A) the monoid of the isomorphism classes of finitely generated projective A-modules. Then one can observe that A has IBN if and only if the submonoid generated by [A] in 𝒱(A) is isomorphic to ℕ. Furthermore, a ring A has type (n, k) if and only if the the submonoid generated by [A] is isomorphic to a free monoid generated by a symbol v subject to nv = (n + k)v (see §1.7).

3.1.2 K_{0}^{gr}-group

Let A be a Γ-graded ring (with identity as usual) and let 𝒱^{gr}(A) denote the monoid of graded isomorphism classes of graded finitely generated projective modules over A with the direct sum as the addition operation. For a graded finitely generated projective A-module P, we denote the graded isomorphism class of P by [P] which is an element of 𝒱^{gr}(A) (see (3.2)). Thus for [P], [Q] ∈ 𝒱^{gr}(A), we have [P] + [Q] = [P ⊕ Q]. Note that for α ∈ Γ, the α-suspension functor 𝒯_{α} : Gr-A → Gr-A, M ↦ M(α) is an isomorphism with the property 𝒯_{α}𝒯_{β} = 𝒯_{α+β}, α, β ∈ Γ. Furthermore, 𝒯_{α} restricts to Pgr-A, the category of graded finitely generated projective A-modules. Thus the abelian group Γ acts on 𝒱^{gr}(A) via

(α, [P]) ↦ [P (α)].  (3.5)

The graded Grothendieck group, K_{0}^{gr}(A), is defined as the group completion of the monoid 𝒱^{gr}(A) which naturally inherits the Γ-module structure via (3.5). This makes K_{0}^{gr}(A) a ℤ[Γ]-module. In particular if A is a ℤ-graded then K_{0}^{gr}(A) is a ℤ[x, x^{−1}]-module. This extra structure plays a crucial role in the applications of graded Grothendieck groups.

For a graded finitely generated projective A-module P, we denote the image of [P] ∈ 𝒱^{gr}(A) under the natural homomorphism 𝒱^{gr}(A) → K_{0}^{gr}(A) by [P] again (see (3.4)). When the ring has graded stable rank 1, this map is injective (see Corollary 3.1.8).

Example 3.1.5. In the following “trivial” cases one can determine the graded Grothendieck group based on the Grothendieck group of the ring.

Trivial group grading: Let A be a ring and Γ be a trivial group. Then A is a Γ-graded ring in an obvious way, Gr-A = Mod- A and K_{0}^{gr}(A) = K_{0}(A) as a ℤ[Γ]-module. However here ℤ[Γ] ≅ ℤ and K_{0}^{gr}(A) does not have any extra module structure. This shows that the statements we prove in the graded setting will cover the results in the classical non-graded setting, by considering rings with trivial gradings from the outset.

Trivial grading: Let A be a ring and Γ be a group. Consider A with the trivial Γ-grading, i.e., A is concentrated in degree zero (see §1.1.1). Then one can observe that Gr-A = ⨁_{Γ} Mod- A. (Here if 𝒞 is an additive category, ⨁_{I} 𝒞, where I is a nonempty index set, is defined in the obvious manner, with objects ⨁_{i∈I} M_{i}, where M_{i} are objects of 𝒞.). Consequently, K_{0}^{gr}(A) ≅ ⨁_{Γ} K_{0}(A). Considering the shifting which induces a ℤ[Γ]-module structure, we have K_{0}^{gr}(A) ≅ K_{0}(A)[Γ] as ℤ[Γ]-module.

Remark 3.1.6. \underline{The group completion of all projective modules is trivial}

One reason to restricting ourselves to the finitely generated projective modules is that the group completion of the monoid of the isomorphism classes of (all) projective modules gives a trivial group. To see this, first observe that for a monoid V, V^{+} is the trivial group if and only if for any x, y ∈ V, there is a z ∈ V such that x + z = y + z. Now consider graded projective modules P and Q. Then for M = ⨁_{∞}(P ⊕ Q), we have P ⊕ M ≅ Q ⊕ M. Indeed,

Q ⊕ M ≅ Q ⊕ (P ⊕ Q) ⊕ (P ⊕ Q) ⊕ ⋯
≅ (Q ⊕ P) ⊕ (Q ⊕ P) ⊕ ⋯
≅ (P ⊕ Q) ⊕ (P ⊕ Q) ⊕ ⋯
≅ P ⊕ (Q ⊕ P) ⊕ (Q ⊕ P) ⊕ ⋯
≅ P ⊕ M.

Lemma 3.1.7. Let A be a Γ-graded ring.

(1) Each element of K_{0}^{gr}(A) is of the form [P] − [Q] for some graded finitely generated projective A-modules P and Q.

(2) Each element of K_{0}^{gr}(A) is of the form [P]−[A^{n}(\overline{α})] for some graded finitely generated projective A-module P and some \overline{α} = (α_{1}, …, α_{n}).

(3) Let P, Q be graded finitely generated projective A-modules. Then [P] = [Q] in K_{0}^{gr}(A) if and only if P ⊕ A^{n}(\overline{α}) ≅_{gr} Q ⊕ A^{n}(\overline{α}), for some \overline{α} = (α_{1}, …, α_{n}).

Proof. (1) This follows immediately from Lemma 3.1.1(2) by considering 𝒱^{gr}(A) as the monoid and the fact that [P] also represent the image of [P] ∈ 𝒱^{gr}(A) in K_{0}^{gr}(A).

(2) This follows from (1) and the fact that for a graded finitely generated projective A-module Q, there is a graded finitely generated projective module Q′ such that Q ⊕ Q′ ≅_{gr} A^{n}(\overline{α}), for some \overline{α} = (α_{1}, …, α_{n}) (see Proposition 1.2.12 and (1.27)).

(3) Suppose [P] = [Q] in K_{0}^{gr}(A). Then by Lemma 3.1.1(1) (for V = 𝒱^{gr}(A)) there is a graded finitely generated projective A-module T such that P ⊕ T ≅_{gr} Q ⊕ T. Since T is graded finitely generated projective, there is an S such that T ⊕S ≅_{gr} A^{n}(\overline{α}), for some \overline{α} = (α_{1}, …, α_{n}) (see (1.27)). Thus

P ⊕ A^{n}(\overline{α}) ≅_{gr} P ⊕ T ⊕ S ≅_{gr} Q ⊕ T ⊕ S ≅_{gr} Q ⊕ A^{n}(\overline{α}).

Since K_{0}^{gr}(A) is a group, the converse is immediate.

For graded finitely generated projective A-modules P and Q, P is called graded stably isomorphic to Q, if [P] = [Q] in K_{0}^{gr}(A) or equivalently by Lemma 3.1.7(3), if P ⊕ A^{n}(\overline{α}) ≅_{gr} Q ⊕ A^{n}(\overline{α}), for some \overline{α} = (α_{1}, …, α_{n}).

Corollary 3.1.8. Let A be a Γ-graded ring with the graded stable rank 1. Then the natural map 𝒱^{gr}(A) → K_{0}^{gr}(A) is injective.

Proof. Let P and Q be graded finitely generated projective A-modules such that [P] = [Q] in K_{0}^{gr}(A). Then by Lemma 3.1.7(3), P ⊕ A^{n}(\overline{α}) ≅_{gr} Q ⊕ A^{n}(\overline{α}), for some \overline{α} = (α_{1}, …, α_{n}). Now by Corollary 1.8.4, P ≅_{gr} Q. Thus [P] = [Q] in 𝒱^{gr}(A).

3.1.3 The graded Grothendieck group of strongly graded rings

Let A be a strongly Γ-graded ring. By Dade’s Theorem 1.5.1 and Remark 1.5.4, the functor (−)_{0} : Pgr-A → Pr- A_{0}, M ↦ M_{0}, is an additive functor with an inverse − ⊗_{A_{0}} A : Pr- A_{0} → Pgr-A so that it induces an equivalence between the category of graded finitely generated projective Amodules and the category of finitely generated A_{0}-module. This implies that

K_{0}^{gr}(A) ≅ K_{0}(A_{0}).  (3.6)

(In fact, this implies that K_{i}^{gr}(A) ≅ K_{i}(A_{0}), for all i ≥ 0, where K_{i}^{gr}(A) and K_{i}(A_{0}) are Quillen’s K-groups.) Furthermore, since A_{α} ⊗_{A_{0}} A_{β} ≅ A_{α+β} as A_{0}-bimodule, the functor 𝒯_{α} on gr-A induces a functor on the level of mod-A_{0}, 𝒯_{α} : mod-A_{0} → mod-A_{0}, M ↦ M ⊗_{A_{0}} A_{α} such that 𝒯_{α}𝒯_{β} ≅ 𝒯_{α+β}, α, β ∈ Γ, so that the following diagram is commutative up to isomorphism.

\xymatrix{Pgr-A \ar[r]^{𝒯_{α}} \ar[d]_{(−)_{0}} & Pgr-A \ar[d]^{(−)_{0}} \\ Pr- A_{0} \ar[r]^{𝒯_{α}} & Pr- A_{0}}  (3.7)

Therefore K_{i}(A_{0}) is also a ℤ[Γ]-module and

K_{i}^{gr}(A) ≅ K_{i}(A_{0}),  (3.8)

as ℤ[Γ]-modules.

Also note that if A is a graded commutative ring then the isomorphism (3.8) is a ring isomorphism.

Example 3.1.9. \underline{K_{0}^{gr} of crossed products}

Let A be a Γ-graded crossed-product ring. Thus A = A_{0}_{ψ}^{ϕ}[Γ] (see §1.1.3). By Proposition 1.1.16(3), A is strongly graded, and so Gr-A ≈ Mod- A_{0} and K_{0}^{gr}(A) ≅ K_{0}(A_{0}) (see (3.6)).

On the other hand, by Corollary 1.3.12(4), the restriction of the shift functor on Pgr-A, 𝒯_{α} : Pgr-A → Pgr-A, is isomorphic to the trivial functor. This shows that the action of Γ on K_{0}^{gr}(A) ≅ K_{0}(A_{0}) (and indeed on all K-groups K_{i}^{gr}(A)) is trivial.

Example 3.1.10. \underline{K_{0}^{Γ} verses K_{0}^{Γ/Ω}}

Let A be a Γ-graded ring and Ω be a subgroup of Γ. Recall the construction of Γ/Ω-graded ring A from Example 1.1.8. The canonical forgetful functor U : Gr^{Γ}-A → Gr^{Γ/Ω}-A is an exact functor (see §1.2.8). Proposition 1.2.12 guarantees that U restricts to the categories of graded finitely generated projective modules, i.e., U : Pgr^{Γ}-A → Pgr^{Γ/Ω}-A. This induces a group homomorphism

θ : K_{0}^{Γ}(A) ⟶ K_{0}^{Γ/Ω}(A),

such that for α ∈ Γ and a ∈ K_{0}^{Γ}(A), we have θ(αa) = (Ω + α)θ(a). Here, to distinguish the graded groups, we denote by K_{0}^{Γ} and K_{0}^{Γ/Ω} the graded Grothendieck groups of A as Γ and Γ/Ω-graded rings, respectively (we will use this notation again in §6).

Now let A be a strongly Γ-graded ring. Then A is also a strongly Γ/Ω-graded (Example 1.1.18). Using (3.6), we have

K_{0}^{Γ}(A) ≅ K_{0}(A_{0}),
K_{0}^{Γ/Ω}(A) ≅ K_{0}(A_{Ω}),

where A_{Ω} = ⨁_{γ∈Ω} A_{γ}.

Example 3.1.11. \underline{K_{0}^{Γ} verses K_{0}^{Ω}}

Let A be a Γ-graded ring and Ω be a subgroup of Γ such that Γ_{A} ⊆ Ω. By Theorem 1.2.8 the category Gr^{Γ}-A is equivalent to ⨁_{Γ/Ω} Gr^{Ω}-A_{Ω}. Since this equivalence preserves the projective modules, we have

K_{0}^{Γ}(A) ≅ ⨁_{Γ/Ω} K_{0}^{Ω}(A).

In fact, the same argument shows that for any (higher) K-groups (see §6), if A is a Γ-graded ring such that for Ω := Γ_{A}, A_{Ω} is strongly Ω-graded, then we have

K_{i}^{Γ}(A) ≅ ⨁_{Γ/Ω} K_{i}^{Ω}(A) ≅ ⨁_{Γ/Ω} K_{i}(A_{0}).  (3.9)

Furthermore, if A_{Ω} is a crossed-product, we can determine the action of Γ on K_{i}^{gr} concretely. First, using Corollary 1.3.12, in (1.20) for any ω ∈ Ω, M(w)_{Ω+α_{i}} ≅ M_{Ω+α_{i}} as graded A_{Ω}-module. Thus the functor \overline{ρ}_{β} in (1.20) reduces to permutations of categories. Representing ⨁_{Γ/Ω} K_{i}(A_{0}) by the additive group of group ring K_{i}(A_{0})[Γ/Ω], by Theorem 1.2.8, the ℤ[Γ]-module structure of K_{i}^{Γ}(A), can be descried as K_{i}(A_{0})[Γ/Ω] with the natural ℤ[Γ]-module structure.

This can be used to calculate the (lower) graded K-theory of graded division algebras (see Proposition 3.6.1, Remark 3.6.2 and Example 3.10.3).

3.1.4 The reduced graded Grothendieck group K̃_{0}^{gr}

For a ring A, the element [A] ∈ K_{0}(A) generates the “obvious” part of the Grothendieck group, i.e., all the elements presented by ±[A^{n}] for some n ∈ ℕ. The reduced Grothendieck group, denoted by K̃_{0}(A) is the quotient of K_{0}(A) by this cyclic subgroup. There is a unique ring homomorphism ℤ → A, which induces the group homomorphism K_{0}(ℤ) → K_{0}(A). Since the ring of integers ℤ is a PID and finitely generated projective modules over a PID ring are free of unique rank (see [82, Theorem 1.3.1]), we have

K̃_{0}(A) = coker(K_{0}(ℤ) → K_{0}(A)).  (3.10)

This group appears quite naturally. For example when A is a Dedekind domain, K̃_{0}(A) coincides with C(A), the class group of A (see [82, Chap. 1,§4]).

We can thus write an exact sequence,

0 ⟶ ker θ ⟶ K_{0}(ℤ) ⟶^{θ} K_{0}(A) ⟶ K̃_{0}(A) ⟶ 0.

For a commutative ring A, one can show that ker θ = 0 and the exact sequence is split, i.e.,

K_{0}(A) ≅ ℤ ⨁ K̃_{0}(A).  (3.11)

In fact, ker θ determines if a ring has IBN (see §1.7). Namely, ker θ = 0 if and only if A has IBN. For, suppose ker θ = 0. If A^{n} ≅ A^{m} as A-modules, for n, m ∈ ℕ, then (n − m)[A] = 0. It follows n = m. Thus A has IBN. On the other hand suppose A has IBN. If n ∈ ker θ, then we can consider n ≥ 0. Then [A^{n}] = 0 implies A^{n+k} ≅ A^{k}, for some k ∈ ℕ. Thus n = 0 and so ker θ = 0.

Here we develop the graded version of the concept of reduced Grothendieck group. Let A be a Γgraded ring. In this setting the obvious part of K_{0}^{gr}(A) is not only the subgroup generated by [A] but by all the shiftings of [A], i.e., the ℤ[Γ]-module generated by [A]. The reduced graded Grothendieck group of the Γ-graded ring A, denoted by K̃_{0}^{gr}(A), is the ℤ[Γ]-module defined by the quotient of K_{0}^{gr}(A) by the submodule generated by [A]. Similarly to the non-graded case, considering ℤ as a Γ-graded ring concentrated in degree zero, the unique graded ring homomorphism ℤ → A, indices a ℤ[Γ]-module homomorphism K_{0}^{gr}(ℤ) → K_{0}^{gr}(A) (see Example 3.1.5(2)). Then

K̃_{0}^{gr}(A) = coker(K_{0}^{gr}(ℤ) → K_{0}^{gr}(A)).  (3.12)

In order to obtain a graded version of the splitting formula (3.11), we need to take out a part of Γ which its action on [A] via shifting would not change [A] in K_{0}^{gr}(A).

First note that the ℤ[Γ]-module homomorphism K_{0}^{gr}(ℤ) → K_{0}^{gr}(A) can be written as

ϕ : ℤ[Γ] ⟶ K_{0}^{gr}(A),  (3.13)
∑_{α} n_{α}α ⟼ ∑_{α} n_{α}[A(α)].

Furthermore, since by Corollary 1.3.11, A(α) ≅_{gr} A(β) as a right A-module if and only if α − β ∈ Γ_{A}^{∗}, the above map induces

ψ : ℤ[Γ/Γ_{A}^{∗}] ⟶ K_{0}^{gr}(A),
∑_{α} n_{α}(Γ_{A}^{∗} + α) ⟼ ∑_{α} n_{α}[A(α)],

so that the following diagram is naturally commutative.

\xymatrix{ℤ[Γ] \ar[rr]^{ϕ} \ar[dr]_{π} &  & K_{0}^{gr}(A), \\  & ℤ[Γ/Γ_{A}^{∗}] \ar[ur]_{ψ}}  (3.14)

Since π is surjective, K̃_{0}^{gr}(A) is also the cokernel of the map ψ.

In Example 3.6.8 we calculate the reduced Grothendieck group of the graded division algebras. Further in Example 3.8.6 we show that the reduced Grothendieck group of a strongly graded ring does not necessarily coincide with the Grothendieck group of its ring of the zero homogeneous part.

3.1.5 The graded Grothendieck group as a ℤ[Γ]-algebra

Let A be a Γ-graded commutative ring. Then, as in the non-graded case, K_{0}^{gr}(A) forms a commutative ring, with multiplication defined on generators by tensor products, i.e., [P].[Q] = [P ⊗_{A} Q] (see §1.2.6). Consider ℤ[Γ] as a group ring. To get the natural group ring operations, here we use the multiplicative notation for the group Γ. With this convention, since for any α, β ∈ Γ, A(α) ⊗_{A} A(β) ≅_{gr} A(αβ), the map

ℤ[Γ] ⟶ K_{0}^{gr}(A),
∑_{α} n_{α}α ⟼ ∑_{α} n_{α}[A(α)]

induces a ring homomorphism. This makes K_{0}^{gr}(A) a ℤ[Γ]-algebra. In fact the argument before Diagram 3.14 shows that K_{0}^{gr}(A) can be considered as ℤ[Γ/Γ_{A}^{∗}]-algebra.

In general, the category of graded finitely generated projective modules is an exact category with the usual notion of (split) short exact sequence. The Quillen K-groups of this category (see [79, §2] for the construction of K-groups of an exact category) is denoted by K_{i}^{gr}(A), i ≥ 0. The group Γ acts on this category via (α, P) ↦ P (α). By functoriality of K-groups this equips K_{i}^{gr}(A) with the structure of a ℤ[Γ]-module. When i = 0, Quillen’s construction coincides with the above construction via the group completion.

3.2 K_{0}^{gr} from idempotents

One can describe the K_{0}-group of a ring A with identity in terms of idempotent matrices of A. This description is quite helpful as we can work with the conjugate classes of idempotent matrices instead of isomorphism classes of finitely generated projective modules. For example we will use this description to show that the graded K_{0} is a continuous map (Theorem 3.2.4.) Also, this description allows us to define the monoid 𝒱 for rings without identity in a natural way (see §3.4). We briefly recall the construction of K_{0} from idempotents here.

Let A be a ring with identity. In the following, we can always enlarge matrices of different sizes over A by adding zeros in the lower right hand corner, so that they can be considered in a ring 𝕄_{k}(A) for a suitable k ∈ ℕ. This means that we work in the matrix 𝕄_{∞}(A) = \underrightarrow{lim}_{n} 𝕄_{n}(A), where the connecting maps are the non-unital ring homomorphism

𝕄_{n}(A) ⟶ 𝕄_{n+1}(A)
p ⟼ (\begin{matrix}p & 0 \\ 0 & 0 \end{matrix}) .  (3.15)

Any idempotent matrix p ∈ 𝕄_{n}(A) (i.e., p^{2} = p) gives rise to the finitely generated projective right A-module pA^{n}. On the other hand any finitely generated projective right module P gives rise to an idempotent matrix p such that pA^{n} ≅ P. We say two idempotent matrices p and q are equivalent if (after suitably enlarging them) there are matrices x and y such that xy = p and yx = q. One can show that p and q are equivalent if and only if they are conjugate if and only if the corresponding finitely generated projective modules are isomorphic. Therefore K_{0}(A) can be defined as the group completion of the monoid of equivalence classes of idempotent matrices with

addition defined by, [p] + [q] = [ (\begin{matrix}p & 0 \\ 0 & q \end{matrix}) ]. In fact, this is the definition that one adapts for 𝒱(A)

when the ring A does not have identity (see for example [28, Chapter 1] or [68, p.296]).

A similar construction can be given in the setting of graded rings. This does not seem to be documented in literature and we provide the details here.

Let A be a Γ-graded ring with identity and let \overline{α} = (α_{1}, …, α_{n}), where α_{i} ∈ Γ. Recall that 𝕄_{n}(A)(\overline{α}) is a graded ring (see §1.3). In the following if we need to enlarge a homogeneous matrix p ∈ 𝕄_{n}(A)(\overline{α}), by adding zeroes in the lower right hand corner (and call it p), then we add zeros in the right hand side of \overline{α} = (α_{1}, …, α_{n}) as well accordingly (and call it \overline{α} again) so that p is a homogeneous matrix in 𝕄_{k}(A)(\overline{α}), where k ≥ n. Recall the definition of 𝕄_{k}(A)[\overline{α}][\overline{δ}] from §1.3.2 and note that if x ∈ 𝕄_{k}(A)[\overline{α}][\overline{δ}] and y ∈ 𝕄_{k}(A)[\overline{δ}][\overline{α}], then by Lemma 1.3.9 and (1.43), xy ∈ 𝕄_{k}(−\overline{α})_{0} and yx ∈ 𝕄_{k}(−\overline{δ})_{0}.

Definition 3.2.1. Let A be a Γ-graded ring, \overline{α} = (α_{1}, …, α_{n}) and \overline{δ} = (δ_{1}, …, δ_{m}), where α_{i}, δ_{j} ∈ Γ. Let p ∈ 𝕄_{n}(A)(\overline{α})_{0} and q ∈ 𝕄_{m}(A)(\overline{δ})_{0} be idempotent matrices. Then p and q are graded equivalent, denoted by p ∼ q, if (after suitably enlarging them) there are x ∈ 𝕄_{k}(A)[−\overline{α}][−\overline{δ}] and y ∈ 𝕄_{k}(A)[−\overline{δ}][−\overline{α}] such that xy = p and yx = q. Furthermore, we say p and q are graded conjugate if there is an invertible matrix g ∈ 𝕄_{2k}(A)[−\overline{δ}][−\overline{α}] with g^{−1} ∈ 𝕄_{2k}(A)[−\overline{α}][−\overline{δ}] such that gpg^{−1} = q.

The relation ∼ defined above is an equivalence relation. Indeed, if p ∈ 𝕄_{n}(A)(\overline{α})_{0} is a homogeneous idempotent, then considering x = p ∈ 𝕄_{n}(A)[−\overline{α}][−\overline{α}] (see (1.43)) and y = 1 ∈ 𝕄_{n}(A)[−\overline{α}][−\overline{α}], Definition 3.2.1 shows that p ∼ p. Clearly ∼ is reflexive. The following trick shows that ∼ is also transitive. Suppose p ∼ q and q ∼ r. Then p = xy, q = yx and q = vw and r = wv. Thus p = p^{2} = xyxy = xqy = (xv)(wy) and r = r^{2} = wvwv = wqv = (wy)(xv). This shows that p ∼ r.

Let p ∈ 𝕄_{n}(A)(\overline{α})_{0} be a homogeneous idempotent. Then one can easily see that for any

δ = (δ_{1}, …, δ_{m}), the idempotent (\begin{matrix}p & 0 \\ 0 & 0 \end{matrix}) ∈ 𝕄_{n+m}(A)(\overline{α}, \overline{β})_{0} is graded equivalent to p. We call this

element an enlargement of p.

Lemma 3.2.2. Let p ∈ 𝕄_{n}(A)(\overline{α})_{0} and q ∈ 𝕄_{m}(A)(\overline{δ})_{0} be idempotent matrices. Then p and q are graded equivalent if and only if some enlargements of p and q are conjugate.

Proof. Let p and q be graded equivalent idempotent matrices. By Definition 3.2.1, there are x′ ∈ 𝕄_{k}(A)[−\overline{α}][−\overline{δ}] and y′ ∈ 𝕄_{k}(A)[−\overline{δ}][−\overline{α}] such that x′y′ = p and y′x′ = q. Let x = px′q and y = qy′p. Then xy = px′qy′p = p(x′y′)^{2}p = p and similarly yx = q. Furthermore x = px = xq and y = yp = qy. Using Lemma 1.3.9, one can check that x ∈ 𝕄_{k}(A)[−\overline{α}][−\overline{δ}] and y ∈ 𝕄_{k}(A)[−\overline{δ}][−\overline{α}]. We now use the standard argument, taking into account the shifting of the matrices. Consider the matrix

(\begin{matrix}1 − p & x \\ y & 1 − q \end{matrix}) ∈ 𝕄_{2k}(A)[−\overline{α}, −\overline{δ}][−\overline{α}, −\overline{δ}].

This matrix has order two and thus is its own inverse. Then

(\begin{matrix}1 − p & x \\ y & 1 − q \end{matrix}) (\begin{matrix}p & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix}1 − p & x \\ y & 1 − q \end{matrix}) = (\begin{matrix}1 − p & x \\ y & 1 − q \end{matrix}) (\begin{matrix}0 & x \\ 0 & 0 \end{matrix}) = (\begin{matrix}0 & 0 \\ 0 & q \end{matrix}) .

Furthermore,

(\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix}0 & 0 \\ 0 & q \end{matrix}) (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) = (\begin{matrix}q & 0 \\ 0 & 0 \end{matrix}) .

Now considering the enlargement (\begin{matrix}p & 0 \\ 0 & 0 \end{matrix}) ∈ 𝕄_{2k}(A)(\overline{α}, \overline{δ})_{0} of p and (\begin{matrix}q & 0 \\ 0 & 0 \end{matrix}) ∈ 𝕄_{2k}(A)(\overline{δ}, \overline{α})_{0} of q,

we have g (\begin{matrix}p & 0 \\ 0 & 0 \end{matrix}) g^{−1} = (\begin{matrix}q & 0 \\ 0 & 0 \end{matrix}), where

g = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix}1 − p & x \\ y & 1 − q \end{matrix}) ∈ 𝕄_{2k}(A)[−\overline{δ}, −\overline{α}][−\overline{α}, −\overline{δ}].

This shows that some enlargements of p and q are conjugates. Conversely, suppose some enlargements of p and q are conjugates. Thus there is a g ∈ 𝕄_{2k}(A)[−\overline{δ}, −\overline{μ}][−\overline{α}, −\overline{β}] such that

g (\begin{matrix}p & 0 \\ 0 & 0 \end{matrix}) g^{−1} = (\begin{matrix}q & 0 \\ 0 & 0 \end{matrix}), where (\begin{matrix}p & 0 \\ 0 & 0 \end{matrix}) ∈ 𝕄_{k}(A)(α, β)_{0} and (\begin{matrix}q & 0 \\ 0 & 0 \end{matrix}) ∈ 𝕄_{k}(A)(δ, μ)_{0}. Now setting

x = g (\begin{matrix}p & 0 \\ 0 & 0 \end{matrix}) and y = g^{−1}, gives that the enlargements of p and q and consequently p and q are

graded equivalent.

The following lemma relates the graded finitely generated projective modules to homogeneous idempotent matrices which eventually leads to an equivalent definition of the graded Grothendieck group by idempotents.

Lemma 3.2.3. Let A be a Γ-graded ring.

(1) Any graded finitely generated projective A-module P gives rise to a homogeneous idempotent matrix p ∈ 𝕄_{n}(A)(\overline{α})_{0}, for some n ∈ ℕ and \overline{α} = (α_{1}, …, α_{n}), such that P ≅_{gr} pA^{n}(−\overline{α}).

(2) Any homogeneous idempotent matrix p ∈ 𝕄_{n}(A)(\overline{α})_{0} gives rise to a graded finitely generated projective A-module pA^{n}(−\overline{α}).

(3) Two homogeneous idempotent matrices are graded equivalent if and only if the corresponding graded finitely generated projective A-modules are graded isomorphic.

Proof. (1) Let P be a graded finitely generated projective (right) A-module. Then there is a graded module Q such that P ⊕ Q ≅_{gr} A^{n}(−\overline{α}) for some n ∈ ℕ and \overline{α} = (α_{1}, …, α_{n}), where α_{i} ∈ Γ (see (1.27)). Identify A^{n}(−\overline{α}) with P ⊕ Q and define the homomorphism p ∈ End_{A}(A^{n}(−\overline{α})) which sends Q to zero and acts as identity on P. Clearly, p is an idempotent and graded homomorphism of degree 0. By (1.10) p ∈ End_{A}(A^{n}(−\overline{α}))_{0}. By (1.34), the homomorphism p can be represented by a matrix p ∈ 𝕄_{n}(A)(\overline{α})_{0} acting from the left so, P ≅_{gr} pA^{n}(−\overline{α}).

(2) Let p ∈ 𝕄_{n}(A)(\overline{α})_{0}, \overline{α} = (α_{1}, …, α_{n}), where α_{i} ∈ Γ. Since for any γ ∈ Γ, pA^{n}(−\overline{α})_{γ} ⊆ A^{n}(−\overline{α})_{γ}, we have

pA^{n}(−\overline{α}) = ⨁_{γ∈Γ} pA^{n}(−\overline{α})_{γ}.

This shows that pA^{n}(−\overline{α}) is a graded finitely generated A-module. Furthermore, 1−p ∈ 𝕄_{n}(A)(\overline{α})_{0} and

A^{n}(−\overline{α}) = pA^{n}(−\overline{α}) ⨁(1 − p)A^{n}(−\overline{α}).

Thus pA^{n}(−\overline{α}) is a graded finitely generated projective A-module.

(3) Let p ∈ 𝕄_{n}(A)(\overline{α})_{0} and q ∈ 𝕄_{m}(A)(\overline{δ})_{0} be graded equivalent idempotent matrices. The first part is similar to the proof of Lemma 3.2.2. By Definition 3.2.1, there are x′ ∈ 𝕄_{k}(A)[−\overline{α}][−\overline{δ}] and y′ ∈ 𝕄_{k}(A)[−\overline{δ}][−\overline{α}] such that x′y′ = p and y′x′ = q. Let x = px′q and y = qy′p. Then xy = px′qy′p = p(x′y′)^{2}p = p and similarly yx = q. Furthermore x = px = xq and y = yp = qy. Now the left multiplication by x and y induce graded right A-homomorphisms qA^{k}(−\overline{δ}) → pA^{k}(−\overline{α}) and pA^{k}(−\overline{α}) → qA^{k}(−\overline{δ}), respectively, which are inverse of each other. Therefore pA^{k}(−\overline{α}) ≅_{gr} qA^{k}(−\overline{δ}).

On the other hand if f : pA^{k}(−\overline{α}) ≅_{gr} qA^{k}(−\overline{δ}), then extend f to A^{k}(−\overline{α}) by sending (1 − p)A^{k}(−\overline{α}) to zero and thus define a map

θ : A^{k}(−\overline{α}) = pA^{k}(−\overline{α}) ⊕ (1 − p)A^{k}(−\overline{α}) ⟶ qA^{k}(−\overline{δ}) ⊕ (1 − q)A^{k}(−\overline{δ}) = A^{k}(−\overline{δ}).

Similarly, extending f^{−1} to A^{k}(−\overline{δ}), we get a map

ϕ : A^{k}(−\overline{δ}) = qA^{k}(−\overline{δ}) ⊕ (1 − q)A^{k}(−\overline{δ}) ⟶ pA^{k}(−\overline{α}) ⊕ (1 − p)A^{k}(−\overline{α}) = A^{k}(−\overline{α})

such that θϕ = p and ϕθ = q. It follows θ ∈ 𝕄_{k}(A)[−\overline{α}][−\overline{δ}] whereas ϕ ∈ 𝕄_{k}(A)[−\overline{δ}][−\overline{α}] (see §1.3.2). This gives that p and q are graded equivalent.

For a homogeneous idempotent matrix p of degree zero, we denote the graded equivalence class

of p, by [p] (see Definition 3.2.1) and we define [p] + [q] = [ (\begin{matrix}p & 0 \\ 0 & q \end{matrix}) ]. This makes the set of

equivalence classes of homogeneous idempotent matrices of degree zero a monoid. Lemma 3.2.3 shows that this monoid is isomorphic to 𝒱^{gr}(A), via [p] ↦ [pA^{n}(−\overline{α})], where p ∈ 𝕄_{n}(A)(\overline{α})_{0} is a homogeneous matrix. Thus K_{0}^{gr}(A) can be defined as the group completion of this monoid. In fact, this is the definition we adapt for 𝒱^{gr} when the graded ring A does not have identity (see §3.4).

3.2.1

Suppose p ∈ 𝕄_{n}(A)(\overline{α})_{0} and q ∈ 𝕄_{m}(A)(\overline{δ})_{0} are homogeneous idempotent matrices of degree zero such that [p] = [q] in K_{0}^{gr}(A). As in Definition 3.2.1, we add zeros to the lower right hand corner of p and q so that p, q can be considered as matrices in 𝕄_{k}(A) for some k ∈ ℕ. Now [p] represents the isomorphism class [pA^{k}(−\overline{α})] and [q] represents [qA^{k}(−\overline{δ})] in K_{0}^{gr}(A) (see Lemma 3.2.3). Since [p] = [q], [pA^{k}(−\overline{α})] = [qA^{k}(−\overline{δ})] which by Lemma 3.1.7(3), implies pA^{k}(−\overline{α})⊕A^{n}(\overline{β}) ≅_{gr} qA^{k}(−\overline{δ})⊕

A^{n}(\overline{β}), for some n ∈ ℕ. Again by Lemma 3.2.3 this implies p ⊕ I_{n} = (\begin{matrix}p & 0 \\ 0 & I_{n} \end{matrix}) is equivalent to

q⊕I_{n} = (\begin{matrix}q & 0 \\ 0 & I_{n} \end{matrix}) , where I_{n} is the identity element of the ring 𝕄_{n}(A). (This can also be seen directly

from Definition 3.2.1.) This observation will be used in Theorem 3.2.4 and later in Lemma 5.1.5.

3.2.2 Action of Γ on idempotents

We define the action of Γ on the idempotent matrices as follows: For γ ∈ Γ and p ∈ 𝕄_{n}(A)(\overline{α})_{0}, γp is represented by the same matrix as p but considered in 𝕄_{n}(A)(\overline{α} − γ)_{0} where \overline{α} − γ = (α_{1} − γ, ⋯, α_{n} − γ). Note that if p ∈ 𝕄_{n}(A)(\overline{α})_{0} and q ∈ 𝕄_{m}(A)(\overline{δ})_{0} are equivalent, then there are x ∈ 𝕄_{k}(A)[−\overline{α}][−\overline{δ}] and y ∈ 𝕄_{k}(A)[−\overline{δ}][−\overline{α}] such that xy = p and yx = q (Definition 3.2.1). Since x ∈ 𝕄_{k}(A)[γ − \overline{α}][γ − \overline{δ}] and y ∈ 𝕄_{k}(A)[γ − \overline{δ}][γ − \overline{α}], it follows that γp is equivalent to γq. Thus K_{0}^{gr}(A) becomes a ℤ[Γ]-module with this definition.

Now a quick inspection of the proof of Lemma 3.2.3 shows that the action of Γ is compatible in both definitions of K_{0}^{gr}.

Let A and B be Γ-graded rings and ϕ : A → B be a Γ-graded homomorphism. Using the graded homomorphism ϕ, one can consider B as a graded A − B-bimodule in a natural way (§1.2.5). Furthermore, if P is a graded right A-module, then P ⊗_{A}B is a graded B-module (§1.2.6). Furthermore, if P is finitely generated projective, so is P ⊗_{A} B. Thus one can define a group homomorphism \overline{ϕ} : K_{0}^{gr}(A) → K_{0}^{gr}(B), where [P] ↦ [P ⊗_{A} B] and extended to all K_{0}^{gr}(A). On the other hand, if p ∈ 𝕄_{n}(A)(\overline{α})_{0} is an idempotent matrix over A then ϕ(p) ∈ 𝕄_{n}(B)(\overline{α})_{0} is an idempotent matrix over B obtained by applying ϕ to each entry of p. This also induces a homomorphism on the level of K_{0}^{gr} using the idempotent presentations. Since

pA^{n}(−\overline{α}) ⊗_{A} B ≅_{gr} ϕ(p)B^{n}(−\overline{α}),

we have the following commutative diagram

\xymatrix{[p] \ar[r] \ar[d] & [ϕ(p)] \ar[d] \\ [pA^{n}(−\overline{α})] \ar[r] & [pA^{n}(−\overline{α}) ⊗_{A} B].}  (3.16)

This shows that the homomorphisms induced on K_{0}^{gr} by ϕ are compatible, whether using the idempotent presentation or the module presentation for the graded Grothendieck groups.

3.2.3 K_{0}^{gr} is a continuous functor

Recall the construction of direct limit of graded rings from Example 1.1.10. We are in a position to determine their graded Grothendieck groups.

Theorem 3.2.4. Let A_{i}, i ∈ I, be a direct system of Γ-graded rings and A = \underrightarrow{lim} A_{i} be a Γ-graded ring. Then K_{0}^{gr}(A) ≅ \underrightarrow{lim} K_{0}^{gr}(A_{i}) as ℤ[Γ]-modules.

Proof. First note that K_{0}^{gr}(A_{i}), i ∈ I, is a direct system of abelian groups so \underrightarrow{lim} K_{0}^{gr}(A_{i}) exists with ℤ[Γ]-module homomorphisms ϕ_{i} : K_{0}^{gr}(A_{i}) → \underrightarrow{lim} K_{0}^{gr}(A_{i}). On the other hand, for any i ∈ I, there is a map ψ_{i} : A_{i} → A which induces \overline{ψ}_{i} : K_{0}^{gr}(A_{i}) → K_{0}^{gr}(A). Due to the universality of the direct limit, we have a ℤ[Γ]-module homomorphism ϕ : \underrightarrow{lim} K_{0}^{gr}(A_{i}) → K_{0}^{gr}(A) such that for any i ∈ I, the diagram

\xymatrix{\underrightarrow{lim} K_{0}^{gr}(A_{i}) \ar[r]^{ϕ} & K_{0}^{gr}(A) \\ K_{0}^{gr}(A_{i}) \ar[u]^{ϕ_{i}} \ar@{-->}[ur]_{_{\overline{ψ}_{i}}}}  (3.17)

is commutative. We show that ϕ is in fact an isomorphism. We use the idempotent description of K_{0}^{gr} group to show this. Note that if p is an idempotent matrix over A_{i} for some i ∈ I, which gives the element [p] ∈ K_{0}^{gr}(A_{i}), then \overline{ψ}_{i}([p]) = [ψ(p)], where ψ(p) is an idempotent matrix over A obtained by applying ψ to each entry of p.

Let p be an idempotent matrix over A = \underrightarrow{lim} A_{i}. Then p has a finite number of entires and each is coming from some A_{i}, i ∈ I. Since I is directed, there is a j ∈ I, such that p is the image of an idempotent matrix in A_{j}. Thus the class [p] in K_{0}^{gr}(A) is the image of an element of K_{0}^{gr}(A_{j}). Since the Diagram 3.17 is commutative, there is an element in \underrightarrow{lim} K_{0}(A_{i}) which maps to [p] in K_{0}^{gr}(A). Since K_{0}^{gr}(A) is generated by elements [p], this shows that ϕ is surjective. We are left to show that ϕ is injective. Suppose x ∈ \underrightarrow{lim} K_{0}^{gr}(A_{i}) such that ϕ(x) = 0. Since there is j ∈ I such that x is the image of an element of K_{0}^{gr}(A_{j}), we have [p]−[q] ∈ K_{0}^{gr}(A_{j}) such that ϕ_{j}([p]−[q]) = x, where p and q are idempotent matrices over A_{j}. Again, since the Diagram 3.17 is commutative, we have \overline{ψ}_{j}([p] − [q]) = 0. Thus [ψ_{j}(p)] = \overline{ψ}_{j}([p]) = \overline{ψ}_{j}([q]) = [ψ_{j}(q)] in K_{0}^{gr}(A). This shows that

a = (\begin{matrix}ψ_{j}(p) & 0 \\ 0 & I_{n} \end{matrix}) is equivalent to b = (\begin{matrix}ψ_{j}(q) & 0 \\ 0 & I_{n} \end{matrix}) in R (see 3.2.3). Thus there are matrices x

and y over R such that xy = a and yx = b. Since the entires of x and y are finite, one can find

k ≥ j such that x and y are images of matrices from R_{k}. Thus (\begin{matrix}ψ_{jk}(p) & 0 \\ 0 & I_{n} \end{matrix}) is equivalent to

(\begin{matrix}ψ_{jk}(q) & 0 \\ 0 & I_{n} \end{matrix}) in R_{k}. This shows that \overline{ψ}_{jk}([p]) = \overline{ψ}_{jk}([q]), i.e., the image of [p] − [q] in K_{0}^{gr}(A_{j}) is

zero. Thus x being the image of this element, is also zero. This finishes the proof.

3.2.4 Hattori-Stallings (Chern) trace map

Recall that, for a ring A, one can relate K_{0}(A) to the Hochschild homology HH_{0}(A) = A/[A, A], where [A, A] is the subgroup generated by additive commutators ab − ba, a, b ∈ A. (Or more generally, if A is a k-algebra, where k is a commutative ring, then HH_{0}(A) is a k-module.) The construction is as follows.

Let P be a finitely generated projective right A-module. Then as in the introduction of §3.2, there is an idempotent matrix p ∈ 𝕄_{n}(A) such that P ≅ pA^{n}. Define

T : 𝒱(A) ⟶ A/[A, A],
[P] ⟼ [A, A] + Tr(p),

where Tr is the trace map of the matrices. Note that if P is isomorphic to Q then the idempotent matrices associated to them, call them p and q, are equivalent, i.e, p = xy and q = yx. This shows that Tr(p)−Tr(q) ∈ [A, A], so the map T is a well-define homomorphism of groups. Since K_{0} is the universal group completion (§3.1.1), the map T induces a map on the level of K_{0}, which is called T again, i.e., T : K_{0}(A) → A/[A, A]. This map is called Hattori-Stallings trace map or Chern map.

We carry out a similar construction in the graded setting. Let A be a Γ-graded ring and P be a finitely generated graded projective A-module. Then there is a homogeneous idempotent p ∈ 𝕄_{n}(A)(\overline{α})_{0}, where α = (α_{1}, …, α_{n}), such that P ≅_{gr} pA^{n}(−\overline{α}). Note that by (1.36), Tr(p) ∈ A_{0}. Set [A, A]_{0} := A_{0} ∩ [A, A] and define

T : 𝒱^{gr}(A) ⟶ A_{0}/[A, A]_{0},
[P] ⟼ [A, A]_{0} + Tr(p).

Similar to the non-graded case, Lemma 3.2.3 applies to show that T is a well-defined homomorphism. Note that the description of action of Γ on idempotents (§3.2.2) shows that T ([P]) = T ([P (α)]) for any α ∈ Γ. Again, this map induces a group homomorphism T : K_{0}^{gr}(A) → A_{0}/[A, A]_{0}. Further, the forgetful functor U : Gr-A → Mod- A (§1.2.7) induces the right hand side of the following commutative diagram, whereas the left hand side is the natural map induces by inclusion A_{0} ⊆ A,

\xymatrix{K_{0}^{gr}(A) \ar[r]^{T} \ar[d]_{U} & A_{0}/[A, A]_{0} \ar[d] \\ K_{0}(A) \ar[r]^{T} & A/[A, A].}

3.3 Relative K_{0}^{gr}-group

Let A be a ring and I be a two sided ideal of A. The canonical epimorphism f : A → A/I induces a homomorphism on the level of K-groups. For example on the level of Grothendieck groups, we have \overline{f} : K_{0}(A) → K_{0}(A/I), \overline{f}([P] − [Q]) = [P/P I] − [Q/QI]. In order to complete this into a long exact sequence, one needs to introduce the relative K-groups, K_{i}(A, I), i ≥ 0. This is done for lower K-groups by Bass and Milnor and the following long exact sequence has been established (see [82, Theorem 4.3.1]),

K_{2}(A) → K_{2}(A/I) → K_{1}(A, I) → K_{1}(A) → K_{1}(A/I) → K_{0}(A, I) → K_{0}(A) → K_{0}(A/I).

In this section, we define the relative K_{0}^{gr} group and establish the sequence only on the level of graded Grothendieck groups. As we will see this requires a careful arrangements of the degrees of the matrices.

Let A be a Γ graded ring and I be a graded two sided ideal of A. Define the graded double ring of A along I

D^{Γ}(A, I) = {(a, b) ∈ A × A | a − b ∈ I}.

One can check that D^{Γ}(A, I) is a Γ-graded ring with

D^{Γ}(A, I)_{γ} = {(a, b) ∈ A_{γ} × A_{γ} | a − b ∈ I_{γ}},

for any γ ∈ Γ.

Let π_{1}, π_{2} : D^{Γ}(A, I) → A, i = 1, 2, be the projections of the first and second components of D^{Γ}(A, I) to A, respectively. The maps π_{i}, i = 1, 2, are Γ-graded ring homomorphisms and induce ℤ[Γ]-module homomorphisms \overline{π}_{1}, \overline{π}_{2} : K_{0}^{gr}(D^{Γ}(A, I)) → K_{0}^{gr}(A), i = 1, 2. Define the relative graded Grothendieck group of the ring A with respect to I as

K_{0}^{gr}(A, I) := ker (K_{0}^{gr}(D^{Γ}(A, I)) ⟶^{\overline{π}_{1}} K_{0}^{gr}(A)).  (3.18)

The restriction of \overline{π}_{2} to K_{0}^{gr}(A, I) gives a ℤ[Γ]-module homomorphism \overline{π}_{2} : K_{0}^{gr}(A, I) → K_{0}^{gr}(A). The following theorem relates these groups.

Theorem 3.3.1. Let A be a Γ-graded ring and I be a graded two sided ideal of A. Then there is an exact sequence of ℤ[Γ]-modules

K_{0}^{gr}(A, I) ⟶^{\overline{π}_{2}} K_{0}^{gr}(A) ⟶^{\overline{f}} K_{0}^{gr}(A/I).

Proof. We will use the presentation of K_{0}^{gr} by idempotents (§3.2) to prove the theorem. If p is a matrix over the ring A, we will denote by \overline{p} the image of p under the canonical graded homomorphism f : A → A/I. Let [p] − [q] ∈ K_{0}^{gr}(A, I). By the construction of D^{Γ}(A, I), p = (p_{1}, p_{2}), where p_{1}, p_{2} are homogeneous idempotent matrices of A such that p_{1} − p_{2} is a matrix over I (in fact over I_{0}), namely \overline{p}_{1} = \overline{p}_{2}. Similarly q = (q_{1}, q_{2}), where q_{1}, q_{2} are homogeneous idempotent matrices and \overline{q}_{1} = \overline{q}_{2}. Furthermore, since [(p_{1}, p_{2})] − [(q_{1}, q_{2})] ∈ K_{0}^{gr}(A, I), by the definition of the relative K-group, [p_{1}] − [q_{1}] = 0. So

\overline{f}([p_{1}] − [q_{1}]) = [\overline{p}_{1}] − [\overline{q}_{1}] = 0  (3.19)

Taking into account that \overline{p}_{1} = \overline{p}_{2} and \overline{q}_{1} = \overline{q}_{2} we have

\overline{f}\overline{π}_{2}([p] − [q]) = \overline{f}\overline{π}_{2}([(p_{1}, p_{2})] − [(q_{1}, q_{2})]) = [\overline{p}_{2}] − [\overline{q}_{2}] = [\overline{p}_{1}] − [\overline{q}_{1}] = 0.

This shows that Im(\overline{π}_{2}) ⊆ ker \overline{f}.

Next we show that ker \overline{f} ⊆ Im(\overline{π}_{2}). Let p ∈ 𝕄_{n}(A)(\overline{α})_{0} and q ∈ 𝕄_{m}(A)(\overline{δ})_{0} be idempotent matrices with \overline{α} = (α_{1}, …, α_{n}) and \overline{δ} = (δ_{1}, …, δ_{m}), where α_{i}, δ_{j} ∈ Γ. Suppose x = [p] − [q] ∈ K_{0}^{gr}(A) and \overline{f}(x) = \overline{f}([p] − [q]) = [\overline{p}] − [\overline{q}] = 0. Since [\overline{p}] = [\overline{q}] in K_{0}^{gr}(A/I), there is an l ∈ ℕ, such that \overline{p} ⊕ 1_{l} is graded equivalent to \overline{q} ⊕ 1_{l} in A/I (see §3.2.1). We can replace p by p ⊕ 1_{l} and q by q ⊕ 1_{l} without changing x, and consequently we get that \overline{p} is graded equivalent to \overline{q}. By Lemma 3.2.2 there is an invertible matrix g ∈ 𝕄_{2k}(A/I)[−\overline{δ}][−\overline{α}] such that

\overline{q} = g\overline{p}g^{−1}.  (3.20)

The following standard trick let us lift an invertible matrix over A/I to an invertible matrix over A. Consider the invertible matrix

(\begin{matrix}g & 1 \\ 0 & g^{−1} \end{matrix}) ∈ 𝕄_{4k}(A/I)[−\overline{δ}, −\overline{α}][−\overline{α}, −\overline{δ}].  (3.21)

This matrix is a product of the following matrices, each can be lifted from A/I to an invertible matrix over A with the same shifting.

(\begin{matrix}1 & g \\ 0 & 1 \end{matrix}) ∈ 𝕄_{4k}(A/I)[−\overline{δ}, −\overline{α}][−\overline{δ}, −\overline{α}],  (\begin{matrix}1 & 0 \\ −g^{−1} & 1 \end{matrix}) ∈ 𝕄_{4k}(A/I)[−\overline{δ}, −\overline{α}][−\overline{δ}, −\overline{α}],
(\begin{matrix}1 & g \\ 0 & 1 \end{matrix}) ∈ 𝕄_{4k}(A/I)[−\overline{δ}, −\overline{α}][−\overline{δ}, −\overline{α}],  (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) ∈ 𝕄_{4k}(A/I)[−\overline{δ}, −\overline{α}][−\overline{α}, −\overline{δ}],

and

(\begin{matrix}g & 1 \\ 0 & g^{−1} \end{matrix}) = (\begin{matrix}1 & g \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 \\ −g^{−1} & 1 \end{matrix}) (\begin{matrix}1 & g \\ 0 & 1 \end{matrix}) (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) .

Let h ∈ 𝕄_{4k}(A)[−\overline{δ}, −\overline{α}][−\overline{α}, −\overline{δ}] be a matrix that lifts (3.21) with h^{−1} ∈ 𝕄_{4k}(A)[−\overline{α}, −\overline{δ}][−\overline{δ}, −\overline{α}].

Consider the enlargements of p and q as (\begin{matrix}p & 0 \\ 0 & 0 \end{matrix}) ∈ 𝕄_{4k}(A)(\overline{α}, \overline{δ})_{0} and (\begin{matrix}q & 0 \\ 0 & 0 \end{matrix}) ∈ 𝕄_{4k}(A)(\overline{δ}, α)_{0}.

By Lemma 3.2.2, h (\begin{matrix}p & 0 \\ 0 & 0 \end{matrix}) h^{−1} and (\begin{matrix}q & 0 \\ 0 & 0 \end{matrix}) are graded equivalent to p and q, respectively, so

replacing them does not change x. Replacing p and q with the new representatives, from (3.20), it follows \overline{p} = \overline{q}. This means p − q is a matrix over I, so (p, q) is an idempotent matrix over D^{Γ}(A, I). Now [(p, p)] − [(p, q)] ∈ K_{0}^{gr}(D^{Γ}(A, I)) which maps to x under \overline{π}_{2}. This completes the proof.

3.4 K_{0}^{gr} of non-unital rings

Let A be a ring which does not necessarily have identity. Let R be a ring with identity such that A is a two sided ideal of R. Then 𝒱(A) is defined as

𝒱(A) := ker (𝒱(R) ⟶ 𝒱(R/A)).

It is easy to see that

𝒱(A) = {[P] | P is a finitely generated projective R-module and P A = P },

where [P] is the class of R-modules isomorphic to P and addition is defined via direct sum as before (compare this with (3.1)). One can show that this definition is independent of the choice of the ring R by interpreting P as an idempotent matrix of 𝕄_{k}(A) for a suitable k ∈ ℕ.

In order to define K_{0}(A) for a non-unital ring A, consider the unitization ring Ã = ℤ × A. The addition is component-wise and multiplication is defined as follows

(n, a)(m, b) = (nm, ma + nb + ab),  (3.22)

where m, n ∈ ℤ and a, b ∈ A. This is a ring with (1, 0) as the identity element and A a two sided ideal of Ã such that Ã/A ≅ ℤ. The canonical epimorphism Ã → Ã/A gives a natural homomorphism on the level of K_{0} and then K_{0}(A) is defined as

K_{0}(A) := ker (K_{0}(Ã) ⟶ K_{0}(Ã/A)).  (3.23)

This construction extends the functor K_{0} from the category of rings with identity, to the category of rings (not necessarily with identity) to the category of abelian groups.

Remark 3.4.1. Notice that, for a ring A without unit, we didn’t define K_{0}(A) as the group completion of 𝒱(A) (as defined in the unital case). This is because the group completion is not in general a left exact functor. For a non-unital ring A, let R be a ring with identity containing A as a two sided ideal. Then we have the following exact sequence of K-theory:

K_{1}(R) ⟶ K_{1}(R/A) ⟶ K_{0}(A) ⟶ K_{0}(R) ⟶ K_{0}(R/A),

(see §3.3 and [82, Theorem 2.5.4]). There is yet another construction, the relative Grothendieck group of R and A, K_{0}(R, A), which one can show is isomorphic to K_{0}(A) (see [9, §7] for comparisons between the groups K_{0}(A) and 𝒱(A)^{+} in the non-graded setting).

A similar construction can be carried over to the graded setting as follows. Let A be a Γ-graded ring which does not necessarily have identity (Remark 1.1.15). Let R be a Γ-graded ring with identity such that A is a graded two-sided ideal of R. For example, consider Ã = ℤ × A with multiplication given by (3.22). Furthermore, Ã = ℤ × A is a Γ-graded with

Ã_{0} = ℤ × A_{0},  (3.24)
Ã_{γ} = 0 × A_{γ}, for γ ≠ 0.

Define

𝒱^{gr}(A) = {[P] | P is a graded finitely generated projective R-module and P A = P }, where [P] is the class of graded R-modules, graded isomorphic to P and addition is defined via direct sum. Parallel to the non-graded setting we define

𝒱^{gr}(A) := ker (𝒱^{gr}(R) ⟶ 𝒱^{gr}(R/A)).  (3.25)

As in the proof of Lemma 3.2.3(1), since P ⊕ Q ≅ R^{n}(−\overline{α}), this gives an idempotent matrix p in 𝕄_{n}(R)(\overline{α})_{0}. However, since P A = P, we have P ⊕ QA = P A ⊕ QA ≅ A^{n}(−\overline{α}). This shows that p ∈ 𝕄_{n}(A)(\overline{α})_{0}. On the other hand, if p ∈ 𝕄_{n}(A)(\overline{α})_{0} ⊆ 𝕄_{n}(R)(\overline{α})_{0} is an idempotent, then pR^{n}(−\overline{α}) is a graded finitely generated projective R-module such that

pR^{n}(−\overline{α})A = pA^{n}(−\overline{α}) = pR^{n}(−\overline{α}).

So [pR^{n}(−\overline{α})] ∈ 𝒱^{gr}(A). Now a repeat of Lemma 3.2.3 shows that 𝒱^{gr}(A) is isomorphic to the monoid of equivalence classes of graded idempotent matrices over A as in §3.2. This shows that the construction of 𝒱^{gr}(A) is independent of the choice of the graded ring R. It also shows that if A has identity, the two constructions (using the graded projective A-modules versus the graded projective R-modules) coincide.

To define the graded Grothendieck group for non-unital rings, we use a similar approach as in (3.23): Let A be a ring (without identity) and let Ã = ℤ × A with the multiplication defined in (3.22), so that A is a graded two sided ideal of Ã. The graded canonical epimorphism Ã → Ã/A gives a natural homomorphism on the level of K_{0}^{gr} and then K_{0}^{gr}(A) is defined as

K_{0}^{gr}(A) := ker (K_{0}^{gr}(Ã) ⟶ K_{0}^{gr}(Ã/A)).  (3.26)

Thus K_{0}^{gr}(A) is a ℤ[Γ]-module. Since the graded homomorphism ϕ : Ã → Ã/A ≅ ℤ, (n, a) ↦ n splits, we obtain the split exact sequence

0 ⟶ K_{0}^{gr}(A) ⟶ K_{0}^{gr}(ℤ × A) ⟶ K_{0}^{gr}(ℤ) ⟶ 0.

Thus

K_{0}^{gr}(ℤ × A) ≅ K_{0}^{gr}(A) ⊕ ℤ[x, x^{−1}].

Interpreting this in the language of the reduced K_{0}^{gr} (§3.1.4), we have

K̃_{0}^{gr}(Ã) = K_{0}^{gr}(A).

This construction extends the functor K_{0}^{gr} from the category of graded rings with identity, to the category of graded rings (not necessarily with identity) to the category of ℤ[Γ]-module.

In Example 3.6.9, we calculate the graded Grothendieck group of the non-unital ring of (countable) square matrices with finite number of nonzero entires using the idempotent representation.

Remark 3.4.2. If Γ-graded ring A has identity, then the ring Ã = ℤ × A defined by the multiplication (3.22) is graded isomorphic to the cartesian product ring ℤ × A, where ℤ is a Γ-graded ring concentrated in degree zero (see Example 1.1.11). Indeed the map

Ã ⟶ ℤ × A
(n, a) ⟼ (n, a + n1_{A})

is a graded ring isomorphism of unital rings. This shows that if A has an identity, the two definitions of K_{0}^{gr} for A coincides. Note also that the unitization ring Ã never is a strongly graded ring.

Similar to the non-graded setting (see from example [82, Theorem 1.5.9]), one can prove that for a graded ideal A of the graded ring R, K_{0}^{gr}(R, A) ≅ K_{0}^{gr}(A) as a ℤ[Γ]-modules. This shows that K_{0}^{gr}(R, A) depends only on the structure of the non-unital ring A.

3.4.1 Graded inner automorphims

Let A be a graded ring and f : A → A an inner-automorphism defined by f(a) = rar^{−1}, where r is a homogeneous and invertible element of A of degree δ. Clearly f is a graded automorphism. Then considering A as a graded left A-module via f, it is easy to observe, for any graded right A-module P, that there is a right graded A-module isomorphism

P (−δ) ⟶ P ⊗_{f} A,
p ⟼ p ⊗ r,

(with the inverse p ⊗ a ↦ pr^{−1}a). This induces an isomorphism between the functors − ⊗_{f} A : Pgr-A → Pgr-A and δ-suspension functor 𝒯_{δ} : Pgr-A → Pgr-A. Recall that a Quillen’s K_{i}-group, i ≥ 0, is a functor from the category of exact categories with exact functors to the category of abelian groups (see §6). Furthermore, isomorphic functors induce the same map on the K-groups [79, p.19]. Thus K_{i}^{gr}(f) = K_{i}^{gr}(𝒯_{δ}). Therefore if r is a homogeneous element of degree zero, i.e., δ = 0, then K_{i}^{gr}(f) : K_{i}^{gr}(A) → K_{i}^{gr}(A) is the identity map.

3.5 K_{0}^{gr} is a pre-ordered module

3.5.1 Γ-pre-ordered modules

An abelian group G is called a pre-ordered abelian group if there is a relation, denoted by ≥, on G which is reflexive and transitive and it respects the group structure. It follows that the set G_{+} := {x ∈ G | x ≥ 0} forms a monoid. Conversely, any monoid C in G, induces a pre-ordering on G, by defining x ≥ y if x − y ∈ C. It follows that with this pre-ordering G_{+} = C which is called the cone of the ordering. (Note that G_{+} should not be confused by G^{+} used for the group completion in §3.1.)

Example 3.5.1. Let V be a monoid and V^{+} its completion (see §3.1). There exist a natural homomorphism ϕ : V → V^{+} (see (3.4)), which makes V^{+} a pre-ordered abelian group with the image of V under this homomorphism as a cone of V^{+}.

Let G has a pre-ordering. An element u ∈ G is called an order-unit if u ≥ 0 and for any x ∈ G, there is n ∈ ℕ, such that nu ≥ x.

For a ring R, by Example 3.5.1, the Grothendieck group K_{0}(R) is a pre-ordered abelian group. Concretely, Consider the set of isomorphism classes of finitely generated projective R-modules in K_{0}(R). This set forms a monoid and thus induces an ordering on K_{0}(R). We check that with this pre-ordering [R] is an order-unit. Let u ∈ K_{0}(R). Then u = [P] − [Q], where P, Q are finitely generated projective R-modules. But there is a finitely generated projective module P′ such that P ⊕ P′ ≅ R^{n} as a right R-module, for some n ∈ ℕ. Then n[R] ≥ u. Indeed,

n[R] − u = n[R] − [P] + [Q] = [R^{n}] − [P] + [Q] = [P ⊕ P′] − [P] + [Q] = [P′] + [Q] ∈ K_{0}(R)_{+}.

Example 3.5.2. Suppose R is a ring such that K_{0}(R) ≠ 0, but the order unit [R] = 0. It follows that all elements of K_{0}(R) are either zero or negative. Furthermore, it is possible that there is an element x ∈ K_{0}(R) such x > 0 and 0 > x simultaneously. One instance of such a ring is constructed in Example 3.8.3

The Grothendieck group of a ring as a pre-ordered group is studied extensively in [38, §15]. In particular it was established that for the so called ultramatricial algebras R, the abelian group K_{0}(R) along with its positive cone and the order-unit [R] is a complete invariant (see [38, Theorem 15.26] and the introduction to §5). This invariant is also called the dimension group in the literature as it coincides with an invariant called the dimension group by Elliot [34] to classify such algebras.

Since we will consider the graded Grothendieck groups, which have an extra ℤ[Γ]-module structure, we need to adopt the above definitions on ordering to the graded setting. For this reason, here we define the category of Γ-pre-ordered modules.

Let Γ be a group and G be a (left) Γ-module. Let ≥ be a reflexive and transitive relation on G which respects the monoid and the module structures, i.e., for γ ∈ Γ and x, y, z ∈ G, if x ≥ y, then x + z ≥ y + z and γx ≥ γy. We call G a Γ-pre-ordered module. We call G a pre-ordered module when Γ is clear from the context. The cone of G is defined as {x ∈ G | x ≥ 0} and denoted by G_{+}. The set G_{+} is a Γ-submonoid of G, i.e., a submonoid which is closed under the action of Γ. In fact, G is a Γ-pre-ordered module if and only if there exists a Γ-submonoid of G. (Since G is a Γ-module, it can be considered as a ℤ[Γ]-module.) An element u ∈ G_{+} is called an order-unit if for any x ∈ G, there are α_{1}, …, α_{n} ∈ Γ, n ∈ ℕ, such that

∑_{i=1}^{n} α_{i}u ≥ x.  (3.27)

As usual, in this setting, we only consider homomorphisms which preserve the pre-ordering, i.e., a Γ-homomorphism f : G → H, such that f(G_{+}) ⊆ H_{+}. We denote by 𝒫_{Γ} the category of pointed Γ-pre-ordered modules with pointed ordered preserving homomorphisms, i.e., the objects are the pairs (G, u), where G is a Γ-pre-ordered module and u is an order-unit, and f : (G, u) → (H, v) is an ordered-preserving Γ-homomorphism such that f(u) = v. Note that when Γ is a trivial group, we are in the classical setting of pre-ordered abelian groups. When Γ = ℤ and so ℤ[Γ] = ℤ[x, x^{−1}], we simply write 𝒫 for 𝒫_{ℤ}.

Example 3.5.3. \underline{K_{0}^{gr}(A) as a Γ-pre-ordered module}

Let A be a Γ-graded ring. Then K_{0}^{gr}(A) is a Γ-pre-ordered module with the set of isomorphic classes of graded finitely generated projective right A-modules as the cone of ordering (i.e., the image of 𝒱^{gr}(A) under the natural homomorphism 𝒱^{gr}(A) → K_{0}^{gr}(A)) and [A] as an order-unit. Indeed, if x ∈ K_{0}^{gr}(A), then by Lemma 3.1.7(1) there are graded finitely generated projective modules P and P′ such that x = [P] − [P′]. But there is a graded module Q such that P ⊕ Q ≅ A^{n}(\overline{α}), where \overline{α} = (α_{1}, …, α_{n}), α_{i} ∈ Γ (see (1.27)). Now

[A^{n}(\overline{α})] − x = [P] + [Q] − [P] + [P′] = [Q] + [P′] = [Q ⊕ P′] ∈ K_{0}^{gr}(A)_{+}.

This shows that ∑_{i=1}^{n} α_{i}[A] = [A^{n}(\overline{α})] ≥ x.

In §5 we will show that for the so called graded ultramatricial algebras R, the ℤ[Γ]-module K_{0}^{gr}(R) along with its positive cone and the order-unit [R] is a complete invariant (see Theorem 5.2.5). We call this invariant the graded dimension module.

Theorem 3.5.4. Let A and B be Γ-graded rings. If A is graded Morita equivalent to B, then there is an order preserving ℤ[Γ]-module isomorphism

K_{0}^{gr}(A) ≅ K_{0}^{gr}(B).

Proof. By Theorem 2.3.5, there is a graded A − B-bimodule Q such that the functor − ⊗_{A} Q : Gr-A → Gr-B is a graded equivalence. The restriction to Pgr-B induces a graded equivalence − ⊗_{A} Q : Pgr-A → Pgr-B. This in return induces an order preserving ℤ[Γ]-module isomorphism ϕ : K_{0}^{gr}(A) → K_{0}^{gr}(B)

If Γ-graded rings A and B are graded Morita equivalent, i.e., Gr-A ≈ Gr-B, then by Theorem 2.3.5, Mod- A ≈ Mod- B and thus not only by Theorem 3.5.4, K_{0}^{gr}(A) ≅ K_{0}^{gr}(B), but also K_{0}(A) ≅ K_{0}(B) as well.

In a more specific case, we have the following.

Lemma 3.5.5. Let A be a Γ-graded ring and P be a A-graded progenerator. Let B = End_{A}(P). Then [P] is an order-unit in K_{0}^{gr}(A) and there is an order preserving ℤ[Γ]-module isomorphism

(K_{0}^{gr}(B), [B]) ≅ (K_{0}^{gr}(A), [P]).

Proof. Since P is a graded generator, by Theorem 2.2.2, there are α_{i} ∈ Γ, 1 ≤ i ≤ n, such that [A] ≤ ∑_{i} α_{i}[P]. Since [A] is an order-unit, this immediately implies that [P] is an order-unit. By (2.5), the functor − ⊗_{B} P : Gr-B → Gr-A is a graded equivalence. The restriction to Pgr-B induces a graded equivalence − ⊗_{B} P : Pgr-B → Pgr-A. This in return induces an order preserving ℤ[Γ]-module isomorphism ϕ : K_{0}^{gr}(B) → K_{0}^{gr}(A), ϕ([Q] − [L]) = [Q ⊗_{B} P] − [L ⊗_{B} P], where Q and L are graded finitely generated projective B-modules. In particular ϕ([B]) = [P]. This completes the proof.

3.6 K_{0}^{gr} of graded division rings

In the case of graded division rings, one can compute the graded Grothendieck group completely, using the description of graded free modules.

Proposition 3.6.1. Let A be a Γ-graded division ring with the support the subgroup Γ_{A}. Then the monoid of isomorphism classes of Γ-graded finitely generated projective A-modules is isomorphic to ℕ[Γ/Γ_{A}] and K_{0}^{gr}(A) ≅ ℤ[Γ/Γ_{A}] as ℤ[Γ]-modules. Furthermore, [A^{n}(δ_{1}, …, δ_{n})] ∈ K_{0}^{gr}(A) corresponds to ∑_{i=1}^{n} \underline{δ}_{i} in ℤ[Γ/Γ_{A}], where \underline{δ}_{i} = Γ_{A} + δ_{i}. In particular if A is a (trivially graded) division ring, Γ a group and A considered as a Γ-graded division ring concentrated in degree zero, then K_{0}^{gr}(A) ≅ ℤ[Γ] and [A^{n}(δ_{1}, …, δ_{n})] ∈ K_{0}^{gr}(A) corresponds to ∑_{i=1}^{n} δ_{i} in ℤ[Γ].

Proof. By Proposition 1.3.10, A(δ_{1}) ≅_{gr} A(δ_{2}) as graded A-module if and only if δ_{1} −δ_{2} ∈ Γ_{A}. Thus any graded free module of rank 1 is graded isomorphic to some A(δ_{i}), where {δ_{i}}_{i∈I} is a complete set of coset representative of the subgroup Γ_{A} in Γ, i.e., {Γ_{A} + δ_{i}, i ∈ I} represents Γ/Γ_{A}. Since any graded finitely generated module M over A is graded free (see §1.1), it follows that

M ≅_{gr} A(δ_{i_{1}})^{r_{1}} ⊕ ⋯ ⊕ A(δ_{i_{k}})^{r_{k}},  (3.28)

where δ_{i_{1}}, … δ_{i_{k}} are distinct elements of the coset representative. Now suppose

M ≅_{gr} A(δ_{i′}_{1})^{s_{1}} ⊕ ⋯ ⊕ A(δ_{i′}_{k′})^{s_{k′}} .  (3.29)

Considering the A_{0}-module M_{−δ_{i_{1}}} , from (3.28) we have M_{−δ_{i_{1}}} = A_{0}^{r_{1}}. This implies one of δ_{i′}_{j}, 1 ≤ j ≤ k′, say, δ_{i′}_{1}, has to be δ_{i_{1}} and so r_{1} = s_{1} as A_{0} is a division ring. Repeating the same argument for each δ_{i_{j}}, 1 ≤ j ≤ k, we see k = k′, δ_{i′}_{j} = δ_{i_{j}} and r_{j} = s_{j}, for all 1 ≤ j ≤ k (possibly after suitable permutation). Thus any graded finitely generated projective A-module can be written uniquely as M ≅_{gr} A(δ_{i_{1}})^{r_{1}} ⊕ ⋯ ⊕ A(δ_{i_{k}})^{r_{k}}, where δ_{i_{1}}, … δ_{i_{k}} are distinct elements of the coset representative. The (well-defined) map

𝒱^{gr}(A) → ℕ[Γ/Γ_{A}]  (3.30)
[A(δ_{i_{1}})^{r_{1}} ⊕ ⋯ ⊕ A(δ_{i_{k}})^{r_{k}}] ↦ r_{1}(Γ_{A} + δ_{i_{1}}) + ⋯ + r_{k}(Γ_{A} + δ_{i_{k}}),

gives a ℕ[Γ]-monoid isomorphism between the monoid of isomorphism classes of Γ-graded finitely generated projective A-modules 𝒱^{gr}(A) and ℕ[Γ/Γ_{A}]. The rest of the proof follows easily.

Remark 3.6.2. One can use Equation 3.9 to calculate the graded K-theory of division algebras as well.

Example 3.6.3. Using Proposition 3.6.1, we calculate the graded K_{0} of two types of graded fields and we determine the action of ℤ[x, x^{−1}] on these groups. These are graded fields obtained from Leavitt path algebras of acyclic and C_{n}-comet graphs, respectively (see Theorem 1.6.15 and 1.6.17).

1. Let K be a field. Consider A = K as a ℤ-graded field with the support Γ_{A} = 0, i.e., A is concentrated in degree 0. By Proposition 3.6.1, K_{0}^{gr}(A) ≅ ℤ[x, x^{−1}] as a ℤ[x, x^{−1}]-module. With this presentation [A(i)] corresponds to x^{i} in ℤ[x, x^{−1}].

2. Let A = K[x^{n}, x^{−n}] be a ℤ-graded field with Γ_{A} = nℤ. By Proposition 3.6.1,

K_{0}^{gr}(A) ≅ ℤ[ℤ/nℤ] ≅ ⨁_{n} ℤ,

is a ℤ[x, x^{−1}]-module. The action of x on (a_{1}, …, a_{n}) ∈ ⨁_{n} ℤ is

x(a_{1}, …, a_{n}) = (a_{n}, a_{1}, …, a_{n−1}).

With this presentation [A] corresponds to (1, 0, …, 0) in ⨁_{n} ℤ. Furthermore, the map

U : K_{0}^{gr}(A) → K_{0}(A)

induced by the forgetful functor (§1.2.7), gives a group homomorphism

U : ⨁_{n} ℤ ⟶ ℤ,
(a_{i}) ⟼ ∑_{i} a_{i}.

In fact, the sequence

⨁_{n} ℤ ⟶^{f} ⨁_{n} ℤ ⟶^{U} ℤ ⟶ 0,

where f(a_{1}, …, a_{n}) = (a_{n} − a_{1}, a_{1} − a_{2}, …, a_{n−1} − a_{n}) is exact. In §6.3 we systematically

related K_{0}^{gr} to K_{0} for certain rings.

Example 3.6.4. Consider the Hamilton quaternion algebra ℍ = ℝ ⊕ ℝi ⊕ ℝj ⊕ ℝk. By Example 1.1.21, ℍ is a ℤ_{2}×ℤ_{2}-graded division ring with Γ_{ℍ} = ℤ_{2}×ℤ_{2}. By Proposition 3.6.1, K_{0}^{gr}(ℍ) ≅ ℤ. In fact since ℍ is a strongly ℤ_{2}×ℤ_{2}-graded division ring, one can deduce the result using the Dade’s theorem (see §3.1.3), i.e., K_{0}^{gr}(ℍ) ≅ K_{0}(ℍ_{0}) = K_{0}(ℝ) ≅ ℤ.

Example 3.6.5. Let (D, v) be a valued division algebra, where v : D^{∗} → Γ is the valuation homomorphism. By Example 1.4.7, there is a Γ-graded division algebra gr(D) associated to D, where Γ_{gr(D)} = Γ_{D}. By Proposition 3.6.1,

K_{0}^{gr}(gr(D)) ≅ Γ/Γ_{D}.

The following example generalises Example 3.6.4 of the Hamilton quaternion algebra ℍ as a ℤ_{2} × ℤ_{2}-graded ring.

Remark 3.6.6. We saw in Example 1.1.21 that ℍ can also be considered as a ℤ_{2}-graded division ring. So ℍ is also strongly ℤ_{2}-graded, and K_{0}^{gr}(ℍ) ≅ K_{0}(ℍ_{0}) = K_{0}(ℂ) ≅ ℤ. Then Z(ℍ) = ℝ, which we can consider as a trivially ℤ_{2}-graded field, so by Proposition 3.6.1, K_{0}^{gr}(ℝ) = ℤ ⊕ ℤ. We note that for both grade groups, ℤ_{2} and ℤ_{2} × ℤ_{2}, we have K_{0}^{gr}(ℍ) ≅ ℤ, but the K_{0}^{gr}(ℝ) are different. So the graded K-theory of a graded ring depends not only on the ring, but also on its grade group.

Example 3.6.7. Let K be a field and let R = K[x^{2}, x^{−2}]. Then R is a ℤ-graded field, with the support 2ℤ, where R can be written as R = ⨁_{n∈ℤ} R_{n}, with R_{n} = Kx^{n} if n is even and R_{n} = 0 if n is odd. Consider the shifted graded matrix ring A = 𝕄_{3}(R)(0, 1, 1), which has support ℤ. Then we will show that A is a graded central simple algebra over R.

It is clear that the centre of A is R, and A is finite dimensional over R. Recall that a graded ideal is generated by homogeneous elements. If J is a non-zero graded ideal of A, then using the elementary matrices, we can show that J = A (see [52, Ex. III.2.9]), so A is graded simple.

By Theorem 1.6.17, A is the Leavitt path algebra of the following graph. Thus by Theorem 1.6.12 A is a strongly ℤ-graded ring.

\xymatrix{E : • \ar[r] & • \ar[r] & • \ar[l]}

Here, we also show that A is a strongly ℤ-graded ring by checking the conditions of Proposition 1.1.16. Since ℤ is finitely generated, it is sufficient to show that I_{3} ∈ A_{1}A_{−1} and I_{3} ∈ A_{−1}A_{1}. But

I_{3} = (\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})(\begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) +(\begin{matrix}0 & 0 & 0 \\ x^{2} & 0 & 0 \\ 0 & 0 & 0 \end{matrix})(\begin{matrix}0 & x^{−2} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})+(\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ x^{2} & 0 & 0 \end{matrix})(\begin{matrix}0 & 0 & x^{−2} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})

and

I_{3} =(\begin{matrix}0 & x^{−2} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})(\begin{matrix}0 & 0 & 0 \\ x^{2} & 0 & 0 \\ 0 & 0 & 0 \end{matrix})+(\begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})(\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})+(\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix})(\begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) .

As in the previous examples, using the Dade’s theorem (see §3.1.3), we have K_{0}^{gr}(A) ≅ K_{0}(A_{0}). Since R_{0} = K, by (1.36) there is a ring isomorphism

A_{0} = (\begin{matrix}R_{0} & R_{1} & R_{1} \\ R_{−1} & R_{0} & R_{0} \\ R_{−1} & R_{0} & R_{0} \end{matrix}) = (\begin{matrix}K & 0 & 0 \\ 0 & K & K \\ 0 & K & K \end{matrix}) ≅ K × 𝕄_{2}(K).

Then

K_{0}^{gr}(A) ≅ K_{0}(A_{0}) ≅ K_{0}(K) ⊕ K_{0}( 𝕄_{2}(K)) ≅ ℤ ⊕ ℤ,

since K_{0} respects Cartesian products and Morita equivalence. Note that

K_{0}(A) = K_{0}( 𝕄_{3}(R)(0, 1, 1)) ≅ K_{0}(R) = K_{0}(K[x^{2}, x^{−2}]) ≅ ℤ.

The isomorphism K_{0}(K[x^{2}, x^{−2}]) ≅ ℤ comes from the fundamental theorem of algebraic K-theory [82, Thm. 3.3.3] (see also [66, p. 484]), K_{0}(K[x, x^{−1}]) ≅ K_{0}(K) ≅ ℤ, and that K[x^{2}, x^{−2}] ≅ K[x, x^{−1}] as rings. So the K-theory of A is isomorphic to one copy of ℤ, which is not the same as the graded K-theory of A.

Example 3.6.8. \underline{Reduced K_{0}^{gr} of graded central simple algebras}

Recall from §3.1.4, that for a Γ-graded ring A, K̃_{0}^{gr} is the cokernel of the homomorphism

ϕ : ℤ[Γ] ⟶ K_{0}^{gr}(A),
∑_{α} n_{α}α ⟼ ∑_{α} n_{α}[A(α)].

Let A be a Γ-graded division ring. We calculate K̃_{0}^{gr}(𝕄_{n}(A)), where 𝕄_{n}(A) is a Γ-graded ring (with no shifting). The ℤ[Γ]-module homomorphism ϕ above takes the form

ℤ[Γ] ⟶ K_{0}^{gr}(𝕄_{n}(A)) ⟶^{≅} K_{0}^{gr}(A) ⟶^{≅} ℤ[Γ/Γ_{A}],
α ⟼ [𝕄_{n}(A)(α)] ⟼ ⨁_{n}[A(α)] ⟼ n(Γ_{A} + α),

where the second map is induced by the Morita theory (see Proposition 2.1.1) and the third map is induced by Proposition 3.6.1. The cokernel of the composition maps gives K̃_{0}^{gr}(𝕄_{n}(A)), which one can immediately calculate

K̃_{0}^{gr}(𝕄_{n}(A)) ≅ \frac{ℤ}{nℤ}[\frac{Γ}{Γ_{A}} ].

In particular, for a graded division ring A, we have

K̃_{0}^{gr}(𝕄_{n}(A)) ≅ 0.

Example 3.6.9. Let K be a field. Consider the following sequence of graded matrix rings

K ⊆ 𝕄_{2}(K)(0, 1) ⊆ 𝕄_{3}(K)(0, 1, 2) ⊆ 𝕄_{4}(K)(0, 1, 2, 3) ⊆ ⋯  (3.31)

where the inclusion comes from the non-unital (graded) ring homomorphism of (3.15). Let R be the graded ring

R = ⋃_{i=1}^{∞} 𝕄_{i}(K)(\overline{α}_{i}),

where \overline{α}_{i} = (0, 1, …, i − 1). Note that R is a non-unital ring. We calculate K_{0}^{gr}(R). Since K_{0}^{gr} respects the direct limit (Theorem 3.2.4), we have

K_{0}^{gr}(R) = K_{0}^{gr}(\underrightarrow{lim} R_{i}) = \underrightarrow{lim} K_{0}^{gr}(R_{i}),

where R_{i} = 𝕄_{i}(K)(\overline{α}_{i}). The non-unital homomorphism ϕ_{i} : R_{i} → R_{i+1} (see (3.15)) induces the homomorphism \overline{ϕ}_{i} : K_{0}^{gr}(R_{i}) → K_{0}^{gr}(R_{i+1}). We will observe that each of these K-groups is ℤ[x, x^{−1}] and each map ϕ_{i} is identity. First, note that the ring R_{i}, i ∈ ℕ, is unital, so using the Morita theory (see Proposition 2.1.1) and Proposition 3.6.1, K_{0}^{gr}(R_{i}) = K_{0}^{gr}(K) ≅ ℤ[x, x^{−1}]. Switching to the idempotent representation of the graded Grothendieck group, observe that the idempotent matrix p_{i} having 1 on the upper left corner and zero everywhere else is in R_{i}_{0} = 𝕄_{i}(K)(0, 1, …, i−1)_{0} and p_{i}R_{i} ≅_{gr} K ⊕ K(1) ⊕ ⋯ ⊕ K(i − 1). Again Proposition 2.1.1 shows that [p_{i}R_{i}] = [K ⊕ K(1) ⊕ ⋯ ⊕ K(i−1)] corresponds to 1 ∈ ℤ[x, x^{−1}]. Now p_{i+1} := ϕ_{i}(p_{i}) gives again a matrix with 1 on the upper left corner and zero everywhere else. So p_{i+1}R_{i+1} ≅_{gr} K ⊕ K(1) ⊕ ⋯ ⊕ K(i) and consequently [p_{i+1}R_{i+1}] = 1. So \overline{ϕ}_{i}(1) = 1. Since \overline{ϕ}_{i} respects the shifting, this shows that \overline{ϕ}_{i} is the identity map.

A similar argument shows that for the graded ring

S = ⋃_{i=1}^{∞} 𝕄_{i}(K)(\overline{α}_{i}),

where \overline{α}_{i} = (0, 1, 1, …, 1), with one zero and 1 repeated i − 1 times, we have an ordered ℤ[x, x^{−1}]- module isomorphism

K_{0}^{gr}(R) ≅ K_{0}^{gr}(S).

Using a similar argument as in the proof of Theorem 1.6.15, one can show that R is isomorphic to the Leavitt path algebra associated to the infinite graph

[figure: E :; •; •; •]

whereas S is the Leavitt path algebra of a graph F consisting of infinite vertices, one in the middle and the rest are connected to this vertex with an edge.

[figure: •; •; F :; •; •; •; •; •]

Clearly the rings R and S are not graded isomorphic, as the support of R is ℤ whereas the support of S is {−1, 0, 1}.

3.7 K_{0}^{gr} of graded local rings

Recall that a ring is a local ring if the set of non-invertible elements form a two sided ideal. When A is a commutative ring, then A is local if and only if A has a unique maximal ideal.

A Γ-graded ring A is called a graded local ring if the two sided ideal M generate by noninvertible homogeneous elements is a proper ideal. One can easily observe that the graded ideal M is the unique graded maximal left, right, and graded two sided ideal of A. When A is a graded commutative ring, then A is graded local if and only if A has a unique graded maximal ideal.

If A is a graded local ring, then the graded ring A/M is a graded division ring. One can further show that A_{0} is a local ring with the unique maximal ideal A_{0} ∩ M. In fact we have the following proposition.

Proposition 3.7.1. Let A be a Γ-graded ring. Then A is a graded local ring if and only if A_{0} is a local ring.

Proof. Suppose A is a graded local ring. Then by definition, the two sided ideal M generated by non-invertible homogeneous elements is a proper ideal. Consider m = A_{0} ∩ M which is a proper ideal of A_{0}. Suppose x ∈ A_{0}\m. Then x is a homogeneous element which is not in M. Thus x has to be invertible. This shows that A_{0} is a local ring with the unique maximal ideal m.

Conversely, suppose A_{0} is a local ring. We first show that any left or right invertible homogeneous element is a two sided invertible element. Let a be a left invertible homogeneous element. Then there is a homogeneous element b such that ba = 1. If ab is not right invertible, then ab ∈ m, where m is the unique maximal ideal of the local ring A_{0}. Thus 1 − ab ∉ m which implies that 1 − ab is invertible. But (1 − ab)a = a − aba = a − a = 0, and since 1 − ab is invertible, we get a = 0 which is a contradiction to the fact that a has a left inverse. Thus a has a right inverse and so is invertible. A similar argument can be written for right invertible elements. Now let M be the ideal generated by all non-invertible homogeneous elements of A. We will show that M is proper, and thus A is a graded local ring. Suppose M is not proper. Thus 1 = ∑_{i} r_{i}a_{i}s_{i}, where a_{i} are non-invertible homogeneous elements and r_{i}, s_{i} are homogeneous elements such that deg(r_{i}a_{i}s_{i}) = 0. If r_{i}a_{i}s_{i} is invertible for some i, using the fact that right and left invertibles are invertibles, it follows that a_{i} is invertible which is a contradiction. Thus r_{i}a_{i}s_{i}, for all i, are homogeneous elements of degree zero and not invertible. So they are all in m. This implies that 1 ∈ m which is a contradiction. Thus M is a proper ideal of A.

For more on graded local rings (graded by a cancellative monoid) see [64].

In Theorem 3.7.4, we will explicitly calculate the graded Grothendieck group of graded local rings. There are two ways to do this. One can prove that a graded finitely generated projective module over a graded local ring is a graded free with a unique rank, as in the case of graded division rings (see §1.4) and adopt the same approach to calculate the graded Grothendieck group (see §3.6). However, there is a more direct way to do so, which we develop here.

Recall the definition of graded Jacobson radical of a graded ring A, J^{gr}(A), from §1.1.5. We also need the graded version of the Nakayama lemma which holds in this setting as well. Namely, if J ⊆ J^{gr}(A) is a graded right ideal of A and P is a graded finitely generated right A-module, and Q is a graded submodule of P such that P = Q + P J, then P = Q (see [73, Corollary 2.9.2]).

We need the following two lemmas in order to calculate the graded Grothendieck group of a graded local ring.

Lemma 3.7.2. Let A be a Γ-graded ring, J ⊆ J^{gr}(A) a homogeneous two sided ideal and P and Q be graded finitely generated projective A-modules. If \overline{P} = P ⊗_{A} A/J = P/P J and \overline{Q} = Q⊗_{A} A/J = Q/QJ are isomorphic as graded A/J-modules, then P and Q are isomorphic as graded A-module.

Proof. Let ϕ : \overline{P} → \overline{Q} be a graded A/J-module isomorphism. Clearly ϕ is also a A-module isomorphism. Consider the Diagram 3.32. Since π_{2} is an epimorphism, and P is a graded finitely generated projective, there is a graded homomorphism ψ : P → Q which makes the diagram commutative.

\xymatrix{P \ar[r]^{π_{1}} \ar@{.>}[d]_{ψ} & \overline{P} \ar[d]^{ϕ} \ar[r] & 0 \\ Q \ar[r]^{π_{2}} & \overline{Q} \ar[r] & 0}  (3.32)

We will show that ψ is a graded A-module isomorphism. Since π_{2}ψ is an epimorphism, Q = ψ(P) + ker π_{2} = ψ(P) + QJ. The Nakayama lemma then implies that Q = ψ(P), i.e., ψ is an epimorphism. Now since Q is graded projective, there is a graded homomorphism i : Q → P such that ψi = 1_{Q} and P = i(Q) ⊕ ker ψ. But

ker ψ ⊆ ker π_{2}ψ = ker ϕπ_{1} = ker π_{1} = P J.

Thus P = i(Q) + P J. Again the Nakayama lemma implies P = i(Q). Thus ker ψ = 0, and so ψ : P → Q is an isomorphism.

Lemma 3.7.3. Let A and B be Γ-graded rings and ϕ : A → B be a graded epimorphism such that ker ϕ ⊆ J^{gr}(A). Then the induced homomorphism \overline{ϕ} : K_{0}^{gr}(A) → K_{0}^{gr}(B) is a ℤ[Γ]-module monomorphism.

Proof. Let J = ker(ϕ). Without the loss of generality we can assume B = A/J. Let x ∈ ker \overline{ϕ}. Write x = [P] − [Q] for two graded finitely generated projective A-modules P and Q. So ϕ(x) = [\overline{P}]−[\overline{Q}] = 0, where \overline{P} = P ⊗_{A}A/J = P/P J and \overline{Q} = Q⊗_{A}A/J = Q/QJ are graded A/J-modules. Since ϕ(x) = 0, by Lemma 3.1.7(3), \overline{P} ⨁ B^{n}(\overline{α}) ≅_{gr} \overline{Q} ⨁ B^{n}(\overline{α}), where \overline{α} = (α_{1}, …, α_{n}). So

\overline{P ⨁ A^{n}(\overline{α})} ≅_{gr} \overline{Q ⨁ A^{n}(\overline{α})}.

By Lemma 3.7.2, P ⨁ A^{n}(\overline{α}) ≅_{gr} Q ⨁ A^{n}(\overline{α}). Therefore x = [P] − [Q] = 0 in K_{0}^{gr}(A). This completes the proof.

For a Γ-graded ring A, recall that Γ_{A} is the support of A and Γ_{A}^{∗} = {α ∈ Γ | A_{α}^{∗} ≠ ∅} is a subgroup of Γ_{A}.

Proposition 3.7.4. Let A be a Γ-graded local ring. Then there is a ℤ[Γ]-module isomorphism

K_{0}^{gr}(A) ≅ ℤ[Γ/Γ_{A}^{∗}].

Proof. Let M = J^{gr}(A) be the unique graded maximal ideal of A. Then by Lemma 3.7.3 the homomorphism

ϕ : K_{0}^{gr}(A) ⟶ K_{0}^{gr}(A/M)
[P] ⟼ [P ⊗_{A} A/M],  (3.33)

is a monomorphism. Since A/M is a graded division ring, by Proposition 3.6.1, K_{0}^{gr}(A/M) ≅ ℤ[Γ/Γ_{A/M}]. Observe that Γ_{A/M} = Γ_{A}^{∗}. On the other hand, for the graded A-module A(α), where α ∈ Γ, we have

A(α) ⊗_{A} A/M ≅_{gr} (A/M)(α),  (3.34)

as the graded A/M-module. Since K_{0}^{gr}(A/M) ≅ ℤ[Γ/Γ_{A}^{∗}] is generated by [(A/M)(α_{i})], where {α_{i}}_{i∈I} is a complete set of coset representative of Γ/Γ_{A}^{∗} (see Proposition 3.6.1), from (3.33) and (3.34) we get ϕ([A(α_{i})] = [(A/M)(α_{i})]. So ϕ is an epimorphism as well. This finishes the proof.

3.8 K_{0}^{gr} of Leavitt path algebras

For a graph E, its associated path algebra 𝒫(E) is a positively graded ring. In §6.1, we will use Quillen’s theorem on graded K-theory of such rings to calculate the graded Grothendieck group of paths algebras (see Theorem 6.1.2).

In this section we calculate the K_{0} and K_{0}^{gr} of Leavitt path algebras. For one thing, they provide very nice examples. We first calculate the non-graded K_{0} of the Leavitt path algebras in §3.8.1 and in §3.8.2, we determine their K_{0}^{gr}-groups.

3.8.1

For a Leavitt path algebra ℒ_{K}(E), the monoid 𝒱(ℒ_{K}(E)) is studied in [6]. In particular using [6, Theorem 3.5], one can calculate the Grothendieck group of a Leavitt path algebra from the adjacency matrix of a graph (see [3, p.1998]). We present the calculation of the Grothendieck group of a Leavitt path algebra here.

Let F be a free abelian monoid generated by a countable set X. The nonzero elements of F can be written as ∑_{t=1}^{n} x_{t}, where x_{t} ∈ X. Let r_{i}, s_{i} be elements of F, where i ∈ I ⊆ ℕ. We define an equivalence relation on F denoted by ⟨r_{i} = s_{i} | i ∈ I⟩ as follows: Define a binary relation → on F \{0}, r_{i} + ∑_{t=1}^{n} x_{t} → s_{i} + ∑_{t=1}^{n} x_{t}, i ∈ I and generate the equivalence relation on F using this binary relation. Namely, a ∼ a for any a ∈ F and for a, b ∈ F \{0}, a ∼ b if there is a sequence a = a_{0}, a_{1}, …, a_{n} = b such that for each t = 0, …, n − 1 either a_{t} → a_{t+1} or a_{t+1} → a_{t}. We denote the quotient monoid by F/⟨r_{i} = s_{i} | i ∈ I⟩. Completing the monoid (see §3.1.1), one can see that there is a canonical group isomorphism

(\frac{F}{⟨r_{i} = s_{i} | i ∈ I⟩})^{+} ≅ \frac{F^{+}}{⟨r_{i} − s_{i} | i ∈ I⟩}.  (3.35)

Let E be a graph (as usual we consider only graphs with no sinks) and A_{E} be the adjacency matrix (n_{ij}) ∈ ℤ^{E^{0}⊕E^{0}}, where n_{ij} is the number of edges from v_{i} to v_{j}. Clearly the adjacency matrix depends on the ordering we put on E^{0}. We usually fix an ordering on E^{0}.

Multiplying the matrix A_{E}^{t} − I from the left defines a homomorphism

ℤ^{E^{0}} ⟶ ℤ^{E^{0}},

where ℤ^{E^{0}} is the direct sum of copies of ℤ indexed by E^{0}. The next theorem shows that the cokernel of this map gives the Grothendieck group of Leavitt path algebras.

Theorem 3.8.1. Let E be finite graph with no sinks and ℒ(E) be the Leavitt path algebra associated to E. Then

K_{0}(ℒ(E)) ≅ coker (A_{E}^{t} − I : ℤ^{E^{0}} ⟶ ℤ^{E^{0}}).  (3.36)

Proof. Let M_{E} be the abelian monoid generated by {v | v ∈ E^{0}} subject to the relations

v = ∑_{{α∈E^{1}|s(α)=v}} r(α),  (3.37)

for every v ∈ E^{0}. The relations (3.37) can be then written as A_{E}^{t}\overline{v}_{i} = I\overline{v}_{i}, where v_{i} ∈ E^{0} and \overline{v}_{i} is the (0, …, 1, 0, …) with 1 in the i-th component. Therefore,

M_{E} ≅ \frac{F}{⟨A_{E}^{t}\overline{v}_{i} = I\overline{v}_{i}, v_{i} ∈ E^{0}⟩},

where F is the free abelian monoid generated by the vertices of E. By [6, Theorem 3.5] there is a natural monoid isomorphism

𝒱(ℒ_{K}(E)) ≅ M_{E}.

So using (3.35) we have,

K_{0}(ℒ(E)) ≅ 𝒱(ℒ_{K}(E))^{+} ≅ M_{E}^{+} ≅ \frac{F^{+}}{⟨(A_{E}^{t} − I)\overline{v}_{i}, v_{i} ∈ E^{0}⟩}.  (3.38)

Now F^{+} is ℤ^{E^{0}} and it is easy to see that the denominator in (3.38) is the image of A_{E}^{t} −I : ℤ^{E^{0}} ⟶ ℤ^{E^{0}}.

Example 3.8.2. Let E be the following graph.

[figure: •; y_{1}; y_{2}; y_{3}; y_{n}]

Then the Leavitt path algebra associated to E, ℒ(E), is the algebra constructed by Leavitt in (1.44). By Theorem 3.8.1,

K_{0}^{gr}(ℒ(E)) ≅ ℤ/(n − 1)ℤ.

Example 3.8.3. Here is an example of a ring R such that K_{0}(R) ≠ 0 but [R] = 0. Let A be the Leavitt path algebra associated to the graph

[figure: •; y_{3}; y_{1}; y_{2}]

Thus as a right A-module, A^{3} ≅ A (see Example 1.3.13). By Example 3.8.2, K_{0}(A) = ℤ/2ℤ and 2[A] = 0. The ring R = 𝕄_{2}(A) is Morita equivalent to A (using the assignment P_{𝕄_{2}(A)} ↦ P ⊗_{𝕄_{2}(A)} A^{2}, see Proposition 2.1.1). Thus K_{0}(R) ≅ K_{0}(A) ≅ ℤ/2ℤ. Under this assignment,

[R] = [𝕄_{2}(A)] is sent to 2[A] which is zero, thus [R] = 0.

3.8.2

Recall that the Leavitt path algebras have a natural ℤ-graded structure (see §1.6.3). In this section we calculate the graded Grothendieck group of Leavitt path algebras. The graded Grothendieck group as a possible invariant for these algebras was first considered in [47]. In the case of finite graphs with no sinks, there is a good description of the action of ℤ on the graded Grothendieck group which we recall here.

Let E be a finite graph with no sinks. Set 𝒜 = ℒ(E) which is a strongly ℤ-graded ring by Theorem 1.6.11. For any u ∈ E^{0} and i ∈ ℤ, u𝒜(i) is a right graded finitely generated projective 𝒜-module and any graded finitely generated projective 𝒜-module is generated by these modules up to isomorphism, i.e.,

𝒱^{gr}(𝒜) = ⟨[u𝒜(i)] | u ∈ E^{0}, i ∈ ℤ⟩.

By §3.1, K_{0}^{gr}(𝒜) is the group completion of 𝒱^{gr}(𝒜). The action of ℕ[x, x^{−1}] on 𝒱^{gr}(𝒜) and thus the action of ℤ[x, x^{−1}] on K_{0}^{gr}(𝒜) is defined on generators by x^{j}[u𝒜(i)] = [u𝒜(i + j)], where i, j ∈ ℤ. We first observe that for i ≥ 0,

x[u𝒜(i)] = [u𝒜(i + 1)] = ∑_{{α∈E^{1}|s(α)=u}}[r(α)𝒜(i)].  (3.39)

First notice that for i ≥ 0, 𝒜_{i+1} = ∑_{α∈E^{1}} α𝒜_{i}. It follows

u𝒜_{i+1} = ⨁_{{α∈E^{1}|s(α)=u}} α𝒜_{i}

as 𝒜_{0}-modules. Using the fact that 𝒜_{n} ⊗_{𝒜_{0}} 𝒜 ≅ 𝒜(n), n ∈ ℤ, and the fact that α𝒜_{i} ≅ r(α)𝒜_{i} as 𝒜_{0}-module, we get

u𝒜(i + 1) ≅ ⨁_{{α∈E^{1}|s(α)=u}} r(α)𝒜(i)

as graded 𝒜-modules. This gives (3.39).

Recall that for a Γ-graded ring A, K_{0}^{gr}(A) is a pre-ordered abelian group with the set of isomorphic classes of graded finitely generated projective right A-modules as the cone of ordering, denoted by K_{0}^{gr}(A)_{+} (i.e., the image of 𝒱^{gr}(A) under the natural homomorphism 𝒱^{gr}(A) → K_{0}^{gr}(A)). Furthermore, [A] is an order-unit. We call the triple, (K_{0}^{gr}(A), K_{0}^{gr}(A)_{+}, [A]) the graded dimension group (see [38, §15] for some background on dimension groups).

In [47] it was conjectured that the graded dimension group is a complete invariant for Leavitt path algebras. Namely, for graphs E and F, ℒ(E) ≅_{gr} ℒ(F) if and only if there is an order preserving ℤ[x, x^{−1}]-module isomorphism

ϕ : K_{0}^{gr}(ℒ(E)) → K_{0}^{gr}(ℒ(F))  (3.40)

such that ϕ([ℒ(E)] = ℒ(F).

3.8.3

For the Leavitt path algebra ℒ(E), the structure of the ring of homogeneous elements of degree zero, ℒ(E)_{0}, is known. Since K_{0}^{gr}(ℒ(E) ≅ K_{0}(ℒ(E)_{0}), we recall the description of ℒ(E)_{0} in the setting of finite graphs with no sinks (see the proof of Theorem 5.3 in [6]). Let A_{E} be the adjacency matrix of E. Let L_{0,n} be the linear span of all elements of the form pq^{∗} with r(p) = r(q) and |p| = |q| ≤ n. Then

ℒ(E)_{0} = ⋃_{n=0}^{∞} L_{0,n},  (3.41)

where the transition inclusion L_{0,n} → L_{0,n+1} is to identify pq^{∗} with r(p) = v by

∑_{{α|s(α)=v}} pα(qα)^{∗}.

Note that since E does not have sinks, for any v ∈ E_{0} the set {α|s(α) = v} is not empty.

For a fixed v ∈ E^{0}, let L_{0,n}^{v} be the linear span of all elements of the form pq^{∗} with |p| = |q| = n and r(p) = r(q) = v. Arrange the paths of length n with the range v in a fixed order p_{1}^{v}, p_{2}^{v}, …, p_{k_{n}^{v}}^{v}, and observe that the correspondence of p_{i}^{v}p_{j}^{v}^{∗} to the matrix unit e_{ij} gives rise to a ring isomorphism L_{0,n}^{v} ≅ 𝕄_{k_{n}^{v}}(K). Furthermore, L_{0,n}^{v}, v ∈ E^{0} form a direct sum. This implies that

L_{0,n} ≅ ⨁_{v∈E^{0}} 𝕄_{k_{n}^{v}}(K),

where k_{n}^{v}, v ∈ E^{0}, is the number of paths of length n with the range v. The inclusion map L_{0,n} → L_{0,n+1} is

A_{E}^{t} : ⨁_{v∈E^{0}} 𝕄_{k_{n}^{v}}(K) ⟶ ⨁_{v∈E^{0}} 𝕄_{k_{n+1}^{v}}(K).  (3.42)

This means (A_{1}, …, A_{l}) ∈ ⨁_{v∈E^{0}} 𝕄_{k_{n}^{v}}(K) is sent to

(∑_{j=1}^{l} n_{j1}A_{j}, …, ∑_{j=1}^{l} n_{jl}A_{j}) ∈ ⨁_{v∈E^{0}} 𝕄_{k_{n+1}^{v}}(K),

where n_{ji} is the number of edges connecting v_{j} to v_{i} and

∑_{j=1}^{l} k_{j}A_{j} = (\begin{matrix}A_{1} \\ & A_{1} \\ & & . & . & . \\ & & & & & A_{l} \end{matrix})

in which each matrix is repeated k_{j} times down the leading diagonal and if k_{j} = 0, then A_{j} is omitted. This shows that ℒ(E)_{0} is an ultramatricial algebra, i.e., it is isomorphic to the union of an increasing chain of a finite product of matrix algebras over a field K (see §5).

Writing ℒ(E)_{0} = \underrightarrow{lim}_{n} L_{0,n}, since the Grothendieck group K_{0} respects the direct limit, we have K_{0}(ℒ(E)_{0}) ≅ \underrightarrow{lim}_{n} K_{0}(L_{0,n}). Since K_{0} of (Artinian) simple algebras are ℤ, the ring homomorphism L_{0,n} → L_{0,n+1} induces the group homomorphism

ℤ^{E^{0}} ⟶^{A_{E}^{t}} ℤ^{E^{0}},

where A_{E}^{t} : ℤ^{E^{0}} → ℤ^{E^{0}} is multiplication from left which is induced by the homomorphism (3.42).

For a finite graph E with no sinks, with n vertices and the adjacency matrix A, by Theorem 1.6.11, K_{0}^{gr}(ℒ(E)) ≅ K_{0}(ℒ(E)_{0}). Thus K_{0}^{gr}(ℒ(E)) is the direct limit of the ordered direct system

ℤ^{n} ⟶^{A^{t}} ℤ^{n} ⟶^{A^{t}} ℤ^{n} ⟶^{A^{t}} ⋯,  (3.43)

where the ordering in ℤ^{n} is defined point-wise.

In general, the direct limit of the system, \underrightarrow{lim}_{A} ℤ^{n}, where A ∈ 𝕄_{n}(ℤ), is an ordered group and can be described as follows. Consider the pair (a, k), where a ∈ ℤ^{n} and k ∈ ℕ, and define the equivalence relation (a, k) ∼ (b, k′) if A^{k′′−k}a = A^{k′′−k′}b for some k′′ ∈ ℕ. Let [a, k] denote the equivalence class of (a, k). Clearly [A^{n}a, n + k] = [a, k]. Then it is not difficult to show that the direct limit \underrightarrow{lim}_{A} ℤ^{n} is the abelian group consists of equivalent classes [a, k], a ∈ ℤ^{n}, k ∈ ℕ, with addition defined by

[a, k] + [b, k′] = [A^{k′}a + A^{k}b, k + k′].  (3.44)

The positive cone of this ordered group is the set of elements [a, k], where a ∈ ℤ^{+}^{n}, k ∈ ℕ. Furthermore, there is automorphism δ_{A} : \underrightarrow{lim}_{A} ℤ^{n} → \underrightarrow{lim}_{A} ℤ^{n} defined by δ_{A}([a, k]) = [Aa, k].

There is another presentation for \underrightarrow{lim}_{A} ℤ^{n} which is sometimes easier to work with. Consider the set

Δ_{A} = {v ∈ A^{n}ℚ^{n} | A^{k}v ∈ ℤ^{n}, for some k ∈ ℕ}.  (3.45)

The set Δ_{A} forms an ordered abelian group with the usual addition of vectors and the positive cone

Δ_{A}^{+} = {v ∈ A^{n}ℚ^{n} | A^{k}v ∈ ℤ^{+}^{n}, for some k ∈ ℕ}.  (3.46)

Furthermore, there is automorphism δ_{A} : Δ_{A} → Δ_{A} defined by δ_{A}(v) = Av. The map

ϕ : Δ_{A} → \underrightarrow{lim}_{A} ℤ^{n}  (3.47)
v ↦ [A^{k}v, k],

where k ∈ ℕ such that A^{k}v ∈ ℤ^{n}, is an isomorphism which respects the action of A and the ordering, i.e., ϕ(Δ_{A}^{+}) = (\underrightarrow{lim}_{A} ℤ^{n})^{+} and ϕ(δ_{A}(v)) = δ_{A}ϕ(v).

Example 3.8.4. Let E be the following graph.

[figure: •; y_{1}; y_{2}; y_{3}; y_{n}]

The non-graded K_{0} of ℒ(E) was computed in Example 3.8.2. The graph E has no sinks, and so by (3.43),

K_{0}^{gr}(ℒ(E)) ≅ \underrightarrow{lim} ℤ,

of the inductive system ℤ ⟶^{n} ℤ ⟶^{n} ℤ ⟶^{n} · · ·. This gives that K_{0}^{gr}(ℒ(E)) ≅ ℤ[1/n].

Example 3.8.5. For the graph

E : [figure: •; •]

with the adjacency A_{E} = ( \begin{matrix}1 & 2 \\ 1 & 0 \end{matrix} ) , the ring of homogeneous element of degree zero, ℒ(E)_{0}, is

the direct limit of the system

K ⊕ K ⟶^{A_{E}^{t}} 𝕄_{2}(K) ⊕ 𝕄_{2}(K) ⟶^{A_{E}^{t}} 𝕄_{4}(K) ⊕ 𝕄_{4}(K) ⟶^{A_{E}^{t}} ⋯
(a, b) ↦ ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) ⊕ ( \begin{matrix}a & 0 \\ 0 & a \end{matrix} )

So K_{0}^{gr}(ℒ(E)) is the direct limit of the direct system

ℤ^{2} ⟶^{A_{E}^{t}} ℤ^{2} ⟶^{A_{E}^{t}} ℤ^{2} ⟶^{A_{E}^{t}} ⋯,

Since det(A_{E}^{t}) = −2, one can easily calculate that

K_{0}^{gr}(ℒ(E)) ≅ ℤ[1/2] ⨁ ℤ[1/2].

Furthermore [ℒ(E)] ∈ K_{0}^{gr}(ℒ(E)) is represented by (1, 1) ∈ ℤ[1/2] ⨁ ℤ[1/2]. Adopting (3.45) for the description of K_{0}^{gr}(ℒ(E)), since the action of x on K_{0}^{gr}(ℒ(E)) represented by action of A_{E}^{t} from the left, we have x(a, b) = (a + b, 2a). Furthermore, considering (3.46) for the positive cone, A_{E}^{t}^{k}(a, b) is eventually positive, if v(a, b) > 0, where v = (2, 1) is the Perron eigenvector of A_{E} (see [53, Lemma 7.3.8]). It follows that

K_{0}^{gr}(ℒ(E))^{+} = Δ_{A_{E}^{t}}^{+} = {(a, b) ∈ ℤ[1/2] ⊕ ℤ[1/2] | 2a + b > 0} ∪ {(0, 0)}.

Example 3.8.6. \underline{Reduced K_{0}^{gr} of strongly graded rings}

When A is a strongly graded ring, the graded K-groups coincide with K-groups of the zero homogeneous part (see (3.8)). However this example shows that this is not the case for the reduced graded Grothendieck groups.

Let A be the Leavitt’s algebra generated by 2n symbols (which is associated to a graph with one vertex and n-loops) (see 1.3.13). By Theorem 1.6.11, this is a strongly graded ring. The homomorphism 3.13 takes the form

ϕ : ℤ[x, x^{−1}] ⟶ K_{0}^{gr}(A) ≅ ℤ[1/n],
∑_{i} n_{i}x^{i} ⟼ ∑_{i} n_{i}[A(i)].

This shows that ϕ is surjective and thus K̃_{0}^{gr}(A) is trivial. On the other hand, by Example 3.8.4, K_{0}(A_{0}) ≅ K_{0}^{gr}(A) ≅ ℤ[1/n]. But

ϕ_{0} : ℤ ⟶ K_{0}(A_{0}) ≅ ℤ[1/n],
n ⟼ n[A_{0}].

This shows that K̃_{0}(A_{0}) is a nontrivial torsion group ℤ[1/n]/ℤ. Thus K̃_{0}^{gr}(A) ≅ ˜K̸_{0}(A_{0}).

Remark 3.8.7. \underline{K_{0}^{gr} of Weyl algebras}

Let A = K(x, y)/⟨xy − yx − 1⟩ be the Weyl algebra, where K is an algebraically closed field of characteristic 0. By Example 1.6.4, this is a ℤ-graded ring. The graded Grothendieck group of this ring is calculated in [86]. It is shown that K_{0}^{gr}(A) ≅ ⨁_{ℤ} ℤ (i.e., a direct sum of a countably many ℤ), K̃_{0}^{gr}(A) = 0 and K_{0}(A) = 0.

3.9 Symbolic dynamics

One of the central objects in the theory of symbolic dynamics is a shift of finite type (i.e., a topological Markov chain). Every finite directed graph E with no sinks and sources gives rise to a shift of finite type X_{E} by considering the set of bi-infinite paths and the natural shift of the paths to the left. This is called an edge shift. Conversely any shift of finite type is conjugate to an edge shift (for a comprehensive introduction to symbolic dynamics see [53]). Several invariant have been proposed in order to classify shifts of finite type, among them Krieger’s dimension group. In this section we see that Krieger’s invariant can be expressible as the graded Grothendieck group of a Leavitt path algebra.

We briefly recall the objects of our interest. Let 𝒜 be a finite alphabet (i.e., a finite set). A full shift space is defined as

𝒜^{ℤ} := {(a_{i})_{i∈ℤ} | a_{i} ∈ 𝒜},

and a shift map σ : 𝒜^{ℤ} → 𝒜^{ℤ} is defined as

σ((a_{i})_{i∈ℤ}) = (a_{i+1})_{i∈ℤ}.

Furthermore, a subshift X ⊆ 𝒜^{ℤ} is a closed σ-invariant subspace of 𝒜^{ℤ}.

Given a finite graph E (see §1.6.3 for terminologies related to graphs), a subshift of finite type associated to E is defined as

X_{E} := {(e_{i})_{i∈ℤ} ∈ (E^{1})^{ℤ} | r(e_{i}) = s(e_{i+1})}.

We say X_{E} is essential if the graph E has no sinks and sources. Furthermore, X_{E} is called irreducible if the adjacency matrix A_{E} is irreducible. For a square nonnegative integer matrix A, we denote by X_{A} the subshift of finite type associated to the graph with the adjacency matrix A. Finally, two shifts of finite type X_{A} and X_{B} are called conjugate (or topologically conjugate of subshifts) and denoted by X_{A} ≅ X_{B}, if there exists a homeomorphism h : X_{A} → X_{B} such that σ_{B} h = h σ_{A}.

The notion of the shift equivalence for matrices was introduced by Williams [96] (see also [53, §7]) in an attempt to provide a computable machinery for determining the conjugacy between two shifts of finite type. Two square nonnegative integer matrices A and B are called elementary shift equivalent, and denoted by A ∼_{ES} B, if there are nonnegative matrices R and S such that A = RS and B = SR. The equivalence relation ∼_{S} on square nonnegative integer matrices generated by elementary shift equivalence is called strong shift equivalence.

Example 3.9.1. Let A = ( \begin{matrix}1 & 2 \\ 1 & 0 \end{matrix} ) and B = ( \begin{matrix}1 & 1 \\ 2 & 0 \end{matrix} ). We show that A is strongly shift

equivalent to B.

Let R_{1} = ( \begin{matrix}1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) and S_{1} = ( \begin{matrix}1 & 1 \\ 0 & 1 \\ 1 & 0 \end{matrix} ). Then A = R_{1}S_{1}. Set E_{1} := S_{1}R_{1} = ( \begin{matrix}1 & 1 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} ).

Let R_{2} = ( \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) and S_{2} = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} ). Then E_{1} = R_{2}S_{2}. Set E_{2} := S_{2}R_{2} =

( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \end{matrix} ). Finally, let R_{3} = ( \begin{matrix}1 & 0 \\ 1 & 0 \\ 1 & 1 \end{matrix} ) and S_{3} = ( \begin{matrix}0 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} ). Then E_{2} = R_{3}S_{3} and

B = S_{3}R_{3}. This shows

A ∼_{ES} E_{1} ∼_{ES} E_{2} ∼_{ES} B.

Thus A ∼_{S} B.

There is a weaker notion, called shift equivalence defined as follows. The nonnegative integer square matrices A and B are called shift equivalent if there are nonnegative matrices R and S such that A^{l} = RS and B^{l} = SR, for some l ∈ ℕ, and AR = RB and SA = BS. Clearly the strongly shift equivalence implies the shift equivalence, but the converse does not hold [53].

Theorem 3.9.2 (Williams [96, 53]). Let A and B be two square nonnegative integer matrices and let E and F be two essential graphs.

(1) X_{A} is conjugate to X_{B} if and only if A is strongly shift equivalent to B.

(2) X_{E} is conjugate to X_{F} if and only if E can be obtained from F by a sequence of in/out-splitting and their converses.

Krieger in [57] defined an invariant for classifying the irreducible shifts of finite type up to shift equivalence. Later Wagoner systematically used this invariant to relate it with higher K-groups. Surprisingly. Krieger’s dimension group and Wagoner’s dimension module in symbolic dynamics turn out to be expressible as the graded Grothendieck groups of Leavitt path algebras. Here, we briefly describe this relation.

In general, a nonnegative integral n × n matrix A gives rise to a stationary system. This in turn gives a direct system of order free abelian groups with A acting as an order preserving group homomorphism as follows

ℤ^{n} ⟶^{A} ℤ^{n} ⟶^{A} ℤ^{n} ⟶^{A} ⋯,

where the ordering in ℤ^{n} is defined point-wise (i.e., the positive cone is ℕ^{n}). The direct limit of this system, Δ_{A} := \underrightarrow{lim}_{A} ℤ^{n}, (i.e, the K_{0} of the stationary system,) along with its positive cone, Δ^{+}, and the automorphism which induced by A on the direct limit, δ_{A} : Δ_{A} → Δ_{A}, is the invariant considered by Krieger, now known as Krieger’s dimension group. Following [53], we denote this triple by (Δ_{A}, Δ_{A}^{+}, δ_{A}).

The following theorem was proved by Krieger ([57, Theorem 4.2], and [53, Theorem 7.5.8], see also [53, §7.5] for a detailed algebraic treatment).

Theorem 3.9.3. Let A and B be two square nonnegative integer matrices. Then A and B are shift equivalent if and only if

(Δ_{A}, Δ_{A}^{+}, δ_{A}) ≅ (Δ_{B}, Δ_{B}^{+}, δ_{B}).

Wagoner noted that the induced structure on Δ_{A} by the automorphism δ_{A} makes Δ_{A} a ℤ[x, x^{−1}]- module which was systematically used in [92, 93] (see also [23, §3]).

Recall that the graded Grothendieck group of a ℤ-graded ring has a natural ℤ[x, x^{−1}]-module structure and the following observation (Theorem 3.9.4) shows that the graded Grothendieck group of the Leavitt path algebra associated to a matrix A coincides with the Krieger dimension group of the shift of finite type associated to A^{t}, i.e., the graded dimension group of a Leavitt path algebra coincides with Krieger’s dimension group,

(K_{0}^{gr}(ℒ(E)), (K_{0}^{gr}(ℒ(E))^{+}) ≅ (Δ_{A^{t}}, Δ_{A^{t}}^{+}).

This will provide a link between the theory of Leavitt path algebras and symbolic dynamics.

Theorem 3.9.4. Let E be a finite graph with no sinks with the adjacency matrix A. Then there is an isomorphism ϕ : K_{0}^{gr}(ℒ(E)) ⟶ Δ_{A^{t}} such that ϕ(xα) = δ_{A^{t}}ϕ(α), α ∈ ℒ(E), x ∈ ℤ[x, x^{−1}] and ϕ(K_{0}^{gr}(ℒ(E))^{+}) = Δ_{A^{t}}^{+}.

Proof. Since by Theorem 1.6.11, ℒ(E) strongly graded, there is an ordered isomorphism K_{0}^{gr}(ℒ(E)) → K_{0}(ℒ(E)_{0}). Thus the ordered group K_{0}^{gr}(ℒ(E)) coincides with the ordered group Δ_{A^{t}} (see (3.43)). We only need to check that their module structures are compatible. It is enough to show that the action of x on K_{0}^{gr} coincides with the action of A^{t} on K_{0}(ℒ(E)_{0}), i.e., ϕ(xα) = δ_{A^{t}}ϕ(α).

Set 𝒜 = ℒ(E). Since graded finitely generated projective modules are generated by u𝒜(i), where u ∈ E^{0} and i ∈ ℤ, it suffices to show that ϕ(x[u𝒜]) = δ_{A^{t}}ϕ([u𝒜]). Since the image of u𝒜 in K_{0}(𝒜_{0}) is [u𝒜_{0}], and 𝒜_{0} = ⋃_{n=0}^{∞} L_{0,n}, (see (3.41)) using the presentation of K_{0} given in (3.44), we have

ϕ([u𝒜]) = [u𝒜_{0}] = [uL_{0,0}, 1] = [u, 1].

Thus

δ_{A^{t}}ϕ([u𝒜]) = δ_{A^{t}}([u, 1]) = [A^{t}u, 1] = ∑_{{α∈E^{1}|s(α)=u}}[r(α), 1].

On the other hand,

ϕ(x[u𝒜]) = ϕ([u𝒜(1)]) = ϕ( ∑_{{α∈E^{1}|s(α)=u}}[r(α)𝒜]) =

∑_{{α∈E^{1}|s(α)=u}}[r(α)𝒜_{0}] = ∑_{{α∈E^{1}|s(α)=u}}[r(α)L_{0,0}, 1] = ∑_{{α∈E^{1}|s(α)=u}}[r(α), 1]. (3.48)

Thus ϕ(x[u𝒜]) = δ_{A^{t}}ϕ([u𝒜]). This finishes the proof.

It is easy to see that two matrices A and B are shift equivalent if and only if A^{t} and B^{t} are shift equivalent. Combining this with Theorem 3.9.4 and the fact that Krieger’s dimension group is a complete invariant for shift equivalent we have the following corollary.

Corollary 3.9.5. Let E and F be finite graphs with no sinks and A_{E} and A_{F} be their adjacency matrices, respectively. Then A_{E} is shift equivalent to A_{F} if and only if there is an order preserving ℤ[x, x^{−1}]-module isomorphism K_{0}^{gr}(ℒ(E)) ≅ K_{0}^{gr}(ℒ(F)).

3.10 K_{1}^{gr}-theory

For a Γ-graded ring A, the category of finitely generated Γ-graded projective right A-modules, Pgr^{Γ}-A, is an exact category. Thus using Quillen’s Q-construction one defines

K_{i}^{gr}(A) := K_{i}(Pgr^{Γ}-A), i ≥ 0.

Furthermore, the shift functors induce auto-equivalences (in fact, automorphisms) 𝒯_{α} : Pgr-A → Pgr-A. These in return give a group homomorphism Γ → Aut(K_{i}^{gr}(A)) or equivalently, a ℤ[Γ]- module on K_{i}^{gr}(A). This general construction will be used in §6.

In this section we concretely construct the graded K_{1}-group, using a graded version of Bass’ construction of K_{1}-group. For a concise introduction of the groups K_{0} and K_{1}, see [59].

Definition 3.10.1. K_{0} and K_{1} of an exact category

Let 𝒫 be an exact category, i.e., a full additive subcategory of an abelian category 𝒜 such that, if

0 ⟶ P_{1} ⟶ P ⟶ P_{2} ⟶ 0,

is an exact sequence in 𝒜 and P_{1}, P_{2} ∈ 𝒫, then P ∈ 𝒫 (i.e., 𝒫 is closed under extension). Furthermore, we assume 𝒫 has a small skeleton, i.e., 𝒫 has a full subcategory 𝒫_{0} which is small and 𝒫_{0} ↪ 𝒫 is an equivalence.

The groups K_{0}(𝒫) and K_{1}(𝒫) are defined as follows.

1. K_{0}(𝒫) is the free abelian group generated by objects of 𝒫_{0}, subject to the relation [P] = [P_{1}] + [P_{2}] if there is an exact sequence

0 ⟶ P_{1} ⟶ P ⟶ P_{2} ⟶ 0,

in 𝒫.

2. K_{1}(𝒫) is the free abelian group generated by pairs (P, f), where P is an object of 𝒫_{0} and f ∈ Aut(P), subject to the relations

[P, f] + [P, g] = [P, fg],

and

[P, f] = [P_{1}, g] + [P_{2}, h],

if there is a commutative diagram in 𝒫_{0}

\xymatrix{0 \ar[r] & P_{1} \ar[r]^{i} \ar[d]^{g} & P \ar[r]^{π} \ar[d]^{f} & P_{2} \ar[r] \ar[d]^{h} & 0 \\ 0 \ar[r] & P_{1} \ar[r]^{i} & P \ar[r]^{π} & P_{2} \ar[r] & 0.}

Note that from the relations of K_{1} it follows

[P, fg] = [P, gf],
[P, fg] = [P ⊕ P, f ⊕ g].

If Γ is a group and 𝒯_{α} : 𝒫 → 𝒫, α ∈ Γ, are auto-equivalences such that 𝒯_{β}𝒯_{α} ≅ 𝒯_{α+β}, then K_{0}(𝒫) and K_{1}(𝒫) have a Γ-module structures.

One can easily see that K_{0}^{gr}(A) = K_{0}(Pgr^{Γ}-A). Following Bass, one defines K_{1}^{gr}(A) = K_{1}(Pgr^{Γ}-A). Since the category Gr-A is an abelian category and Pgr-A is an exact category, the main

theorems of K-theory are valid for the grade Grothendieck group, such as Dévissage, Resolution theorem and the localisation exact sequences of K-theory (see §6.2.1 and [82, Chapter 3] and [79, 94]).

Example 3.10.2. Let F be a field and F [x_{1}, …, x_{r}] be the polynomial ring with r variables, which is considered as a ℤ-grade ring with support ℕ (i.e., deg(x_{i}) = 1, 1 ≤ i ≤ r). Then we will prove in §6 (see Proposition 6.1.1) that

K_{1}^{gr}(F [x_{1}, …, x_{r}]) = F^{∗} ⊗_{ℤ} ℤ[x, x^{−1}].

Example 3.10.3. \underline{K_{1}^{gr} of graded division algebras}

Let A be a Γ-graded division ring. Then A_{0} is a division ring and Ω := Γ_{A} is a group (§1.1.4). By (3.9), for i = 1, and the description of K_{1}-group of division ring due to Dieudonné ([33, § 20]), we have

K_{1}^{Γ}(A) ≅ ⨁_{Γ/Ω} K_{1}^{Ω}(A) ≅ ⨁_{Γ/Ω} K_{1}(A_{0}) = ⨁_{Γ/Ω} A_{0}^{∗}/[A_{0}^{∗}, A_{0}^{∗}],  (3.49)

where A_{0}^{∗} is the group of invertible elements of division ring A_{0} and [A_{0}^{∗}, A_{0}^{∗}] is the multiplicative commutator subgroup. Representing ⨁_{Γ/Ω} K_{1}(A_{0}) as the additive group of the group ring K_{1}(A_{0})[Γ/Ω], (see Example 3.1.11) the action of Γ can be described as follows: for β ∈ Γ,

β( ⨁_{Ω+α∈Γ/Ω} K_{1}(A_{0})(Ω + α)) = ⨁_{Ω+α∈Γ/Ω} K_{1}(A_{0})(Ω + α + β).

As a concrete example, let A = K[x^{n}, x^{−n}] be a ℤ-graded field with Γ_{A} = nℤ. Then by (3.49),

K_{1}^{gr}(A) ≅ ⨁_{n} K^{∗},

is a ℤ[x, x^{−1}]-module. The action of x on (a_{1}, …, a_{n}) ∈ ⨁_{n} K^{∗} is

x(a_{1}, …, a_{n}) = (a_{n}, a_{1}, …, a_{n−1}).

Compare this with the computation of K_{0}^{gr}(A) in Example 3.6.3(2).

Remark 3.10.4. \underline{The matrix description of K_{1}^{gr}-group} Let A be a strongly Γ-graded ring. Then the map

K_{1}^{gr}(A) ⟶ K_{1}(A_{0}),  (3.50)
[P, f] ⟼ [P_{0}, f_{0}]

is an isomorphism of groups (see §1.5). Here for a graded isomorphism f : P → P, we denote by f_{α}, the restriction of f to P_{α}, where α ∈ Γ. Note that K_{1}^{gr}(A) is a ℤ[Γ]-graded module, where the action of Γ on the generators is defined by α[P, f] = [P (α), f].

Since K_{1}(A_{0}) has a matrix description ([82, Theorem 3.1.7]), from (3.50) we get a matrix representation

K_{1}^{gr}(A) ≅ K_{1}(A_{0}) ≅ GL(A_{0})/E(A_{0}).

We don’t know whether for an arbitrary graded ring, one can give a matrix description for K_{1}^{gr}(A). For some work in this direction see [99].

Graded Picard Groups

Let A be a commutative ring. If M is a finitely generated projective A-module of constant rank 1, then there is an A-module N such that M ⊗_{A} N ≅ A. In fact this is an equivalent condition. The module M above is called an invertible module. The isomorphism classes of invertible modules with the tensor product form a group, denoted by Pic(A) and called the Picard group of A. Since A is commutative, K_{0}(A) is a ring with the multiplication defined by the tensor product and one can prove that there is an exact sequence

1 ⟶ Pic(A) ⟶^{ϕ} K_{0}(A)^{∗},  (4.1)

where K_{0}(A)^{∗} is the group of invertible elements of K_{0}(A) and ϕ([A]) = [A].

When A is a graded commutative ring, a parallel construction, using the graded modules, gives the graded Picard group Pic^{gr}(A). As one expects when A is strongly graded commutative ring then, using Dade’s theorem 1.5.1 one immediately gets

Pic^{gr}(A) ≅ Pic(A_{0}).  (4.2)

However when A is a noncommutative ring, the situation is substantially more involved. One needs to define the invertible bimodules in order to define the Picard group. Furthermore, since for bimodules M and N, M ⊗_{A} N is not necessarily isomorphic to N ⊗_{A} M as bimodules, the Picard group is not abelian. Even when A is strongly graded, the identities such as (4.2) does not hold in the noncommutative setting (see [41]).

4.1 Pic^{gr} of a graded commutative ring

Let A be a commutative Γ-graded ring. A graded A-module P is called a graded invertible module if there is a graded module Q such that P ⊗_{A} Q ≅_{gr} A as graded A-modules. It is clear that if P is a graded invertible module, so is P (α) for any α ∈ Γ. The graded Picard group, Pic^{gr}(A), is defined as the set of graded isomorphism classes of graded invertible A-modules with tensor product as multiplication. This is a well-defined binary operation and makes Pic^{gr}(A) an abelian group with the isomorphism class of A as identity element. Since a graded invertible module is an invertible module, we have a group homomorphism Pic^{gr}(A) → Pic(A), [P] ↦ [P], where [P] represents the isomorphism class of P in both groups.

Since for any α, β ∈ Γ, A(α) ⊗_{A} A(β) ≅_{gr} A(α + β) (see §1.2.6), the map

ϕ : Γ → Pic^{gr}(A),  (4.3)
α ↦ [A(α)]

is a group homomorphism. This map will be used in the next lemma to calculate the graded Picard group of graded fields.

For a Γ-graded ring A, recall from §1.1.2 that Γ_{A} is the support of A and Γ_{A}^{∗} = {α ∈ Γ | A_{α}^{∗} ≠ ∅}. Furthermore, for a graded field A, Γ_{A} is a subgroup of Γ.

Proposition 4.1.1. Let A be a Γ-graded field with the support Γ_{A}. Then Pic^{gr}(A) ≅ Γ/Γ_{A}.

Proof. Consider the map ϕ : Γ → Pic^{gr}(A) from (4.3). Since any graded invertible module is graded projective and graded projective modules over graded fields are graded free (Proposition 1.4.1), it follows that the graded invertible modules have to be of the form A(α) for some α ∈ Γ. This shows that ϕ is an epimorphism. Now if ϕ(α) = [A(α)] = [A], then A(α) ≅_{gr} A, which by Corollary 1.3.11, it follows that α ∈ Γ_{A}. Conversely, if α ∈ Γ_{A}, then there is an element of degree α, which has to be invertible, as A is a graded field. Thus A(α) ≅_{gr} A, again by Corollary 1.3.11. This shows that the kernel of ϕ is Γ_{A}. This completes the proof.

The graded Grothendieck group of a graded local ring was determined in Proposition 3.7.4. Here we determine its graded Picard group.

Proposition 4.1.2. Let A be a commutative Γ-graded local ring with the support Γ_{A}. Then Pic^{gr}(A) ≅ Γ/Γ_{A}^{∗}.

Proof. Let M be the unique graded maximal ideal of A. By Lemma 3.7.2, if for graded projective A-modules P and Q, \overline{P} = P/P M is isomorphic to \overline{Q} = Q/QM as graded A/M-modules, then P is isomorphic to Q as graded A-modules. This immediately implies that the natural map

ϕ : Pic^{gr}(A) ⟶ Pic^{gr}(A/M),
[P] ⟼ [\overline{P}]

is a monomorphism. But by the proof of Proposition 4.1.1, any graded invertible A/M-module is of the form (A/M)(α) for some α ∈ Γ. Since ϕ([A(α)]) = [(A/M)(α)], ϕ is an isomorphism. Since Γ_{A/M} = Γ_{A}^{∗}, by Proposition 4.1.1, Pic^{gr}(A/M) = Γ/Γ_{A}^{∗}. This completes the proof.

In the next theorem (Theorem 4.1.5) we will be using the calculus of exterior algebras in the graded setting. Recall that if M is an A-module, the n-th exterior power of M is the quotient of the tensor product of n copies of M over A, denoted by ⨂^{n} M (or T_{n}(M) as in Example 1.1.3), by the submodule generated by m_{1} ⊗ ⋯ ⊗ m_{n}, where m_{i} = m_{j} for some 1 ≤ i ≠ j ≤ n. The n-th exterior power of M is denoted by ⋀^{n} M. We set ⋀^{0} M = A and clearly ⋀^{1} M = M.

If A is a commutative Γ-graded ring and M is a graded A-module, then ⨂^{n} M is a graded A-module (§1.2.6) and the submodule generated by m_{1} ⊗ ⋯ ⊗ m_{n}, where m_{i} = m_{j} for some 1 ≤ i ≠ j ≤ n coincides with the submodule generated by m_{1} ⊗ ⋯ ⊗ m_{n}, where all m_{i} are homogeneous and m_{i} = m_{j} for some 1 ≤ i ≠ j ≤ n, and m_{1} ⊗ ⋯ ⊗ m_{n} + m′_{1} ⊗ ⋯ ⊗ m′_{n}, where all m_{i} and m′_{i} are homogeneous, m_{i} = m′_{i} for all 1 ≤ i ≤ n except for two indices i, j, where i ≠ j and m_{i} = m′_{j} and m_{j} = m′_{i}. Thus this submodule is a graded submodule of ⨂^{n} M and therefore ⋀^{n} M is a graded A-module as well. We will use the following isomorphism which is valid in the non-graded setting and it is easy to see it respects the grading as well.

⋀^{n}(M ⊕ N) ≅_{gr} ⨁_{r=0}^{n} ( ⋀^{r} M ⊗_{A} ⋀^{n−r} N).  (4.4)

Lemma 4.1.3. Let A be a commutative Γ-graded ring. Then

⋀^{n} A^{m}(α_{1}, …, α_{m}) ≅_{gr} ⨁_{1≤i_{1}<i_{2}<···<i_{n}≤m} A(α_{i_{1}} + α_{i_{2}} + ⋯ + α_{i_{n}}).  (4.5)

Proof. We prove the lemma by induction on m. For m = 1 and n = 1 we clearly have ⋀^{1} A(α_{1}) ≅_{gr} A(α_{1}). For n ≥ 2, since ⋀^{2} A = 0 it follows that ⋀^{n} A(α) = 0. This shows that (4.5) is valid for m = 1. Now using the induction, by (4.4), since ⋀^{n} A(α) = 0 for any α ∈ Γ and n ≥ 2, we have

⋀^{n} A^{m}(α_{1}, …, α_{m}) ≅_{gr} ⋀^{n} (A(α_{1}) ⊕ A^{m−1}(α_{2}, …, α_{m}))

≅_{gr} A ⊗_{A} ⋀^{n} A^{m−1}(α_{2}, …, α_{m}) ⨁ A(α_{1}) ⊗_{A} ⋀^{n−1} A^{m−1}(α_{2}, …, α_{m})

≅_{gr} ⨁_{2≤i_{1}<i_{2}<···<i_{n}≤m} A(α_{i_{1}} + α_{i_{2}} + ⋯ + α_{i_{n}}) ⨁

A(α_{1}) ⊗_{A} ⨁_{2≤i_{1}<i_{2}<···<i_{n−1}≤m} A(α_{i_{1}} + α_{i_{2}} + ⋯ + α_{i_{n}})

≅_{gr} ⨁_{2≤i_{1}<i_{2}<···<i_{n}≤m} A(α_{i_{1}} + α_{i_{2}} + ⋯ + α_{i_{n}}) ⨁

⨁_{2≤i_{1}<i_{2}<···<i_{n−1}≤m} A(α_{1} + α_{i_{1}} + α_{i_{2}} + ⋯ + α_{i_{n}})

≅_{gr} ⨁_{1≤i_{1}<i_{2}<···<i_{n}≤m} A(α_{i_{1}} + α_{i_{2}} + ⋯ + α_{i_{n}}).

The following corollary is immediate and will be used in Theorem 4.1.5.

Corollary 4.1.4. Let A be a commutative Γ-graded ring. Then

⋀^{n} A^{n}(α_{1}, …, α_{n}) ≅_{gr} A(α_{1} + α_{2} + ⋯ + α_{n}).

Recall from §3.1.5 that if A is a Γ-graded commutative ring, then K_{0}^{gr}(A) is a commutative ring. Denote by K_{0}^{gr}(A)^{∗} the group of invertible elements of this ring. The following theorem establishes the graded version of the exact sequence (4.1).

Theorem 4.1.5. Let A be a commutative Γ-graded ring. Then there is an exact sequence,

1 ⟶ Pic^{gr}(A) ⟶^{ϕ} K_{0}^{gr}(A)^{∗},

where ϕ([P]) = [P].

Proof. It is clear that ϕ : Pic^{gr}(A) → K_{0}^{gr}(A)^{∗} is a well-defined group homomorphism. We only need to show that ϕ is injective. Suppose ϕ([P]) = ϕ([Q]). Thus [P] = [Q] in K_{0}^{gr}(A). By Lemma 3.1.7, P ⊕ A^{n}(\overline{α}) ≅_{gr} Q ⊕ A^{n}(\overline{α}), where \overline{α} = (α_{1}, …, α_{n}). Since P and Q are graded invertible, they are in particular invertible, and so are of constant rank 1. Thus ⋀^{n}(P) = 0, for n ≥ 2. Now by (4.4)

⋀^{n+1} (P ⊕ A^{n}(\overline{α})) ≅_{gr} ⨁_{i+j=n+1} ⋀^{i} A^{n}(\overline{α}) ⊗ ⋀^{j} P

≅_{gr} ⋀^{n} A^{n}(\overline{α}) ⊗ ⋀^{1} P

≅_{gr} A(α_{1} + ⋯ + α_{n}) ⊗ P ≅_{gr} P (α_{1} + ⋯ + α_{n}),

thanks to Corollary 4.1.4. Similarly, ⋀^{n+1} (Q ⊕ A^{n}(\overline{α})) ≅_{gr} Q(α_{1} + ⋯ + α_{n}). Thus

P (α_{1} + ⋯ + α_{n}) ≅_{gr} Q(α_{1} + ⋯ + α_{n})

which implies P ≅_{gr} Q. So ϕ is an injective map.

4.2 Pic^{gr} of a graded noncommutative ring

When A is a noncommutative ring, the definition of the (graded) Picard group is more involved (see [11, Chapter 2], [15], [29, §55], and [36] for non-graded Picard groups of noncommutative rings). Note that if P are Q are A − A-bimodules then P ⊗_{A} Q is not necessarily isomorphic to Q ⊗_{A} P as A-bimodules. This is an indication that the Picard group, in this setting, is not necessarily an abelian group.

Let A and B be Γ-graded rings and P be a graded A − B-bimodule. Then P is called a graded invertible A−B-bimodule, if there is a graded B −A-bimodule Q such that P ⊗_{B} Q ≅_{gr} A as A−Abimodules and Q ⊗_{A} P ≅_{gr} B as B − B-bimodules and the following diagrams are commutative.

\xymatrix{P ⊗_{B} Q ⊗_{A} P \ar[r] \ar[d] & A ⊗_{A} P \ar[d] & Q ⊗_{A} P ⊗_{B} Q \ar[r] \ar[d] & B ⊗_{B} Q \ar[d] \\ P ⊗_{B} B \ar[r] & P & Q ⊗_{A} A \ar[r] & Q.}  (4.6)

As in the commutative case (§4.1), for a noncommutative graded ring A, the graded Picard group, Pic^{gr}(A), is defined as the set of graded isomorphism classes of graded invertible A − A-bimodules with tensor product as multiplications. The graded isomorphism class of the graded bimodule P is denoted by [P]. Since P is invertible, it has an inverse [Q] in Pic^{gr}(A) and the Diagram 4.6 guarantees that ([P][Q])[P] = [P]([Q][P]) = [P].

Theorem 4.2.1. Let A and B be Γ-graded rings. If A is graded Morita equivalent to B, then

Pic^{gr}(A) ≅ Pic^{gr}(B).

Proof. Let Gr-A ≈_{gr} Gr-B. Then there is a graded equivalence ϕ : Gr-A → Gr-B with an inverse ψ : Gr-B → Gr-A. By Theorem 2.3.5 (and its proof), ψ(B) = P is a graded B − A-bi-module and ϕ ≅ − ⊗_{A} P^{∗} and ψ ≅ − ⊗_{B} P. Now one can easily check that the map

Pic^{gr}(A) ⟶ Pic^{gr}(B),
[M] ⟼ [P ⊗_{A} M ⊗_{A} P^{∗}]

is an isomorphism of groups.

Let A be a Γ-graded ring. Consider the group Aut^{gr}(A) of all the graded automorphisms and its subgroup Inn_{A_{0}}^{gr}(A) which consists of graded inner automorphisms induced by the homogeneous elements of degree zero as follows.

Inn_{A_{0}}^{gr}(A) = {f : A → A | f(x) = uxu^{−1}, u ∈ A_{0}^{∗}}.

The graded A-bimodule structure on A induced by f, g ∈ Aut^{gr}(A) will be denoted by_{f}A_{g}. Namely, A acts on _{f}A_{g} as follows, a.x.b = f(a)xg(b). One can prove the following graded A-bimodule isomorphisms.

fA_{g} ≅_{gr} _{hf}A_{hg}
fA_{1} ⊗_{A} _{g}A_{1} ≅_{gr} _{fg}A_{1}
fA_{1} ⊗_{A} _{1}A_{f} ≅_{gr} _{1}A_{f} ⊗_{A} _{f}A_{1} ≅_{gr} A.  (4.7)

The following theorem provides two exact sequences between the groups Γ, Inn_{A_{0}}^{gr}(A) and Aut^{gr}(A) and the graded Picard group. The first sequence belongs to the graded setting, whereas the second sequence is a graded version of a similar result in the non-graded setting (see [29, Theorem 55.11]).

Recall that Z(A) stands for the centre of the ring A which is a graded subring of A (when the grade group Γ is abelian, see Example 1.1.25). Furthermore, it is easy to see that the map

ϕ : Γ ⟶ Pic^{gr}(A),  (4.8)
α ⟼ [A(α)]

(which was considered in the case of commutative graded rings in (4.3)) is well-defined and is a homomorphism.

Theorem 4.2.2. Let A be a Γ-graded ring. Then the following sequences are exact.

(1) The sequence 1 ⟶ Γ_{Z(A)}^{∗} ⟶ Γ ⟶^{ϕ} Pic^{gr}(A), where ϕ(α) = [A(α)].

(2) The sequence 1 ⟶ Inn_{A_{0}}^{gr}(A) ⟶ Aut^{gr}(A) ⟶^{ϕ} Pic^{gr}(A), where ϕ(f) = [_{f}A_{1}].

Proof. (1) Consider the group homomorphism ϕ defined in (4.8). If u ∈ Z(A)^{∗} is a homogeneous element of degree α, then the map ψ : A → A(α), a ↦ ua, is a graded A-bimodule isomorphism. This shows that Γ_{Z(A)}^{∗} ⊆ ker(ϕ). On the other hand if ϕ(α) = [A(α)] = 1_{Pic^{gr}(A)} = [A], then there is a graded A-bimodule isomorphism ψ : A → A(α). From this it follows that there is an invertible homogeneous element u ∈ Z(A) of degree α such that ψ(x) = ux (see also Corollary 1.3.11 and Proposition 1.3.10). This completes the proof.

(2) The fact that the map ϕ is well-defined and is a homomorphism follows from (4.7). We only need to show that ker ϕ coincides with Inn_{A_{0}}^{gr}(A). Let f ∈ Inn_{A_{0}}^{gr}(A), so that f(x) = uxu^{−1}, for any x ∈ A, where u ∈ A_{0}^{∗}. Then the A-graded bimodules _{f}A_{1} and A are isomorphic. Indeed for the map θ : _{f}A_{1} → A, x ↦ u^{−1}x, we have

θ(a.x.b) = θ(uau^{−1}xb) = au^{−1}xb = a.θ(x).b,

which gives a graded A-bimodule isomorphisms. This shows that Inn_{A_{0}}^{gr}(A) ⊆ ker ϕ. Conversely, suppose ϕ(f) = [_{f}A_{1}] = [A]. Thus there is a graded A-bimodule isomorphism θ : _{f}A_{1} → A such that

θ(a.x.b) = θ(f(a)xb) = aθ(x)b.  (4.9)

Since θ is bijective, there is a u ∈ A such that θ(u) = 1. Also, since θ is graded and 1 ∈ A_{0} it follows that u ∈ A_{0}. But A = θ(A) = θ(A1) = Au and similarly A = uA. This implies u ∈ A_{0}^{∗}. Plugging x = a = 1 in (4.9), we have θ(b) = ub, for any b ∈ A. Using this identity, by plugging x = b = 1 in (4.9), we obtain uf(a) = au. Since u is invertible, we get f(a) = u^{−1}au, so f ∈ Inn_{A_{0}}^{gr}(A) and we are done.

For a graded noncommutative ring A the problem of determining the Picard group Pic(A) is difficult. There are several cases in the literature where Picent(A) := Pic_{R}(A) is determined, where R is a centre of A and Pic_{R}(A) is a subgroup of Pic(A) consisting of all isomorphism classes of invertible bimodules P of A which are centralised by R, (i.e., rp = pr, for all r ∈ R and p ∈ P) i.e., P is a A ⊗_{R} A^{op}-left module (see [36], and [29, §55]). For a graded division algebra A (i.e., a graded division ring which is finite dimensional over its centre) we will be able to determine Pic^{gr}(A). In fact since graded division algebras are graded Azumaya algebras, we first determine Pic_{R}^{gr}(A), where A is a graded Azumaya algebra over its centre R. The approach follows the idea in Bass [15, §3, Corollary 4.5], where it is shown that for an Azumaya algebra A over R, the Picard group Pic_{R}(A) coincides with Pic(R).

Recall that a graded algebra A over a commutative graded ring R is called a graded Azumaya algebra if A is a graded faithfully projective R-module and the natural map ϕ : A ⊗_{R} A^{op} → End_{R}(A), ϕ(a ⊗ b)(x) = axb, where a, x ∈ A and b ∈ A^{op}, is a graded R-isomorphism. This implies that A is an Azumaya algebra over R. Conversely, if A is a graded algebra over graded ring R and A is an Azumaya algebra over R, then A is faithfully projective as an R-module and the natural map ϕ : A ⊗_{R} A^{op} → End_{R}(A) is naturally graded. Thus A is also a graded Azumaya algebra.

A graded division algebra is a graded central simple algebra. We first define these type of rings and show that they are graded Azumaya algebras.

A graded algebra A over a graded commutative ring R is said to be a graded central simple algebra over R if A is graded simple ring, i.e., A does not have any proper graded two-sided ideals, Z(A) ≅_{gr} R and A is finite dimensional as an R-module. Note that since A is graded simple, the centre of A (identified with R), is a graded field. Thus A is graded free as a graded module over its centre by Proposition 1.4.1, so the dimension of A over R is uniquely defined.

Proposition 4.2.3. Let A be a Γ-graded central simple algebra over a graded field R. Then A is a graded Azumaya algebra over R.

Proof. Since A is graded free of finite dimension over R, it follows that A is faithfully projective over R. Consider the natural graded R-algebra homomorphism ψ : A ⊗_{R} A^{op} → End_{R}(A) defined by ψ(a ⊗ b)(x) = axb where a, x ∈ A, b ∈ A^{op}. Since graded ideals of A^{op} coincide with graded ideals of A, A^{op} is also graded simple. Thus A⊗A^{op} is also graded simple (see [51, Proposition 1.1], so ψ is injective. Hence the map is surjective by dimension count, using Theorem 1.4.3. This shows that A is an Azumaya algebra over R, as required.

Let A be a graded algebra over a graded commutative ring R. For a graded A-bimodule P centralised by R, define

P^{A} = {p ∈ P | ap = pa, for all a ∈ A}.

Denote by Gr-A_{R} -Gr the category of graded A-bimodules centralised by R. The functors

Gr-A_{R} -Gr ⟶ Gr-R,  Gr-R ⟶ Gr-A_{R} -Gr,
P ⟼ P^{A}  N ⟼ N ⊗_{R} A

are inverse equivalence of categories which graded projective modules correspond to graded projective modules and furthermore, invertible modules correspond to invertible modules (see [24, Proposition III.4.1]). This immediately implies that

Pic_{R}^{gr}(A) ≅ Pic^{gr}(R).  (4.10)

Lemma 4.2.4. Let A be a Γ-graded division algebra with centre R. Then Pic_{R}^{gr}(A) ≅ Γ/Γ_{R}.

Proof. By Proposition 4.2.3, A is a graded Azumaya algebra over R. By (4.10), Pic_{R}^{gr}(A) ≅ Pic^{gr}(R). Since R is a graded field, by Proposition 4.1.1, Pic^{gr}(R) = Γ/Γ_{R}.

Example 4.2.5. Recall from Example 1.1.21 that the Hamilton quaternion algebra ℍ = ℝ ⊕ ℝi ⊕ ℝj ⊕ ℝk is a graded division algebra with two different gradings, i.e., ℤ_{2} and ℤ_{2} × ℤ_{2}, respectively. Since in both gradings, the centre ℝ is concentrated in degree 0, by Lemma 4.2.4, Pic_{ℝ}^{gr}(ℍ) = ℤ_{2} or Pic_{ℝ}^{gr}(ℍ) = ℤ_{2} ⊕ ℤ_{2}, depending on the grading. This also shows that contrary to the graded Grothendieck group, Pic_{ℝ}^{gr}(ℍ) ≅ P̸ic(ℍ_{0}), despite the fact that in both gradings, ℍ is a strongly graded ring.

Example 4.2.6. Let (D, v) be a tame valued division algebra over a henselian field F, where v : D^{∗} → Γ is the valuation homomorphism. By Example 1.4.7, there is a Γ-graded division algebra gr(D) associated to D with the centre gr(F), where Γ_{gr(D)} = Γ_{D} and Γ_{gr(F)} = Γ_{F}. Since by Lemma 4.2.3, graded division algebras are graded Azumaya, by Lemma 4.2.4,

Pic_{gr(F)}^{gr} ( gr(D)) = Γ/Γ_{F} .

Let A be a strongly graded Γ-graded ring. Then for any α ∈ Γ, A_{α} is a finitely generated projective invertible A_{0}-bimodule (see Theorem 1.5.11) and the map ψ : Γ → Pic(A_{0}), α ↦ [A_{α}] is a group homomorphism. We then have a natural commutative diagram

\xymatrix{Γ \ar[dr]_{ψ} \ar[r]^{ϕ} & Pic^{gr}(A) \ar[d]^{(−)_{0}} \\  & Pic(A_{0})}  (4.11)

Remark 4.2.7. \underline{Relating Pic^{gr}(A) to Pic(A_{0}) for a strongly graded ring A}

Example 4.2.5 shows that for a strongly graded ring A, Pic^{gr}(A) is not necessarily isomorphic to Pic(A_{0}). However one can relate these two groups with an exact sequence as follows.

1 ⟶ H^{1}(Γ, Z(A_{0})^{∗}) ⟶ Pic^{gr}(A) ⟶^{Ψ} Pic(A_{0})^{Γ} ⟶ H^{2}(Γ, Z(A_{0})^{∗}).

Here Γ acts on Pic(A_{0}) by γ[P] = [A_{γ} ⊗_{A_{0}} P ⊗_{A_{0}} A_{−γ}], so the notation Pic(A_{0})^{Γ} refers to the group of Γ-invariant elements of Pic(A_{0}). Furthermore, Z(A_{0})^{∗} denotes the units of the centre of A_{0} and H^{1}, H^{2} denote the first and second cohomology groups. The map Ψ is defined by Ψ([P]) = [P_{0}] (see [67] for details. Also see [16]).

We include a result from [75] which describes if A_{0} has IBN, then A has gr-IBN, based on a condition on Pic(A_{0}). In general, one can produce an example of strongly graded ring A such that A_{0} has IBN whereas A is a non-IBN ring.

For a ring R, define Pic_{n}(R) = {[X] ∈ Pic(R) | X^{n} ≅ R^{n} as right R-module}. This is a subgroup of Pic(R) and Pic_{n}(R) and Pic_{m}(R) are subgroups of Pic_{nm}(R). Thus Pic_{∞}(R) = ⋃_{n≥1} Pic_{n}(R) is a subgroup of Pic(R).

Proposition 4.2.8. Let A be a strongly Γ-graded ring such that {[A_{α}] | α ∈ Γ} ⊆ Pic_{∞}(A_{0}). If A_{0} has IBN then A has gr-IBN. Furthermore, if Γ is finite, then A has IBN if and only if A_{0} has IBN.

Proof. Let A^{n}(\overline{α}) ≅_{gr} A^{m}(\overline{β}) as graded right A-modules, where \overline{α} = (α_{1}, …, α_{n}) and \overline{β} = (β_{1}, …, β_{m}). Using Dade’s theorem 1.5.1, we have

A_{α_{1}} ⊕ ⋯ ⊕ A_{α_{n}} ≅ A_{β_{1}} ⊕ ⋯ ⊕ A_{β_{m}},  (4.12)

as right A_{0}-modules. Since each [A_{α_{i}}] and [A_{β_{j}}] is in some Pic_{n_{i}}(A_{0}) and Pic_{n_{j}}(A_{0}), respectively, there is a large enough t such that [A_{α_{i}}], [A_{β_{j}}] ∈ Pic_{t}(A_{0}), for all 1 ≤ i ≤ n and 1 ≤ j ≤ m. Thus A_{α_{i}} ≅ A_{0}^{t} and A_{β_{j}} ≅ A_{0}^{t}, for all i and j. Replacing this into (4.12) we get A_{0}^{nt} ≅ A_{0}^{mt}. Since A_{0} has IBN, it follows n = m.

Now suppose Γ is finite. If A_{0} has IBN, then we will prove that A has IBN. From what we proved above, it follows that A has gr-IBN. If A^{n} ≅ A^{m}, then U(A^{n}) ≅_{gr} U(A^{m}), where U is the forgetful functor (see §1.2.7). Since U(A^{n}) = U(A)^{n} and U(A) = ⨁_{γ∈Γ} A(γ), using the fact that A has gr-IBN, it follows immediately that n = m. Conversely, suppose A has IBN. If A_{0}^{n} ≅ A_{0}^{m}, then tensoring with A over A_{0} we have A^{n} ≅_{gr} A_{0}^{n} ⊗_{A_{0}} A ≅ A_{0}^{m} ⊗_{A_{0}} A ≅_{gr} A^{m}. So n = m and we are done.

Graded Ultramatricial Algebras, Classification via K_{0}^{gr}

Let F be a field. An F -algebra is called an ultramatricial algebra, if it is isomorphic to the union of an increasing chain of a finite product of matrix algebras over F. When F is the field of complex numbers, these algebras are also called locally semisimple algebras (or LS-algebras for short), as they are isomorphic to a union of a chain of semisimple ℂ-algebras. An important example of such rings is the group ring ℂ[S_{∞}], where S_{∞} is the infinite symmetric group (see Example 5.2.2). These rings appeared in the setting of C^{∗}-algebras and then von Neumann regular algebras in the work of Grimm, Brattelli, Elliott, Goodearl, Handelman and many others after them.

Despite their simple constructions, the study of ultramatricial algebras are far from over. As it is noted in [90] “The current state of the theory of LS-algebras and its applications should be considered as the initial one; one has discovered the first fundamental facts and noted a general circle of questions. To estimate it in perspective, one must consider the enormous number of diverse and profound examples of such algebras. In addition one can observe the connections with a large number of areas of mathematics.”

One of the sparkling examples of the Grothendieck group as a complete invariant is in the setting of ultramatricial algebras (and AF C^{∗}-algebras). It is by now a classical result that the K_{0}-group along with its positive cone and the position of identity is a complete invariant for such algebras ([38, Theorem 15.26]). To be precise, let R and S be (unital) ultramatricial algebras. Then R ≅ S if and only if there is an order isomorphism (K_{0}(R), [R]) ≅ (K_{0}(S), [S]).

The theory has also been worked out for the non-unital ultramatricial algebras (see [39, Chapter XII]). Two valuable surveys on ultramatricial algebras and their relations with other branches of mathematics are [90, 97]. The lecture notes by Effros [35] is also an excellent detailed account of this theory.

In this section we initiate the graded version of this theory. We define the graded ultramatricial algebras. We then show that the graded Grothendieck group, equipped with its module structure and its ordering is a complete invariant for such algebras (Theorem 5.2.5). Again, when the graded group considered to be trivial, we retrieve the Brattelli-Elliott Theorem (Corollary 5.2.7).

5.1 Graded matricial algebras

We begin with the graded version of matricial algebras.

Definition 5.1.1. Let A be a Γ-graded field. A Γ-graded matricial A-algebra is a graded A-algebra of the form

𝕄_{n_{1}}(A)(\overline{δ}_{1}) × ⋯ × 𝕄_{n_{l}}(A)(\overline{δ}_{l}),

where \overline{δ}_{i} = (δ_{1}^{(i)}, …, δ_{n_{i}}^{(i)}), δ_{j}^{(i)} ∈ Γ, 1 ≤ j ≤ n_{i} and 1 ≤ i ≤ l.

If Γ is a trivial group, then Definition 5.1.1 reduces to the definition of matricial algebras ([38, §15]). Note that if R is a graded matricial A-algebra, then R_{0} is a matricial A_{0}-algebra (see §1.4.1).

In general, when two graded finitely generated projective modules represent the same element in the graded Grothendieck group, then they are not necessarily graded isomorphic but rather graded stably isomorphic (see Lemma 3.1.7(3)). However for graded matricial algebras one can prove that the graded stably isomorphic modules are in fact graded isomorphic. Later in Lemma 5.1.5 we show this is also the case in the larger category of graded ultramatricial algebras.

Lemma 5.1.2. Let A be a Γ-graded field and R be a Γ-graded matricial A-algebra. Let P and Q be graded finitely generated projective R-modules. Then [P] = [Q] in K_{0}^{gr}(R), if and only if P ≅_{gr} Q.

Proof. Since the functor K_{0}^{gr} respects the direct sum, it suffices to prove the statement for a graded matricial algebra of the form R = 𝕄_{n}(A)(\overline{δ}). Let P and Q be graded finitely generated projective R-modules such that [P] = [Q] in K_{0}^{gr}(R). By Proposition 2.1.2, R = 𝕄_{n}(A)(\overline{δ}) is graded Morita equivalent to A. So there are equivalent functors ψ and ϕ such that ψϕ ≅ 1 and ϕψ ≅ 1, which also induce an isomorphism K_{0}^{gr}(ψ) : K_{0}^{gr}(R) → K_{0}^{gr}(A) such that [P] ↦ [ψ(P)]. Now since [P] = [Q], it follows [ψ(P)] = [ψ(Q)] in K_{0}^{gr}(A). But since A is a graded field, by the proof of Proposition 3.6.1, any graded finitely generated projective A-module can be written uniquely as a direct sum of appropriate shifted A. Writing ψ(P) and ψ(Q) in this form, the homomorphism (3.30) shows that ψ(P) ≅_{gr} ψ(Q). Now applying the functor ϕ to this we obtain P ≅_{gr} Q.

Let A be a Γ-graded field (with the support Γ_{A}) and 𝒞 be a category consisting of Γ-graded matricial A-algebras as objects and A-graded algebra homomorphisms as morphisms. We consider the quotient category 𝒞^{out} obtained from 𝒞 by identifying homomorphisms which differ up to a degree zero graded inner-automorphim. That is, the graded homomorphisms ϕ, ψ ∈ Hom_{𝒞}(R, S) represent the same morphism in 𝒞^{out} if there is an inner-automorpshim θ : S → S, defined by θ(s) = xsx^{−1}, where deg(x) = 0, such that ϕ = θψ. The following theorem shows that K_{0}^{gr} “classifies” the category of 𝒞^{out}. This is a graded analog of a similar result for matricial algebras (see [38, Lemma 15.23]).

Theorem 5.1.3. Let A be a Γ-graded field and 𝒞^{out} be the category consisting of Γ-graded matricial A-algebras as objects and A-graded algebra homomorphisms modulo graded inner-automorphisms as morphisms. Then K_{0}^{gr} : 𝒞^{out} → 𝒫 is a fully faithful functor. Namely,

(1) (well-defined and faithful) For any graded matricial A-algebras R and S and ϕ, ψ ∈ Hom_{𝒞}(R, S), we have ϕ(r) = xψ(r)x^{−1}, r ∈ R, for some invertible homogeneous element x of degree 0 in S, if and only if K_{0}^{gr}(ϕ) = K_{0}^{gr}(ψ).

(2) (full) For any graded matricial A-algebras R and S and the morphism

f : (K_{0}^{gr}(R), [R]) → (K_{0}^{gr}(S), [S])

in 𝒫, there is ϕ ∈ Hom_{𝒞}(R, S) such that K_{0}^{gr}(ϕ) = f.

Proof. (1) (well-defined.) Let ϕ, ψ ∈ Hom_{𝒞}(R, S) such that ϕ = θψ, where θ(s) = xsx^{−1} for some invertible homogeneous element x of S of degree 0. Then K_{0}^{gr}(ϕ) = K_{0}^{gr}(θψ) = K_{0}^{gr}(θ)K_{0}^{gr}(ψ) = K_{0}^{gr}(ψ) since K_{0}^{gr}(θ) is the identity map (see §3.4.1).

(faithful.) The rest of the proof is similar to the non-graded version with an extra attension given to the grading (cf. [38, p.218]). We give it here for the completeness. Let K_{0}^{gr}(ϕ) = K_{0}^{gr}(ψ). Let R = 𝕄_{n_{1}}(A)(\overline{δ}_{1}) × ⋯ × 𝕄_{n_{l}}(A)(\overline{δ}_{l}) and set g_{jk}^{(i)} = ϕ(e_{jk}^{(i)}) and h_{jk}^{(i)} = ψ(e_{jk}^{(i)}) for 1 ≤ i ≤ l

and 1 ≤ j, k ≤ n_{i}, where e_{jk}^{(i)} are the standard basis for 𝕄_{n_{i}}(A). Since ϕ and ψ are graded

homomorphism, deg(e_{jk}^{(i)}) = deg(g_{jk}^{(i)}) = deg(h_{jk}^{(i)}) = δ_{j}^{(i)} − δ_{k}^{(i)}.

Since e_{jj}^{(i)} are pairwise graded orthogonal idempotents (of degree zero) in R and ∑_{i=1}^{l} ∑_{j=1}^{n_{i}} e_{jj}^{(i)} =

1, the same is also the case for g_{jj}^{(i)} and h_{jj}^{(i)}. Then

[g_{11}^{(i)}S] = K_{0}^{gr}(ϕ)([e_{11}^{(i)}R]) = K_{0}^{gr}(ψ)([e_{11}^{(i)}R]) = [h_{11}^{(i)}S].

By Lemma 5.1.2, g_{11}^{(i)}S ≅_{gr} h_{11}^{(i)}S. Thus there are homogeneous elements of degree zero x_{i} and y_{i} such that x_{i}y_{i} = g_{11}^{(i)} and y_{i}x_{i} = h_{11}^{(i)} (see §1.2.10). Let x = ∑_{i=1}^{l} ∑_{j=1}^{n_{i}} g_{j1}^{(i)}x_{i}h_{1j}^{(i)} and y =

∑_{i=1}^{l} ∑_{j=1}^{n_{i}} h_{j1}^{(i)}y_{i}g_{1j}^{(i)}. Note that x and y are homogeneous elements of degree zero. One can easily check that xy = yx = 1. Now for 1 ≤ i ≤ l and 1 ≤ j, k ≤ n_{i} we have

xh_{jk}^{(i)} = ∑_{s=1}^{l} ∑_{t=1}^{n_{s}} g_{t1}^{(s)}x_{s}h_{1t}^{(s)}h_{jk}^{(i)}

= g_{j1}^{(i)}x_{i}h_{1j}^{(i)}h_{jk}^{(i)} = g_{jk}^{(i)}g_{k1}^{(i)}x_{i}h_{1k}^{(i)}

= ∑_{s=1}^{l} ∑_{t=1}^{n_{s}} g_{jk}^{(i)}g_{t1}^{(s)}x_{s}h_{1t}^{(i)} = g_{jk}^{(i)}x

Let θ : S → S be the graded inner-automorphism θ(s) = xsy. Then

θψ(e_{jk}^{(i)})) = xh_{jk}^{(i)}y = g_{jk}^{(i)} = ϕ(e_{jk}^{(i)}).

Since e_{jk}^{(i)}, 1 ≤ i ≤ l and 1 ≤ j, k ≤ n_{i}, form a homogeneous A-basis for R, θψ = ϕ.

(2) Let R = 𝕄_{n_{1}}(A)(\overline{δ}_{1}) × ⋯ × 𝕄_{n_{l}}(A)(\overline{δ}_{l}). Consider R_{i} = 𝕄_{n_{i}}(A)(\overline{δ}_{i}), 1 ≤ i ≤ l. Each R_{i} is a graded finitely generated projective R-module, so f([R_{i}]) is in the cone of K_{0}^{gr}(S), i.e., there is a graded finitely generated projective S-module P_{i} such that f([R_{i}]) = [P_{i}]. Then

[S] = f([R]) = f([R_{1}] + ⋯ + [R_{l}]) = [P_{1}] + ⋯ + [P_{l}] = [P_{1} ⊕ ⋯ ⊕ P_{l}].

Since S is a graded matricial algebra, by Lemma 5.1.2, P_{1} ⊕ ⋯ ⊕ P_{l} ≅_{gr} S as a right S-module. So there are homogeneous orthogonal idempotents g_{1}, …, g_{l} ∈ S such that g_{1} + ⋯ + g_{l} = 1 and g_{i}S ≅_{gr} P_{i} (see §1.2.10). Note that each of R_{i} = 𝕄_{n_{i}}(A)(\overline{δ}_{i}) are graded simple algebras. Set \overline{δ}_{i} = (δ_{1}^{(i)}, …, δ_{n}^{(i)}) (here n = n_{i}). Let e_{jk}^{(i)},1 ≤ j, k ≤ n, be matrix units of R_{i} and consider the

graded finitely generated projective (right) R_{i}-module, V = e_{11}^{(i)}R_{i} = A(δ_{1}^{(i)} − δ_{1}^{(i)}) ⊕ A(δ_{2}^{(i)} − δ_{1}^{(i)}) ⊕ ⋯ ⊕ A(δ_{n}^{(i)} − δ_{1}^{(i)}). Then (1.33) shows

R_{i} ≅_{gr} V (δ_{1}^{(i)} − δ_{1}^{(i)}) ⊕ V (δ_{1}^{(i)} − δ_{2}^{(i)}) ⊕ ⋯ ⊕ V (δ_{1}^{(i)} − δ_{n}^{(i)}),

as graded R-module. Thus

[P_{i}] = [g_{i}S] = f([R_{i}]) = f([V (δ_{1}^{(i)} − δ_{1}^{(i)})]) + f([V (δ_{1}^{(i)} − δ_{2}^{(i)})]) + ⋯ + f([V (δ_{1}^{(i)} − δ_{n}^{(i)})]). (5.1)

There is a graded finitely generated projective S-module Q such that f([V]) = f([V (δ_{1}^{(i)} − δ_{1}^{(i)})]) = [Q]. Since f is a ℤ[Γ]-module homomorphism, for 1 ≤ k ≤ n,

f([V (δ_{1}^{(i)} − δ_{k}^{(i)})]) = f((δ_{1}^{(i)} − δ_{k}^{(i)})[V]) = (δ_{1}^{(i)} − δ_{k}^{(i)})f([V]) = (δ_{1}^{(i)} − δ_{k}^{(i)})[Q] = [Q(δ_{1}^{(i)} − δ_{k}^{(i)})].

From (5.1) and Lemma 5.1.2 now follows

g_{i}S ≅_{gr} Q(δ_{1}^{(i)} − δ_{1}^{(i)}) ⊕ Q(δ_{1}^{(i)} − δ_{2}^{(i)}) ⊕ ⋯ ⊕ Q(δ_{1}^{(i)} − δ_{n}^{(i)}).  (5.2)

Let g_{jk}^{(i)} ∈ End(g_{i}S) ≅_{gr} g_{i}Sg_{i} maps the j-th summand of the right hand side of (5.2) to its k-th

summand and everything else to zero. Observe that deg(g_{jk}^{(i)}) = δ_{j}^{(i)} − δ_{k}^{(i)} and g_{jk}^{(i)}, 1 ≤ j, k ≤ n,

form the matrix units. Furthermore, g_{11}^{(i)} + ⋯ + g_{nn}^{(i)} = g_{i} and g_{11}^{(i)}S = Q(δ_{1}^{(i)} − δ_{1}^{(i)}) = Q. Thus [g_{11}^{(i)}S] = [Q] = f([V]) = f([e_{11}^{(i)}R_{i}]).

Now for 1 ≤ i ≤ l, define the A-algebra homomorphism R_{i} → g_{i}Sg_{i}, e_{jk}^{(i)} ↦ g_{jk}^{(i)}. This is a graded

homomorphism, and induces a graded homorphism ϕ : R → S such that ϕ(e_{jk}^{(i)}) = g_{jk}^{(i)}. Clearly

K_{0}^{gr}(ϕ)([e_{11}^{(i)}R_{i}]) = [ϕ(e_{11}^{(i)})S] = [g_{11}^{(i)}S] = f([e_{11}^{(i)}R_{i}]),

for 1 ≤ i ≤ l. Now K_{0}^{gr}(R) is generated by [e_{11}^{(i)}R_{i}], 1 ≤ i ≤ l, as ℤ[Γ]-module. This implies that K_{0}^{gr}(ϕ) = f.

Remark 5.1.4. In Theorem 5.1.3, both parts (1) and (2) are valid when the ring S is a graded ultramatricial algebra. In fact, in the proofs of (1) and (2), the only property of S which is used is that if for two graded finitely generated projective S-modules P and Q, we have [P] = [Q] in K_{0}^{gr}(S), then P ≅_{gr} Q. Lemma 5.1.5 shows that this is the case for ultramatricial algebras.

Lemma 5.1.5. Let A be a Γ-graded field and R be a Γ-graded ultramatricial A-algebra. Let P and Q be graded finitely generated projective R-modules. Then [P] = [Q] in K_{0}^{gr}(R), if and only if P ≅_{gr} Q.

Proof. We will use the description of K_{0}^{gr} based on homogeneous idempotents to prove this Lemma (see §3.2). Let p and q be homogeneous idempotents matrices over R corresponding to the graded finitely generated R-modules P and Q, respectively (see Lemma 3.2.3(2)). Suppose [P] = [Q] in K_{0}^{gr}(R). We will show that p and q are graded equivalent in R, which by Lemma 3.2.3(3), implies that P ≅_{gr} Q. Since by Definition 5.2.1, R = ⋃_{i∈I} R_{i}, there is j ∈ I such that p and q are homogeneous idempotent matrices over R_{j}. But since [p] = [q] in K_{0}^{gr}(R), there is an n ∈ ℕ, such that p⊕1_{n} is graded equivalent to q ⊕1_{n} in R (see §3.2.1). So there is k ∈ I, k ≥ j, such that p⊕1_{n} and q ⊕1_{n} are graded equivalent in R_{k}. Thus [p⊕1_{R_{k}}] = [q ⊕1_{R_{k}}] in K_{0}^{gr}(R_{k}). This implies [p] = [q] in K_{0}^{gr}(R_{k}). Since R_{k} is a graded matricial algebra, by Lemma 5.1.5, p is graded equivalent to q in R_{k}. So p is graded equivalent to q in R and consequently P ≅_{gr} Q as R-module. The converse is immediate.

5.2 Graded ultramatricial algebras, classification via K_{0}^{gr}

The direct limit of a direct system of graded rings is a ring with a graded structure (see Example 1.1.10). In this section, we study a particular case of such graded rings, namely the direct limit of graded matricial algebras. Recall from Definition 5.1.1 that a Γ-graded matricial algebra over the graded field A is of the form

𝕄_{n_{1}}(A)(\overline{δ}_{1}) × ⋯ × 𝕄_{n_{l}}(A)(\overline{δ}_{l}),

for some shifts \overline{δ}_{i}, 1 ≤ i ≤ l.

Definition 5.2.1. Let A be a Γ-graded field. Then the ring R is called a Γ-graded ultramatricial Aalgebra if R = ⋃_{i=1}^{∞} R_{i}, where R_{1} ⊆ R_{2} ⊆ … is a sequence of graded matricial A-subalgebras. Here the inclusion respects the grading, i.e., R_{i}_{α} ⊆ R_{i+1}_{α}. Furthermore, under the inclusion R_{i} ⊆ R_{i+1}, we have 1_{R_{i}} = 1_{R_{i+1}}.

Clearly R is also a Γ-graded A-algebra with R_{α} = ⋃_{i=1}^{∞} R_{i}_{α}. If Γ is a trivial group, then Definition 5.2.1 reduces to the definition of ultramatricial algebras ([38, §15]). Note that if R is a graded ultramatricial A-algebra, then R_{0} is a ultramatricial A_{0}-algebra (see §1.4.1).

Example 5.2.2. Let ℂ be a field of complex numbers. Furthermore let Γ_{1} ⊆ Γ_{2} ⊆ … be a chain of finite abelian groups. Consider the locally finite group Γ := ⋃_{i=1}^{∞} Γ_{i}. Consider ℂ as Γ-graded field concentrated in degree 0. Then the group ring ℂ[Γ] is a Γ-graded ultramatricial ℂ-algebra. The construction is as follows. For each i, the group ring ℂ[Γ_{i}] is considered as a Γ-graded ring with the support Γ_{i} (with the standard grading of group rings as in Example 1.1.7). Then the inclusion ℂ[Γ_{i}] ⊆ ℂ[Γ_{i+1}] gives a sequence of graded ℂ-algebras. By Maschke’s theorem, each of ℂ[Γ_{i}] is a matricial algebra with the grading coming from the group ring. This makes ℂ[Γ] = ⋃_{i} ℂ[Γ_{i}] a graded ultramatricial ℂ-algebra.

Example 5.2.3. Let A be a ring. We identify 𝕄_{n}(A) as a subring of 𝕄_{2n}(A) under the monomor-

phism X ∈ 𝕄_{n}(A) ↦ (\begin{matrix}X & 0 \\ 0 & X \end{matrix}). With this identification, we have a sequence

𝕄_{2}(A) ⊆ 𝕄_{4}(A) ⊆ ….

Now let A be a Γ-graded field, α_{1}, α_{2} ∈ Γ and consider the sequence of graded subalgebras

𝕄_{2}(A)(α_{1}, α_{2}) ⊆ 𝕄_{4}(A)(α_{1}, α_{2}, α_{1}, α_{2}) ⊆ ….

Then

R = ⋃_{i=1}^{∞} 𝕄_{2^{i}}(A)((α_{1}, α_{2})^{2^{i−1}}),

where (α_{1}, α_{2})^{k} stands for k copies of (α_{1}, α_{2}), is a graded ultramatricial algebra.

Example 5.2.4. Let A = K[x^{3}, x^{−3}] be a ℤ-graded field, where K is a field. Consider the following two sequences of graded matricial algebras

𝕄_{2}(A)(0, 1) ⊆ 𝕄_{4}(A)(0, 1, 0, 1) ⊆ 𝕄_{8}(A)(0, 1, 0, 1, 0, 1, 0, 1) ⊆ …  (5.3)
𝕄_{2}(A)(0, 1) ⊆ 𝕄_{4}(A)(0, 1, 1, 2) ⊆ 𝕄_{8}(A)(0, 1, 1, 2, 0, 1, 1, 2) ⊆ …  (5.4)

Let R and S be graded ultramatricial algebras constructed as in Example 5.2.3, from the union of matricial algebras of sequences 5.3 and 5.4, respectively. We calculate K_{0}^{gr}(R) and K_{0}^{gr}(S). Since K_{0}^{gr} respects the direct limit (Theorem 3.2.4), we have

K_{0}^{gr}(R) = K_{0}^{gr}(\underrightarrow{lim} R_{i}) = \underrightarrow{lim} K_{0}^{gr}(R_{i}),

where R_{i} corresponds to the i-th algebra in the sequence 5.3. Since all the matricial algebras R_{i} are strongly graded, we get \underrightarrow{lim} K_{0}^{gr}(R_{i}) = \underrightarrow{lim} K_{0}(R_{i_{0}}).

Recall that (see Proposition 1.4.1) if

T ≅_{gr} 𝕄_{m} (K[x^{n}, x^{−n}])(p_{1}, …, p_{m}),

then letting d_{l}, 0 ≤ l ≤ n − 1, be the number of i such that p_{i} represents \overline{l} in ℤ/nℤ, we have

T ≅_{gr} 𝕄_{m} (K[x^{n}, x^{−n}])(0, …, 0, 1, …, 1, …, n − 1, …, n − 1),

where 0 ≤ l ≤ n − 1 occurring d_{l} times. It is now easy to see

T_{0} ≅ 𝕄_{d_{0}}(K) × ⋯ × 𝕄_{d_{n−1}}(K).

Using this the zero homogeneous ring of the sequence 5.3 becomes

K ⊕ K ⊆ 𝕄_{2}(K) ⊕ 𝕄_{2}(K) ⊆ ⋯
(x, y) ↦ (\begin{matrix}x & 0 \\ 0 & x \end{matrix}) ⊕ (\begin{matrix}y & 0 \\ 0 & y \end{matrix}) ↦ ⋯

Thus

K_{0}^{gr}(R) ≅ \underrightarrow{lim} K_{0}(R_{i_{0}}) ≅ ℤ[1/2] ⊕ ℤ[1/2].

Similarly, the zero homogeneous ring of the sequence 5.4 becomes

K ⊕ K ⊆ K ⊕ 𝕄_{2}(K) ⊕ K ⊆ 𝕄_{2}(K) ⊕ 𝕄_{4}(K) ⊕ 𝕄_{2}(K)
(x, y) ↦ (x, (\begin{matrix}y & 0 \\ 0 & x \end{matrix}) , y)

(x, y, z) ↦ ((\begin{matrix}x & 0 \\ 0 & x \end{matrix}) , (\begin{matrix}y & 0 \\ 0 & y \end{matrix}) , (\begin{matrix}z & 0 \\ 0 & z \end{matrix})).

Thus

K_{0}^{gr}(S) ≅ \underrightarrow{lim} K_{0}(S_{i_{0}}) ≅ ℤ[1/2] ⊕ ℤ[1/2] ⊕ ℤ[1/2].

We are now in a position to classify the graded ultramatricial algebras via K_{0}^{gr}-group. The following theorem shows that the dimension module is a complete invariant for the category of graded ultramatricial algebras.

Theorem 5.2.5. Let R and S be Γ-graded ultramatricial algebras over a graded field A. Then R ≅_{gr} S as graded A-algebras if and only if there is an order preserving ℤ[Γ]-module isomorphism

(K_{0}^{gr}(R), [R]) ≅ (K_{0}^{gr}(S), [S]).

Proof. One direction is clear. Let R = ⋃_{i=1}^{∞} R_{i} and S = ⋃_{i=1}^{∞} S_{i}, where R_{1} ⊆ R_{2} ⊆ … and S_{1} ⊆ S_{2} ⊆ … are sequences of graded matricial A-algebras. Let ϕ_{i} : R_{i} → R and ψ_{i} : S_{i} → S, i ∈ ℕ, be inclusion maps. In order to show that the isomorphism

f : (K_{0}^{gr}(R), [R]) ⟶ (K_{0}^{gr}(S), [S])

between the graded Grothendieck groups gives the isomorphism between the rings R and S, we will find a sequence n_{1} < n_{2} < ⋯ of positive numbers and graded A-module injections ρ_{k} : R_{n_{k}} → S such that ρ_{k+1} is an extension of ρ_{k} and ⋃_{k=1}^{∞} R_{n_{k}} = R. To achieve this we repeatedly use Theorem 5.1.3 (and Remark 5.1.4) (a “local” version of this theorem) and the fact that since R_{n}, n ∈ ℕ, is a finite dimensional A-algebra, for any A-graded homomorphism ρ : R_{n} → S, we have ρ(R_{n}) ⊆ S_{i}, for some positive number i. Throughout the proof, for simplicity, we write \overline{θ} for the ℤ[Γ]-homomorphism K_{0}^{gr}(θ) induced by a graded A-algebra homomorphism θ : R → S.

We first prove two auxiliary facts.

I. If σ : S_{k} → R_{n} is a graded A-algebra such that

\overline{ϕ}_{n}\overline{σ} = f̄^{−1}\overline{ψ}_{k},  (5.5)

(see Diagram 5.6) then there exist an integer j > k and a graded A-algebra homomorphism ρ : R_{n} → S_{j} such that ψ_{j}ρσ = ψ_{k} and \overline{ψ}_{j}\overline{ρ} = \overline{f} \overline{ϕ}_{n}.

[figure: S_{k}; σ; ψ_{k}; R_{n}; ϕ_{n}; ρ; S_{j}; ψ_{j}; S; R.; K_{0}^{gr}(S_{k}); \overline{σ}; _{\overline{ψ}_{k}}; K_{0}^{gr}(R_{n}); _{\overline{ϕ}_{n}}; \overline{ρ}; K_{0}^{gr}(S_{j}); _{\overline{ψ}_{j}}; K_{0}^{gr}(S); f̄^{−1}; K_{0}^{gr}(R).]  (5.6)

Proof of I. Consider \overline{f} \overline{ϕ}_{n} : K_{0}^{gr}(R_{n}) → K_{0}^{gr}(S). By Theorem 5.1.3 (and Remark 5.1.4), there is an A-graded homomorphism ρ′ : R_{n} → S such that \overline{ρ′} = \overline{f} \overline{ϕ}_{n}. Since R_{n} (a matricial A-algebra) is a finite dimensional A-algebra, ρ′(R_{n}) ⊆ S_{i} for some i. Thus ρ′ gives a graded homomorphism ρ′′ : R_{n} → S_{i} such that ψ_{i}ρ′′ = ρ′ (recall that ψ_{i} is just an inclusion). Furthermore

\overline{ψ}_{i}\overline{ρ′′} = \overline{ρ′} = \overline{f} \overline{ϕ}_{n}.  (5.7)

Then, using (5.5), \overline{ψ}_{i}\overline{ρ′′}\overline{σ} = \overline{f} \overline{ϕ}_{n}\overline{σ} = \overline{ψ}_{k}. Theorem 5.1.3 implies that there is a graded inner automorphism θ of S such that

θψ_{i}ρ′′σ = ψ_{k}.  (5.8)

The restriction of θ on S_{i} gives θ|_{S_{i}} : S_{i} → S. Since S_{i} is finite dimensional A-algebra, it follows that there is j such that θ(S_{i}) ⊆ S_{j}. So θ gives a graded homomorphism θ′ : S_{i} → S_{j} such that ψ_{j}θ′ = θψ_{i}. Set ρ = θ′ρ′′ : R_{n} → S_{i}. We get, using (5.8)

ψ_{j}ρσ = ψ_{j}θ′ρ′′σ = θψ_{i}ρ′′σ = ψ_{k}.

This gives the first part of I. Using Theorem 5.1.3 and (5.7) we have

\overline{ψ}_{j}\overline{ρ} = \overline{ψ}_{j}\overline{θ}′\overline{ρ}′′ = \overline{θ} \overline{ψ}_{i}\overline{ρ}′′ = \overline{ψ}_{i}\overline{ρ}′′ = \overline{f} \overline{ϕ}_{n}.

This completes the proof of I.

The second auxiliary fact we need is the to replace R_{i}’s and S_{i}’s in I as follows. The proof is similar to I.

II. If ρ : R_{n} → S_{k} is a graded A-algebra such that

\overline{ψ}_{k}\overline{ρ} = f̄\overline{ϕ}_{n}.

Then there exist an integer m > n and a graded A-algebra homomorphism σ : S_{k} → R_{m} such that ϕ_{m}σρ = ϕ_{n} and \overline{ϕ}_{m}\overline{σ} = f̄^{−1}\overline{ψ}_{k}.

Now we are in a position to construct positive numbers n_{1} < n_{2} < ⋯ and graded A-algebra homomorphisms ρ_{k} : R_{n_{k}} → S such that

1. S_{k} ⊆ ρ_{k}(R_{n_{k}}) and \overline{ρ}_{k} = f̄\overline{ϕ}_{n_{k}}, for all k ∈ ℕ.

2. ρ_{k+1} is an extension of ρ_{k} and for all k ∈ ℕ, i.e, the following diagram commutes.

\xymatrix{R_{n_{k}} \ar@{^{(}->}[d] \ar[r]^{ρ_{k}} & S \\ R_{n_{k+1}} \ar[ur]_{ρ_{k+1}}}

Furthermore, ρ_{k} is injective for all k ∈ ℕ.

Consider the morphism f̄^{−1}\overline{ψ}_{1} : K_{0}^{gr}(S_{1}) → K_{0}^{gr}(R). By Theorem 5.1.3, there is a graded A-algebra homomorphism σ′ : S_{1} → R such that \overline{σ}′ = f̄^{−1}\overline{ψ}_{1}. Since S_{1} is a finite dimensional Aalgebra, σ′(S_{1}) ⊆ R_{n_{1}} for some positive number n_{1}. So σ′ gives a graded A-algebra homomorphism σ : S_{1} → R_{n_{1}} such that ϕ_{n_{1}}σ = σ′ and \overline{ϕ}_{n_{1}} \overline{σ} = f̄^{−1}\overline{ψ}_{1}. Thus σ satisfies the conditions of part I. Therefore there is j > 1 and a graded A-algebra homomorphism ρ : R_{n_{1}} → S_{j} (see Diagram 5.9) such that ψ_{j}ρσ = ψ_{1} and \overline{ψ}_{j}\overline{ρ} = f̄\overline{ϕ}_{n_{1}}.

\xymatrix{S_{1} \ar[rr]^{σ} \ar[dd]_{ψ_{1}} &  & R_{n_{1}} \ar@{.>}[dl]^{ρ} \\  & S_{j} \ar@{.>}[dl]^{ψ_{j}} \\ S}  (5.9)

So ρ_{1} = ψ_{j}ρ : R_{n_{1}} → S is a graded A-homomorphism such that ρ_{1}σ = ψ_{1} and \overline{ρ}_{1} = f̄\overline{ϕ}_{n_{1}}. But

S_{1} = ψ_{1}(S_{1}) = ρ_{1}σ(S_{1}) ⊆ ρ_{1}(R_{n_{1}}).

This proves (1). We now proceed by induction. Suppose there are {n_{1}, …, n_{k}} are {ρ_{1}, …, ρ_{k}}, for some positive integer k such that (1) and (2) above are satisfied. Since R_{n_{k}} is a finite dimensional A-algebra, there is i ≥ k + 1 such that ρ_{k}(R_{n_{k}}) ⊆ S_{i}. So ρ_{k} gives a graded A-homomorphism ρ′ : R_{n_{k}} → S_{i} such that ψ_{i}ρ′ = ρ_{k} and \overline{ψ}_{i}\overline{ρ′} = f̄\overline{ϕ}_{n_{k}}. By II, there is n_{k+1} > n_{k} and σ : S_{i} → R_{n_{k+1}} such that ϕ_{n_{k+1}}σρ′ = ϕ_{n_{k}} and \overline{ϕ}_{n_{k+1}} \overline{σ} = f̄^{−1}\overline{ϕ}_{i}. Since ϕ_{n_{k+1}}σρ′ = ϕ_{n_{k}}, and ϕ_{n_{k}} (being an inclusion) is injective, ρ′ is injective and so ρ_{k} = ψ_{i}ρ′ is injective. Since σ : S_{i} → R_{n_{k+1}} satisfies conditions of Part I, there is j > i such that ρ : R_{n_{k+1}} → S_{j} such that ψ_{j}ρσ = ψ_{i} and \overline{ψ}_{j}\overline{ρ} = f̄\overline{ϕ}_{n_{k+1}}. Thus for ρ_{k+1} = ψ_{j}ρ : R_{n_{k+1}} → S we have \overline{ρ}_{k+1} = \overline{ψ}_{j}\overline{ρ} = f̄\overline{ϕ}_{n_{k+1}}. Since ρ_{k+1}σ = ψ_{j}ρσ = ψ_{i} and i ≥ k + 1, we have

S_{k+1} = ψ_{i}(S_{k+1}) = ρ_{k+1}σ(S_{k+1}) ⊆ ρ_{k+1}σ(S_{i}) ⊆ ρ_{k+1}(R_{n_{k+1}}).

Finally, since ϕ_{n_{k+1}}σρ′ = ϕ_{n_{k}} and ρ_{k+1}σρ′ = ψ_{i}ρ′ = ρ_{k}, it follows that ρ_{k+1} is an extension of ρ_{k}. Then by induction (1) and (2) follows.

We have n_{1} < n_{2} < ⋯ and k ≤ n_{k}. So ⋃ R_{n_{k}} = R. So ρ_{k} induces an injection ρ : R → S such that ρϕ_{n_{k}} = ρ_{k}, for any k. But

S_{k} ⊆ ρ_{k}(R_{n_{k}}) = ρ(R_{n_{k}}) ⊆ ρ(R).

It follows that ρ is an epimorphism as well. This finishes the proof.

Theorem 5.2.6. Let R and S be Γ-graded ultramatricial algebras over a graded field A. Then R and S are graded Morita equivalent if and only if there is an order preserving ℤ[Γ]-module isomorphism K_{0}^{gr}(R) ≅ K_{0}^{gr}(S).

Proof. Let R be graded Morita equivalent to S. Then by Theorem 2.3.6, there is a R-graded progenerator P, such that S ≅_{gr} End_{R}(P). Now using Lemma 3.5.5, we have

(K_{0}^{gr}(S), [S]) ≅ (K_{0}^{gr}(R), [P]).

In particular K_{0}^{gr}(R) ≅ K_{0}^{gr}(S) as ordered ℤ[Γ]-modules. (Note that the ultramatricial assumptions on rings are not used in this direction.)

For the converse, suppose K_{0}^{gr}(S) ≅ K_{0}^{gr}(R) as ordered ℤ[Γ]-modules. Denote the image of [S] ∈ K_{0}^{gr}(S) under this isomorphism by [P]. Then

(K_{0}^{gr}(S), [S]) ≅ (K_{0}^{gr}(R), [P]).

Since [P] is an order-unit (3.27), there is \overline{α} = (α_{1}, …, α_{n}), α_{i} ∈ Γ, such that

∑_{i=1}^{n} α_{i}[P] = [P^{n}(\overline{α})] ≥ [R].

This means there is a graded finitely generated projective A-module, Q such that [P^{n}(\overline{α})] = [R⊕Q]. Since R is a graded ultramatricial algebra, by Lemma 5.1.5, P^{n}(\overline{α}) ≅_{gr} R⊕Q. Using Theorem 2.2.2, it follows that P is a graded progenerator. Let T = End_{R}(P). Using Lemma 3.5.5, we get an order preserving ℤ[Γ]-module isomorphism

(K_{0}^{gr}(T), [T]) ≅ (K_{0}^{gr}(R), [P]) ≅ (K_{0}^{gr}(S), [S]).

Observe that T is also a graded ultramatricial algebras. By the main Theorem 5.2.5, End_{R}(P) = T ≅_{gr} S. By Theorem 2.3.6, this implies that R and S are graded Morita equivalent.

Considering a trivial group as the graded group, we retrieve the classical Brattelli-Elliott Theorem (see the introduction of this chapter).

Corollary 5.2.7 (The Brattelli-Elliott Theorem). Let R and S be ultramatricial algebras over a field F. Then R and S are Morita equivalent if and only if there is an order preserving isomorphism K_{0}(R) ≅ K_{0}(S).

Furthermore, R ≅ S as F -algebras if and only if there is an order preserving isomorphism

(K_{0}(R), [R]) ≅ (K_{0}(S), [S]).

Proof. The corollary follows by considering Γ to be a trivial group in Theorems 5.2.5 and 5.2.6.

Graded Versus Non-graded (Higher) K-Theory

Recall that for a Γ-graded ring A, the category of finitely generated Γ-graded projective right Amodules is denoted by Pgr^{Γ}-A. This is an exact category with the usual notion of (split) short exact sequence. Thus one can apply Quillen’s Q-construction [79] to obtain K-groups

K_{i}(Pgr^{Γ}-A),

for i ≥ 0, which we denote by K_{i}^{gr}(A). If there are more than one grading involve (which is the case in this chapter) we denote this group by K_{i}^{Γ}(A). The group Γ acts on the category Pgr^{Γ}-A from the right via (P, α) ↦ P (α). By functoriality of K-groups this equips K_{i}^{gr}(A) with the structure of a right ℤ[Γ]-module.

The relation between graded K-groups and non-graded K-groups is not always apparent. For example consider the ℤ-graded matrix ring A = 𝕄_{5}(K)(0, 1, 2, 2, 3), where K is a field. Using graded Morita theory one can show that K_{0}^{ℤ}(A) ≅ ℤ[x, x^{−1}] (see Example 3.6.3), whereas K_{0}(A) ≅ ℤ and K_{0}(A_{0}) ≅ ℤ × ℤ × ℤ × ℤ (see Example 1.3.5).

In this chapter we describe a theorem due to van den Bergh [19], relating the graded K-theory to non-graded K-theory. In order to do so, we need a generalisation of a result of Quillen on relating K_{i}^{gr}-groups of a positively graded rings to their non-graded K_{i}-groups (§6.1). We also need to recall a proof of the fundamental theorem of K-theory, i.e., for a regular Noetherian ring A, K_{i}(A[x]) ≅ K_{i}(A) (§6.2). We will see that in the proof of this theorem, graded K-theory appears in a crucial way. Putting all these together we can relate the graded K-theory of regular Noetherian ℤ-graded rings to their non-graded K-theory (§6.3).

As a conseqeunce, we will see that if A is a ℤ-graded right regular Noetherian ring, then one has an exact sequence (Corollary 6.3.2),

\xymatrix{K_{0}^{gr}(A) \ar[r]^{[P]↦[P (1)]−[P]} & K_{0}^{gr}(A) \ar[r]^{U} & K_{0}(A) \ar[r] & 0,}

where U is induced by the forgetful functor (§1.2.7). This shows that

K_{0}^{gr}(A)/⟨[P] − [P (1)]⟩ ≅ K_{0}(A),

where P is a graded finitely generated projective A-module. Further the action of ℤ[x, x^{−1}] on the quotient group is trivial. This means, as soon as we discard the shifting in K_{0}^{gr}, this group reduces to the usual K_{0}-group.

However this is not in general the case. For example consider the group ring ℤ[G], where G is an abelian group. Since ℤ[G] has IBN, the canonical homomorphism θ : K_{0}(ℤ) → K_{0}(ℤ[G]) is injective (see §3.1.4). The augmentation map ℤ[G] → ℤ, g ↦ 1, induces ϑ : K_{0}(ℤ[G]) → K_{0}(ℤ), such that ϑθ = 1. Thus

K_{0}(ℤ[G]) ≅ ℤ ⨁ K̃_{0}(ℤ[G]).

Now since ℤ[G] is a group ring (so a crossed product), K_{0}^{gr}(ℤ[G]) ≅ K_{0}(ℤ) ≅ ℤ and the action of ℤ[x, x^{−1}] on the K_{0}^{gr}(ℤ[G]) is trivial (see Example 3.1.9). So K_{0}^{gr}(ℤ[G])/⟨[P] − [P (1)]⟩ ≅ ℤ. This shows that K_{0}^{gr}(ℤ[G]) does not shed any light on the group K_{0}(ℤ[G]). The Grothendieck group of group rings is far from easy to compute (see for example [14] and [94, Example 2.4]).

In this chapter we compare the graded K-theory to its non-graded counterpart. We start with a generalisation of Quillen’s theorem.

6.1 K_{∗}^{gr} of positively graded rings

For a ℤ-graded ring A with the support in ℕ, in his seminal paper [79], Quillen calculated the graded K-theory of A in terms of K-theory of its zero homogeneous component.

Theorem 6.1.1. [79, §3, Proposition] Let A be a ℤ-graded ring with the support in ℕ. Then for i ≥ 0, there is a ℤ[x, x^{−1}]-module isomorphism,

K_{i}^{gr}(A) ≅ K_{i}(A_{0}) ⊗_{ℤ} ℤ[x, x^{−1}].  (6.1)

Furthermore, the element K_{i}(A_{0}) ⊗ x^{r} is the image of \overline{f}_{r} where f_{r} is the exact functor

f_{r} : Pr- A_{0} ⟶ Pgr-A,
P ⟼ (P ⊗_{A_{0}} A)(−r).

In Theorem 6.1.3, we prove a generalised version of this proposition. Proposition 6.1.1 was used in an essential way to prove the fundamental theorem of K-theory (see §6.2.2). In particular, Proposition 6.1.1 gives a powerful tool to calculate the graded Grothendieck group of positively graded rings. We demonstrate this by calculating K_{0}^{gr} of path algebras.

Theorem 6.1.2. Let E be a finite graph and 𝒫_{K}(E) be the path algebra with coefficients in the field K. Then

K_{0}^{gr}(𝒫_{K}(E)) ≅ ⨁_{|E^{0}|} ℤ[x, x^{−1}].

Proof. Recall from (1.6.3), that 𝒫_{K}(E) is a graded ring with the support ℕ. Furthermore 𝒫_{K}(E)_{0} = ⨁_{|E^{0}|} K. Then by (6.1)

K_{0}^{gr}(𝒫_{K}(E)) ≅ K_{0}(⨁_{|E^{0}|} K) ⊗_{ℤ} ℤ[x, x^{−1}] ≅ ⨁_{|E^{0}|} ℤ ⊗_{ℤ} ℤ[x, x^{−1}] ≅ ⨁_{|E^{0}|} ℤ[x, x^{−1}].

The proof of Proposition 6.1.2 shows that, in the case of i = 0, [A] ∈ K_{i}^{gr}(A) is sent to [A_{0}] ⊗ 1 ∈ K_{i}(A_{0}) ⊗_{ℤ} ℤ[x, x^{−1}] under the isomorphism (6.1). Thus for the graphs

\xymatrix{E : & • \ar[r] & • & F : & • \ar[r] & • \ar[l]}

We have

(K_{0}^{gr}(𝒫_{K}(E), [𝒫_{K}(E)]) ≅ (ℤ[x, x^{−1}] ⊕ ℤ[x, x^{−1}], (1, 1)) ≅ (K_{0}^{gr}(𝒫_{K}(E), [𝒫_{K}(E)]).

This in particular shows that the ordered group K_{0}^{gr} in itself would not classify the path algebras.

Contrary to other fundamental theorems in the subject, such as fundamental theorem of Ktheory (i.e., K_{i}(R[x, x^{−1}]) = K_{i}(R) × K_{i−1}(R), for R a regular ring), one cannot use an easy induction on (6.1) to write a similar statement for “multi-variables” rings. For example, it appears that there is no obvious inductive approach to generalise (6.1) to ℤ^{m} × G-graded rings. However, by generalising Quillen’s argument to take gradings into account on both sides of the isomorphism, such a procedure becomes feasible. The details have been worked out in [49] and we present it here.

We will prove the following statement.

Theorem 6.1.3. Let G be a group, and let A be a ℤ × G-graded ring with support in ℕ × G. Then there is a ℤ[ℤ × G]-module isomorphism

K_{i}^{G}(A_{(0,−)}) ⊗_{ℤ[G]} ℤ[ℤ × G] ≅ K_{i}^{ℤ×G}(A),

where A_{(0,−)} = ⨁_{g∈G} A_{(0,g)}.

By a straightforward induction this now implies:

Corollary 6.1.4. For a ℤ^{m} × G-graded ring A with support in ℕ^{m} × G there is a ℤ[x_{1}^{±1}, ⋯, x_{m}^{±1}]- module isomorphism

K_{i}^{G}(A_{(0,−)}) ⊗_{ℤ} ℤ[x_{1}^{±1}, ⋯, x_{m}^{±1}] ≅ K_{i}^{ℤ^{m}×G}(A) .

For a trivial group G this is a direct generalisation of Quillen’s theorem to ℤ^{m}-graded rings. As in Quillen’s calculation the proof of the Theorem 6.1.3 is based on a version of Swan’s

Theorem, modified to the present situation: it provides a correspondence between isomorphism classes of ℤ × G-graded finitely generated projective A-modules and of G-graded finitely generated projective A_{(0,−)}-modules.

Proposition 6.1.5. Let Γ be a (possibly non-abelian) group. Let A be a Γ-graded ring, A_{0} a graded subring of A and π: A → A_{0} a graded ring homomorphism such that π|_{A_{0}} = 1. (In other words, A_{0} is a retract of A in the category of Γ-graded rings.) We denote the kernel of π by A_{+}.

Suppose that for any graded finitely generated right A-module M the condition MA_{+} = M implies M = 0. Then the natural functor

S = − ⊗_{A_{0}} A: Pgr^{Γ}-A_{0} → Pgr^{Γ}-A

induces a bijective correspondence between the isomorphism classes of graded finitely generated projective A_{0}-modules and of graded finitely generated projective A-modules. An inverse of the bijection is given by the functor

T = − ⊗_{A} A_{0} : Pgr^{Γ}-A → Pgr^{Γ}-A_{0} .

There is a natural isomorphism T ∘ S ≅ id, and for each P ∈ Pgr^{Γ}-A a non-canonical isomorphism S ∘ T (P) ≅ P. The latter is given by

T (P) ⊗_{A_{0}} A → P , x ⊗ a ↦ g(x) · a,  (6.2)

where g is an A_{0}-linear section of the epimorphism P → T (P).

Proof. For any graded finitely generated projective A_{0}-module Q we have a natural isomorphism T S(Q) ≅ Q given by

ν_{Q}: T S(Q) = Q ⊗_{A_{0}} A ⊗_{A} A_{0} → Q , q ⊗ a ⊗ a_{0} ↦ qπ(a)a_{0} .  (6.3)

We will show that for a graded projective A-module P there is a non-canonical graded isomorphism ST (P) ≅_{gr} P. The lemma then follows.

Consider the natural graded A-module epimorphism

f: P → T (P) = P ⊗_{A} A_{0} , p ↦ p ⊗ 1 .

Here T (P) is considered as an A-module via the map π. Since T (P) is a graded projective A_{0}-module, the map f has a graded A_{0}-linear section g: T (P) → P. This section determines an A-linear graded map

ψ: ST (P) = P ⊗_{A} A_{0} ⊗_{A_{0}} A → P , p ⊗ a_{0} ⊗ a ↦ g(p ⊗ a_{0}) · a ,

and we will show that ψ is an isomorphism. First note that the map

T (f): T (P) → T T (P) , p ⊗ a_{0} ↦ f(p) ⊗ a_{0} = p ⊗ 1 ⊗ a_{0}

is an isomorphism (consider T (P) as an A-module via π here). In fact the inverse is given by the isomorphism T T (P) = P ⊗_{A} A_{0} ⊗_{A} A_{0} → P ⊗_{A} A_{0} which maps p ⊗ a_{0} ⊗ b_{0} to p ⊗ (a_{0}b_{0}). Tracing the definitions now shows that both composites

\xymatrix{T ST (P) \ar[r]^{T (ψ)} \ar[r]_{ν_{T (P)}} & T (P) \ar[r]^{T (f)}_{≅} & T T (P)}

map p ⊗ a_{0} ⊗ a ⊗ b_{0} ∈ P ⊗_{A} A_{0} ⊗_{A_{0}} A ⊗_{A} A_{0} = T ST (P) to the element

f(g(p ⊗ a_{0}) · a) ⊗ b_{0} = p ⊗ (a_{0}π(a)b_{0}) ⊗ 1 ∈ T T (P).

This implies that T (ψ) = ν_{T (P)}, which is an isomorphism.

The exact sequence

0 ⟶ ker ψ ⟶ ST (P) ⟶^{ψ} P ⟶ coker ψ ⟶ 0  (6.4)

gives rise, upon application of the right exact functor T, to an exact sequence

T ST (P) ⟶^{T (ψ)} T (P) ⟶ T (coker ψ) ⟶ 0 .

Since T (ψ) is an isomorphism we have T (coker ψ) = coker T (ψ) = 0. Since coker ψ is finitely generated by (6.4) this implies coker ψ = 0 (note that T (M) = MA_{+} for every finitely generated module M). In other words, ψ is surjective and (6.4) becomes the short exact sequence 0 → ker ψ → ST (P) →^{ψ} P → 0. This latter sequence splits since P is projective; this immediately implies that ker ψ is finitely generated, and since T (ψ) is injective we also have T (ker ψ) = ker T (ψ) = 0. The hypotheses guarantee ker ψ = 0 now so that ψ is injective as well as surjective, and thus is an isomorphism as claimed.

Lemma 6.1.6. Let G and Γ be groups, and let A be a G-graded ring. Then, considering A as a Γ × G-graded ring in a trivial way where necessary, the functorial assignment (M, γ) ↦ M(γ, 0) induces a ℤ[Γ × G]-module isomorphism

K_{i}^{Γ×G}(A) ≅ K_{i}^{G}(A) ⊗_{ℤ[G]} ℤ[Γ × G] .

Proof. Let P = ⨁_{(γ,g)∈Γ×G} P_{(γ,g)} be a Γ × G-graded finitely generated projective A-module. Since the support of A is G = 1 × G, there is a unique decomposition P = ⨁_{γ∈Γ} P_{γ}, where the P_{γ} = ⨁_{g∈G} P_{(γ,g)} are finitely generated G-graded projective A-modules. This gives a natural isomorphism of categories (see Corollary 1.2.10)

Ψ: Pgr^{Γ×G}-A ⟶^{≅} ⨁_{γ∈Γ} Pgr^{G}-A .

The natural right action of Γ × G on these categories is described as follows: for a given module P ∈ Pgr^{Γ×G}-A as above and elements (γ, g) ∈ Γ × G we have

P (γ, g)_{(δ,h)} = P_{(γ+δ,g+h)} and Ψ(P)_{δ} = P_{γ+δ}(g)

so that Ψ(P (γ, g)) = Ψ(P)(γ, g). Since K-groups respect direct sums we thus have a chain of ℤ[Γ × G]-linear isomorphisms

K_{i}^{Γ×G}(A) = K_{i}(Pgr^{Γ×G}-A) ≅ K_{i}( ⨁_{γ∈Γ} Pgr^{G}-A)

= ⨁_{γ∈Γ} K_{i}^{G}(A) ≅ K_{i}^{G}(A) ⊗_{ℤ} ℤ[Γ] = K_{i}^{G}(A) ⊗_{ℤ[G]} ℤ[Γ × G] .

We are in a position to prove the Theorem 6.1.3.

Proof of Theorem 6.1.3. Let A be a ℤ × G-graded ring with support in ℕ × G. That is, A comes equipped with a decomposition

A = ⨁_{ω∈ℕ} A_{(ω,−)} where A_{(ω,−)} = ⨁_{g∈G} A_{(ω,g)} .

The ring A has a ℤ × G-graded subring A_{(0,−)} (with trivial grading in ℤ-direction). The projection map A → A_{(0,−)} is a ℤ × G-graded ring homomorphism; its kernel is denoted A_{+}. Explicitly, A_{+} is the two-sided ideal

A_{+} = ⨁_{ω>0} A_{(ω,−)} .

We identify the quotient ring A/A_{+} with the subring A_{(0,−)} via the projection.

If P is a graded finitely generated projective A-module, then P ⊗_{A} A_{(0,−)} is a finitely generated ℤ × G-graded projective A_{(0,−)}-module. Similarly, if Q is a graded finitely generated projective A_{(0,−)}-module then Q ⊗_{A_{(0,−)}} A is a ℤ × G- graded finitely generated projective A-module. We can thus define functors

T = − ⊗_{A} A_{(0,−)} : Pgr^{ℤ×G}-A ⟶ Pgr^{ℤ×G}-A_{(0,−)}
and S = − ⊗_{A_{(0,−)}} A: Pgr^{ℤ×G}-A_{(0,−)} ⟶ Pgr^{ℤ×G}-A .

Since T (P) = P/P A_{+} we see that the support of T (P) is contained in the support of P.

Observe now that if M is a finitely generated ℤ × G-graded A-module and MA_{+} = M then M = 0; for if M ≠ 0 there is a minimal ω ∈ ℤ such that M_{(ω,−)} ≠ 0, but (MA_{+})_{(ω,−)} = 0. It follows from Proposition 6.1.5 that for each graded finitely generated projective A-module P there is a non-canonical isomorphism P ≅ T (P) ⊗_{A_{(0,−)}} A as in (6.2) which respects the ℤ × G-grading. Explicitly, for a given (ω, g) ∈ ℤ × G we have an isomorphism of abelian groups

P_{(ω,g)} ≅ ⨁_{(κ,h)} T (P)_{(κ,h)} ⊗ A_{(−κ+ω,−h+g)} ;  (6.5)

the tensor product T (P)_{(κ,h)} ⊗ A_{(−κ+ω,−h+g)} denotes, by convention, the abelian subgroup of T (P)⊗_{A_{(0,−)}}A generated by primitive tensors of the form x⊗y with homogeneous elements x ∈ T (P) of degree (κ, h) and y ∈ A of degree (−κ + ω, −h + g).

For a ℤ × G-graded A-module P write P = ⨁_{ω∈ℤ} P_{(ω,−)}, where P_{(ω,−)} = ⨁_{g∈G} P_{(ω,g)}. For λ ∈ ℤ let F^{λ}P denote the A-submodule of P generated by the elements of ⋃_{ω≤λ} P_{(ω,−)}; this is ℤ × G-graded again. As an explicit example, we have

F^{λ}A(ω, g) = \begin{cases}A(ω, g) & if λ ≥ −ω , \\ 0 & else. \end{cases}

Suppose that P is a graded finitely generated projective A-module. Since the support of A is contained in ℕ × G there exists n ∈ ℤ such that F^{−n}P = 0 and F^{n}P = P. Write Pgr_{n}^{ℤ×G}-A for the full subcategory of Pgr^{ℤ×G}-A spanned by those modules P which satisfy F^{−n}P = 0 and F^{n}P = P. Then Pgr^{ℤ×G}-A is the filtered union of the Pgr_{n}^{ℤ×G}-A.

Let P ∈ Pgr_{n}^{ℤ×G}-A; we want to identify F^{λ}P. By definition, the A-module F^{λ}P is generated by the elements of P_{(ω,g)} for ω ≤ λ, with P_{(ω,g)} having been identified in (6.5). We remark that the direct summands in (6.5) indexed by κ > ω are trivial as A has support in ℕ × G. On the other hand, for ω ≥ κ a given primitive tensor x ⊗ y ∈ P_{(ω,g)} with x ∈ T (P)_{(κ,h)} and y ∈ A_{(−κ+ω,−h+g)} can always be re-written, using the right A-module structure of T (P) ⊗_{A_{(0,−)}} A, as

x ⊗ y = (x ⊗ 1) · y where x ⊗ 1 ∈ T (P)_{(κ,h)} ⊗ A_{(0,0)} ⊆ P_{(κ,h)} .

That is, the A-module F^{λ}P is generated by those summands of (6.5) with κ = ω ≤ λ. We claim now that F^{λ}P is isomorphic to

M^{(λ)} = ⨁_{κ≤λ} T (P)_{(κ,−)} ⊗_{A_{(0,−)}} A(−κ, 0) ,  (6.6)

considering T (P)_{(κ,−)} as a ℤ×G-graded A_{(0,−)}-module with support in {0}×G. The homogeneous components of M^{(λ)} are given by

M_{(ω,g)}^{(λ)} = ⨁_{κ≤λ} ⨁_{h∈G} T (P)_{(κ,h)} ⊗ A(−κ, 0)_{(ω,−h+g)} .

Now elements of the form

x ⊗ 1 ∈ T (P)_{(κ,h)} ⊗ A(−κ, 0)_{(κ,−h+h)} ⊆ M_{(κ,h)}^{(λ)}

clearly form a set of A-module generators for M^{(λ)} so that, by the argument given above, F^{λ}P and M^{(λ)} have the same generators in the same degrees. The claim follows. The module F^{λ}P is finitely generated (viz., by those generators of P that have ℤ-degree at most λ). Since T (P) is a finitely generated projective A_{(0,−)}-module so is its summand T (P)_{(λ_{k},−)}; consequently, P ↦ F^{k}P is an endofunctor of Pgr_{n}^{ℤ×G}-A. It is exact as can be deduced from the (non-canonical) isomorphism in (6.6), using exactness of tensor products.

From the isomorphism F^{k}P ≅ M^{(λ_{k})}, cf. (6.6), we obtain an isomorphism

F^{k+1}P/F^{k}P ≅ T (P)_{(λ_{k},−)} ⊗_{A_{(0,−)}} A(−λ_{k}, 0) ;  (6.7)

in particular, F^{k+1}P/F^{k}P ∈ Pgr_{n}^{ℤ×G}-A.

The isomorphism (6.7) depends on the isomorphism (6.6), and thus ultimately on (6.2). The latter depends on a choice of a section g of P → T (P). Given another section g_{0} the difference g − g_{0} has image in ker(P → T (P)) = P A_{+}. Since A_{+} consists of elements of positive ℤ-degree only, this implies that the isomorphism F^{k+1}P ≅ M^{(λ_{k+1})} does not depend on g up to elements in F^{k}P; in other words, the quotient F^{k+1}P/F^{k}P is independent of the choice of g. Thus the isomorphism (6.7) is, in fact, a natural isomorphism of functors.

We are now in a position to perform the K-theoretical calculations. First define the exact functor

Θ_{q} : Pgr_{q}^{ℤ×G}-A_{(0,−)} ⟶ Pgr_{q}^{ℤ×G}-A
P = ⨁_{ω} P_{(ω,−)} ↦ ⨁_{ω} P_{(ω,−)} ⊗_{A_{(0,−)}} A(−ω, 0) ;

here Pgr_{q}^{ℤ×G}-A_{(0,−)} denotes the full subcategory of Pgr^{ℤ×G}-A_{(0,−)} spanned by modules with support in [−q, q] × G, and P_{(ω,−)} on the right is considered as a ℤ × G-graded A_{(0,−)}-module with support in {0} × G.

Next define the exact functor

Ψ_{q} : Pgr_{q}^{ℤ×G}-A ⟶ Pgr_{q}^{ℤ×G}-A_{(0,−)}
P ↦ ⨁_{ω∈ℤ} T (P)_{(ω,−)} ;

here T (P)_{(ω,−)} is considered as an A_{(0,−)}-module with support in {ω} × G.

Now Ψ_{q} ∘ Θ_{q} ≅ id; indeed, the composition sends the summand P_{(ω,−)} of P to the κ-indexed direct sum of

T (P_{(ω,−)} ⊗_{A_{(0,−)}} A(−ω, 0))_{(κ,−)} ≅ \begin{cases}P_{(ω,−)} & if κ = ω , \\ 0 & else. \end{cases}

In particular, Ψ_{q} ∘ Θ_{q} induces the identity on K-groups. As for the other composition, we have

Θ_{q} ∘ Ψ_{q}(P) = ⨁_{ω} T (P)_{(ω,−)} ⊗_{A_{(0,−)}} A(−ω, 0) =_{(6.7)} ⨁_{j=−q}^{q−1} F^{j+1}P/F^{j}P .

Since F^{q} = id, additivity for characteristic filtrations [79, p. 107, Corollary 2] implies that Θ_{q} ∘ Ψ_{q} induces the identity on K-groups.

For any P ∈ Pgr^{ℤ×G}-A_{(0,−)} we have

(P ⊗_{A_{(0,−)}} A(−ω, 0))(0, g) = P (0, g) ⊗_{A_{(0,−)}} A(−ω, 0) ,

by direct calculation. Hence the functor Θ_{q} induces a ℤ[G]-linear isomorphism on K-groups. Since K-groups are compatible with direct limits, letting q → ∞ yields a ℤ[G]-linear isomorphism K_{i}^{ℤ×G}(A_{(0,−)}) ≅ K_{i}^{ℤ×G}(A) and thus, by Lemma 6.1.6, a ℤ[ℤ × G]-module isomorphism

K_{i}^{G}(A_{(0,−)}) ⊗_{ℤ[G]} ℤ[ℤ × G] ≅ K_{i}^{ℤ×G}(A) .

6.2 The fundamental theorem of K-theory

6.2.1 Quillen’s K-theory of exact categories

Recall that an exact category 𝒫 is a full additive subcategory of an abelian category 𝒜 which is closed under extension (see Definition 3.10.1). An exact functor ℱ : 𝒫 → 𝒫′, between exact categories 𝒫 and 𝒫′, is an additive functor which preserves the short exact sequences.

From an exact category 𝒫, Quillen constructed abelian groups K_{n}(𝒫), n ≥ 0 (see [79]). In fact, for a fixed n, K_{n} is a functor from the category of exact categories, with exact functors as morphisms to the category of abelian groups. If f : 𝒜 → ℬ is an exact functor, then we denote the corresponding group homomorphism by \overline{f} : K_{n}(𝒜) ⟶ K_{n}(ℬ).

When 𝒫 = Pr- A, the category of finitely generated projective A-module, then K_{0}(𝒫) = K_{0}(A). Quillen proved that basic theorems established for K_{0}-group, such as Dévissage, resolution and localisation theorems ([88, 94]), can be established for these higher K-groups. We briefly mention the main theorems of K-theory which will be used in §6.2.2. The following statements were established by Quillen in [79] (see also [87, 94]).

1. The Grothendieck group. For an exact category 𝒫, the abelian group K_{0}(𝒫) coincides with the construction given in Definition 3.10.1(1). In particular, Quillen’s construction gives the Grothendieck group K_{0}(A) = K_{0}(Pr- A) and graded Grothendieck group K_{0}^{gr}(A) = K_{0}(Pgr-A), for non-graded and graded rings, respectively.

2. Dévissage. Let 𝒜 be an abelian category and ℬ be a non-empty full subcategory closed under subobjects, quotient objects and finite products in 𝒜. Thus ℬ is an abelian category and the inclusion ℬ ↪ 𝒜 is exact. If any element A in 𝒜, has a finite filtration

0 = A_{0} ⊆ A_{1} ⊆ ⋯ ⊆ A_{n} = A,

with A_{i}/A_{i−1} in ℬ, 1 ≤ i ≤ n, then the inclusion ℬ ↪ 𝒜 induces isomorphisms K_{n}(ℬ) ≅ K_{n}(𝒜), n ≥ 0.

3. Resolution. Let ℳ be an exact category and 𝒫 be a full subcategory closed under extensions in ℳ. Suppose that if 0 → M → P → P′ → 0 is exact in ℳ with P and P′ in 𝒫, then M is in 𝒫. Furthermore, for every M in ℳ, there is a finite 𝒫 resolution of M,

0 ⟶ P_{n} ⟶ ⋯ ⟶ P_{1} ⟶ P_{0} ⟶ M ⟶ 0.

Then the inclusion 𝒫 ↪ ℳ induces isomorphisms K_{n}(𝒫) ≅ K_{n}(ℳ), n ≥ 0.

4. Localisation. Let 𝒮 be a Serre subcategory of abelian category 𝒜 (i.e., 𝒮 is an abelian subcategory of 𝒜, closed under subobjects, quotient objects and extensions in 𝒜), and let 𝒜/𝒮 be the quotient abelian category. Then there is a natural long exact sequence

⋯ ⟶ K_{n+1}(𝒜/𝒮) ⟶^{δ} K_{n}(𝒮) ⟶ K_{n}(𝒜) ⟶ K_{n}(𝒜/𝒮) ⟶ · · · .

5. Exact Sequence of Functors. Let 𝒫 and 𝒫′ be exact categories and

0 ⟶ f′ ⟶ f ⟶ f′′ ⟶ 0,

be an exact sequence of exact functors from 𝒫 to 𝒫′. Then

\overline{f} = \overline{f}′ + \overline{f}′′ : K_{n}(𝒫) ⟶ K_{n}(𝒫′).

Remark 6.2.1. \underline{Base change and transfer functors}

Let A be a ring with identity. If A is a right Noetherian then the category of finitely generated right A-modules, mod- A, form an abelian category. Thus one can define the K-groups of this category, G_{n}(A) := K_{n}(mod- A), n ∈ ℕ. If further A is regular, then by resolution theorem (§6.2.1), G_{n}(A) = K_{n}(A).

Let R → S be a ring homomorphism. This makes S an R-module. If S is a finitely generated Rmodule, then any finitely generated S-module can be considered as a finitely generated R-module. This induces an exact functor, called a transfer functor, mod- S → mod- R. Since K-theory is a functor from the category of exact categories with exact functors as morphisms, we get a transfer map G_{n}(S) → G_{n}(R), n ∈ ℕ. A similar argument shows that, if R → S is a graded homomorphism, then we have a transfer map G_{n}^{gr}(S) → G_{n}^{gr}(R), n ∈ ℕ.

If under the ring homomorphism R → S, S is a flat R-module, then the base change functor − ⊗_{R} S : Mod- R → Mod- S is exact, which restricts to − ⊗_{R} S : mod- R → mod- S. This in turn induces the group homomorphism G_{n}(R) → G_{n}(S), n ∈ ℕ. Note that any ring homomorphism R → S, induces an exact functor − ⊗_{R} S : Pr- R → Pr- S, which induces the homomorphism K_{n}(R) → K_{n}(S). However, to prove the fundamental theorem of K-theory (see (6.12)), we need to use the Dévissage and localisation theorems, which are only valid for abelian categories. This forces us to work in the abelian category gr- A (assuming A is right Noetherian), rather than the exact category Pr- A.

Example 6.2.2. \underline{A long localisation exact sequence for graded rings}

Let A be a right regular Noetherian Γ-graded ring, i.e., A is graded right regular and graded right Noetherian. We assume the ring is graded regular so that we could work with K-theory instead of G-theory. We recall the concept of graded regular rings. For a graded right module M, the minimum length of all graded projective resolutions of M is defined as the graded projective dimension of M and denoted by pdim^{gr}(M). If M does not admit a finite graded projective resolution, then we set pdim^{gr}(M) = ∞. The graded right global dimension of the graded ring A is the supremum of the graded projective dimension of all the graded right modules over A and denoted by gldim^{gr}(A). We say A is a graded right regular if the graded right global dimension is finite. As soon as Γ is a trivial group, the above definitions become the standard definitions of projective dimension, denoted by, pdim, and the global dimension, denoted by, gldim, in ring theory. In the case of ℤ-graded ring, we have the following relation between the dimensions (see [72, Theorem II.8.2])

gldim^{gr}(A) ≤ gldim(A) ≤ 1 + gldim^{gr}(A).  (6.8)

This shows, if A is graded regular, then A is regular. This will be used in §6.3.

Let S be a central multiplicative closed subsets of A, consisting of homogeneous elements. Then S^{−1}A is a Γ-graded ring (see Example 1.1.12). Let gr_{S}- A be a category of S-torsion graded finitely generated right A-modules, i.e., a graded finitely generated A-module M such that Ms = 0 for some s ∈ S. Clearly gr_{S}- A is a Serre subcategory of the abelian category gr- A and

gr- A/ gr_{S}- A ≈ gr-(S^{−1}A).  (6.9)

Now let S = {s^{k} | k ∈ ℕ}, where s is a homogeneous element in the centre of A. Then any module M in gr_{S}- A has a filtration

0 = Ms^{k} ⊆ ⋯ ⊆ Ms ⊆ M,

and clearly the consecutive quotients in the filtration are Γ-graded A/sA-modules. Considering gr- A/sA as a subcategory of gr_{S}- A via the transfer map A → A/sA (see Remark 6.2.1), and using Dévissage theorem, we get, for any n ∈ ℕ,

K_{n}(gr_{S}- A) ≅ K_{n}^{gr}(A/sA).

Now using the localisation theorem for (6.9), we obtain a long localisation exact sequence

⋯ ⟶ K_{n+1}^{gr}(A_{s}) ⟶ K_{n}^{gr}(A/sA) ⟶ K_{n}^{gr}(A) ⟶ K_{n}^{gr}(A_{s}) ⟶ ⋯,  (6.10)

where A_{s} := S^{−1}A.

In particular, consider the Γ-graded rings A[y] and A[y, y^{−1}], where deg(y) = α. For A[y] and S = {y^{n} | n ∈ ℕ}, the sequence 6.10 reduces to

⋯ ⟶ K_{n+1}^{gr}(A[y, y^{−1}]) ⟶ K_{n}^{gr}(A) ⟶ K_{n}^{gr}(A[y]) ⟶ K_{n}^{gr}(A[y, y^{−1}]) ⟶ · · · .  (6.11)

This long exact sequence will be used in §6.3 (with deg(y) = 1) to relate K_{n}^{gr}-groups to K_{n}-groups.

6.2.2 The fundamental theorem

Let A be a right Noetherian and regular ring. Consider the polynomial ring A[x]. The fundamental theorem of K-theory gives that, for any n ∈ ℕ,

K_{n}(A[x]) ≅ K_{n}(A).  (6.12)

In this section we prove that if A is a positively graded ℤ-graded ring, then K_{n}(A) ≅ K_{n}(A_{0}), which in particular gives (6.12). We will follow Gersten’s treatment in [37]. The proof shows how the graded K-theory is effectively used to prove a theorem on (non-graded) K-theory.

Let A be a positively ℤ-graded ring, i.e., A = ⨁_{i∈ℕ} A_{i}. Define a ℤ-grading on the polynomial ring A[x] as follows (see Example 1.1.7):

A[x]_{n} = ⨁_{i+j=n} A_{i}x^{j}.  (6.13)

Clearly this is also a positively graded ring and A[x]_{0} = A_{0}.

Throughout the proof of the fundamental theorem we consider two evaluation maps on A[x] as follows.

1. Consider the evaluation homomorphism at 0, i.e.,

e_{0} : A[x] ⟶ A,
f(x) ⟼ f(0).  (6.14)

This is a graded homomorphism (with the grading defined in (6.13)), and A becomes a graded finitely generated A[x]-module. This induces an exact transfer functor (see Remark 6.2.1)

i : gr- A ⟶ gr- A[x],
M ⟼ M  (6.15)

and in turn a homomorphism

\overline{i} : G_{n}^{gr}(A) → G_{n}^{gr}(A[x]).

Let M be a graded A-module. Under the above homomorphism, M is also a grade A[x]- module (see (6.15)). On the other hand, the canonical graded homomorphism A → A[x], gives the graded A[x]-module M ⊗_{A} A[x] ≅_{gr} M[x]. Then we have the following short exact sequence of graded A[x]-modules,

0 ⟶ M[x](−1) ⟶^{x} M[x] ⟶^{e_{0}} M ⟶ 0.  (6.16)

Note that this short exact sequence is not an split sequence.

2. Consider the evaluation homomorphism at 1, i.e.,

e_{1} : A[x] ⟶ A,
f(x) ⟼ f(1).  (6.17)

Note that this homomorphism is not graded. Clearly ker(e_{1}) = (1−x) (i.e., the ideal generated by 1 − x). Thus A[x]/(1 − x) ≅ A. If M is a (graded) A[x]-module, then

M ⊗_{A[x]} A ≅ M ⊗_{A[x]} A[x]/(1 − x) ≅ M/M(1 − x),

is an A-module. Thus the homomorphism e_{1} induces the functor

ℱ := − ⊗_{A[x]} A : gr- A[x] ⟶ mod- A
M ⟼ M/M(1 − x).  (6.18)

We need the following Lemma (see [72, Lemma II.8.1]).

Lemma 6.2.3. Consider the functor ℱ defined in (6.18). Then we have the following.

(1) ℱ is an exact functor.

(2) ℱ(M) = 0 if and only if Mx^{n} = 0 for some n ∈ ℕ.

Proof. (1) Since the tensor product is a right exact functor, we are left to show that if 0 → M′ → M is exact, then by (6.18),

0 ⟶ M′/M′(1 − x) ⟶ M/M(1 − x),

is exact, i.e., M′ ∩ M(1 − x) = M′(1 − x).

Let m′ ∈ M′ ∩ M(1 − x). Then

m′ = m(1 − x),  (6.19)

for some m ∈ M. Since M′ and M are graded modules, m′ = ∑ m′_{i} and m = ∑ m_{i}, with deg(m′_{i}) = deg(m_{i}) = i. Comparing the degrees in the Equation 6.19, we have m_{i} − m_{i−1}x = m′_{i} ∈ M′_{i}. Let j be the smallest i in the support of m such that m_{j} ∉ M′_{j}. Then m_{j−1}x ∈ M′_{j} which implies that m_{j} = m′_{j} − m_{j−1}x ∈ M′_{j}, which is a contradiction. Thus m ∈ M′, and so m′ ∈ M′(1 − x).

(2) The proof is elementary and left to the reader.

By Lemma 6.2.3(2), ker(ℱ) is an (abelian) full subcategory of gr- A[x], with objects the graded A[x]-modules M such that Mx^{n} = 0, for some n ∈ ℕ. Define 𝒞_{1} the (abelian) full subcategory of gr- A[x] with objects the graded A[x]-modules M such that Mx = 0. Thus 𝒞_{1} ↪ ker(ℱ) is an exact functor. Furthermore, for any object M in ker(ℱ), we have a finite filtration of graded A[x]-modules

0 = Mx^{n} ⊆ Mx^{n−1} ⊆ ⋯ ⊆ Mx ⊆ M,

such that Mx^{i}/Mx^{i+1} ∈ 𝒞_{1}. Thus by Dévissage theorem (§6.2.1), we have

K_{n}(𝒞_{1}) ≅ K_{n}(ker(ℱ)).  (6.20)

Next recall the exact functor i : gr- A ⟶ gr- A[x] from (6.15). It is easy to see that this functor induces an equivalence between the categories

i : gr- A ⟶ 𝒞_{1},  (6.21)

and thus on the level of K-groups. Combing this with (6.20) we get

K_{n}^{gr}(A) = K_{n}(𝒞_{1}) ≅ K_{n}(ker(ℱ)).  (6.22)

Since by Lemma 6.2.3, ℱ is exact, ker(ℱ) is closed under subobjects, quotients and extensions, i.e, it is a Serre subcategory of gr- A[x]. We next prove that

gr- A[x]/ ker(ℱ) ≅ mod- A.  (6.23)

Any finitely generated module over a Noetherian ring is finitely presented. We need the following general lemma to invoke the localisation.

Lemma 6.2.4. Let M be a finitely presented A-module. Then there is a finitely presented graded A[x]-module N such that ℱ(N) ≅ M, where ℱ is the functor defined in (6.18).

Proof. Since ℱ = −⊗_{A[x]} A, for any free A-module M of rank k we have ℱ(⨁_{k} A[x]) ≅ ⨁_{k} A ≅ M. Let M be a finitely presented A-module. Then there is an exact sequence

⨁_{m} A ⟶^{f} ⨁_{n} A ⟶ M ⟶ 0.

If we show that there are graded free A[x]-modules F_{1} and F_{2} and a graded homomorphism g : F_{1} → F_{2} such that the following diagram is commutative,

\xymatrix{ℱ(F_{1}) \ar[r]^{ℱ(g)} \ar[d]_{≅} & ℱ(F_{2}) \ar[d]^{≅} \\ ⨁_{m} A \ar[r]^{f} & ⨁_{n} A,}

then since ℱ is right exact, one can complete the diagram

\xymatrix{ℱ(F_{1}) \ar[r]^{ℱ(g)} \ar[d]_{≅} & ℱ(F_{2}) \ar[r] \ar[d]^{≅} & ℱ(coker(g)) \ar[r] \ar@{.>}[d]^{≅} & 0 \\ ⨁_{m} A \ar[r]^{f} & ⨁_{n} A \ar[r] & M \ar[r] & 0.}

This shows ℱ(coker(g)) ≅ M and we will be done.

Thus, suppose f : ⨁_{m} A → ⨁_{n} A is an A-module homomorphism. With the misuse of notation, denote the standard basis for both A-modules A^{m} and A^{n}, by e_{i}. Then

f(e_{i}) = ∑_{j=1}^{n} a_{ij}e_{j},

where 1 ≤ i ≤ m and a_{ij} ∈ A. Decomposing a_{ij} into its homogeneous components, a_{ij} = ∑_{k} a_{ij}_{k}, since the number of a_{ij}, 1 ≤ i ≤ m, 1 ≤ j ≤ n, are finite, there is N ∈ ℕ such that a_{ij}_{N} = 0 for all i, j. Now consider the graded free A[x]-modules ⨁_{m} A[x](−N) and ⨁_{n} A[x], with the standard bases (which are of degrees N and 0 in ⨁_{m} A[x](−N) and ⨁_{n} A[x], respectively). Define

g : ⨁_{m} A[x](−N) ⟶ ⨁_{n} A[x],
e_{i} ⟼ ∑_{j} ( ∑_{k} x^{N−k}a_{ij}_{k})e_{j}.

This is a graded A[x]-module homomorphism. Further, ℱ(g) amount to evaluation of g at x = 1 which coincides with the map f. This finishes the proof.

Remark 6.2.5. The proof of Lemma 6.2.4, in particular, shows that if f ∈ 𝕄_{n}(A), then there is a g ∈ 𝕄_{n}(A[x])(\overline{δ}) such that evaluation of g at 1, gives f.

Lemma 6.2.4 along with standard results of localisation theory (see [88, Theorem 5.11]) implies that

gr- A[x]/ ker(ℱ) ≅ mod- A.

Now we are ready to apply the localisation theorem (§6.2.1) to the sequence

ker(ℱ) ↪ gr- A[x] ⟶ gr- A[x]/ ker(ℱ) ≅ mod- A,

to obtain a long exact sequence

⋯ ⟶ K_{n+1}(A) ⟶ K_{n}(ker(ℱ)) ⟶ K_{n}^{gr}(A[x]) ⟶ K_{n}(A) ⟶ · · · .

Now using (6.22) to replace K_{n}(ker(ℱ)), we get

⋯ ⟶ K_{n+1}(A) ⟶ K_{n}^{gr}(A) ⟶^{\overline{i}} K_{n}^{gr}(A[x]) ⟶ K_{n}(A) ⟶ · · · .  (6.24)

Next, we show that the homomorphism \overline{i} is injective. Note that the homomorphism \overline{i} on the level of K-groups induced by the composition of exact functors (see (6.15) and (6.21)),

Pgr-A ↪ gr- A ⟶ 𝒞_{1} ↪ ker(ℱ) ↪ gr- A[x],

which send a finitely generated graded projective A-module M, to M considered as graded A[x]- module. Define three exact functors ϕ_{i} : Pgr-A → gr- A[x], 1 ≤ i ≤ 3, as follows,

ϕ_{1}(M) = M[x](−1)  (6.25)
ϕ_{2}(M) = M[x]
ϕ_{3}(M) = M.

Now the exact sequence 6.16 and the exact sequence of the functors Theorem (§6.2.1) immediately give,

\overline{ϕ}_{3} = \overline{ϕ}_{2} − \overline{ϕ}_{1}.  (6.26)

Note that \overline{ϕ}_{3} = \overline{i}.

Now invoking Quillen’s Theorem 6.1.1, from exact sequence 6.24, we get

⋯ ⟶ K_{n+1}(A) ⟶ ℤ[x, x^{−1}] ⊗ K_{n}(A_{0}) ⟶^{\overline{i}} ℤ[x, x^{−1}] ⊗ K_{n}(A_{0}) ⟶ K_{n}(A) ⟶ · · · . (6.27)

Further, the maps \overline{ϕ}_{2} and \overline{ϕ}_{1} becomes 1 ⊗ 1 and t ⊗ 1, respectively. Thus from (6.26) we get

\overline{i} = (1 − t) ⊗ 1 : ℤ[x, x^{−1}] ⊗_{ℤ} K_{n}(A_{0}) ⟶ ℤ[x, x^{−1}] ⊗_{ℤ} K_{n}(A_{0}).

This shows that \overline{i} is an injective map. Thus the exact sequence 6.27 reduces to

0 ⟶ ℤ[x, x^{−1}] ⊗_{ℤ} K_{n}(A_{0}) ⟶^{\overline{i}} ℤ[x, x^{−1}] ⊗_{ℤ} K_{n}(A_{0}) ⟶ K_{n}(A) ⟶ 0.  (6.28)

Again, from the description of \overline{i} it follows,

K_{n}(A) ≅ K_{n}(A_{0}).

6.3 Relating K_{∗}^{gr} to K_{∗}

In [19], van den Bergh, following the methods to prove the fundamental theorem of K-theory (§6.2.2), established a long exact sequence relating graded K-theory of a ring to non-graded Ktheory in the case of right regular Noetherian ℤ-graded rings. In this section we will present van den Bergh’s observation. As a consequence, we will see that in the case of graded Grothendieck group, the shifting of modules is all the difference between the K_{0}^{gr} and K_{0}-groups (Corollary 6.3.2).

Let A be a right regular Noetherian ℤ-graded ring. Thus G^{gr}-theory and G-theory become K^{gr}-theory and K-theory (see Example 6.2.2). Consider the ℤ-graded ring A[y], where deg(y) = 1 and ℤ × ℤ-graded ring B = A[t], where the homogeneous components defined by B_{(n,m)} = A_{m}t^{n}. Furthermore, consider the ℤ×ℤ-graded rings B[z] and B[z, z^{−1}], where deg(z) = (1, −1). It is easy to check that

(1) The support of B is ℕ × ℤ and B_{(0,−)} = A as ℤ-graded rings.

(2) The support of B[z] is ℕ × ℤ and B[z]_{(0,−)} = A as ℤ-graded rings.

(3) B[z, z^{−1}] is a ℤ × ℤ-strongly graded ring and Gr^{ℤ×ℤ}-B[z, z^{−1}] ≈ Gr^{ℤ}-A[y].

Proof. We first show that B[z, z^{−1}] is a ℤ × ℤ-strongly graded ring, by proving for any (n, m) ∈ ℤ × ℤ,

1 ∈ B[z, z^{−1}]_{(n,m)}B[z, z^{−1}]_{(−n,−m)}.

But note that if m ≥ 0, then B_{(m+n,0)}z^{−m} ∈ B[z, z^{−1}]_{(n,m)} and B_{(−m−n,0)}z^{m} ∈ B[z, z^{−1}]_{(−n,−m)}. Further, B_{(m+n,0)} = A_{0}t^{m+n} and B_{(−m−n,0)} = A_{0}t^{−m−n}. This shows that

1 ∈ A_{0} ⊆ B[z, z^{−1}]_{(n,m)}B[z, z^{−1}]_{(−n,−m)}.

Now by Example 1.5.8,

Gr^{ℤ×ℤ}-B[z, z^{−1}] ≈ Gr^{ℤ}-B[z, z^{−1}]_{(0,−)}.

But it is easy to see that B[z, z^{−1}]_{(0,−)} ≅_{gr} A[y] as ℤ-graded rings.

By Example 6.2.2, we have a long exact sequence

⋯ ⟶ K_{n+1}^{gr}(B[z, z^{−1}]) ⟶ K_{n}^{gr}(B) ⟶^{\overline{i}} K_{n}^{gr}(B[z]) ⟶ K_{n}^{gr}(B[z, z^{−1}]) ⟶ · · · .  (6.29)

By Corollary 6.1.4 and (1) and (2) above,

K_{n}^{gr}(B) = K_{n}^{ℤ×ℤ}(B) ≅ ℤ[x, x^{−1}] ⊗ K_{n}^{ℤ}(B_{(0,−)}) ≅ ℤ[x, x^{−1}] ⊗ K_{n}^{gr}(A)

and

K_{n}^{gr}(B[z]) = K_{n}^{ℤ×ℤ}(B[z]) ≅ ℤ[x, x^{−1}] ⊗ K_{n}^{ℤ}(B[z]_{(0,−)}) ≅ ℤ[x, x^{−1}] ⊗ K_{n}^{gr}(A).

Furthermore, from (3) we have

K_{n}^{gr}(B[z, z^{−1}]) = K_{n}^{ℤ×ℤ}(B[z, z^{−1}]) ≅ K_{n}^{ℤ}(A[y]) = K_{n}^{gr}(A[y]).

So, from the sequence 6.29, we get

⋯ ⟶ K_{n+1}^{gr}(A[y]) ⟶ ℤ[x, x^{−1}] ⊗ K_{n}^{gr}(A) ⟶^{\overline{i}} ℤ[x, x^{−1}] ⊗ K_{n}^{gr}(A) ⟶ K_{n}^{gr}(A[y]) ⟶ · · · . (6.30)

The rest is similar to the method used to prove the fundamental theorem in §6.2.2. First note that we have a short exact sequence of ℤ × ℤ-graded S[z]-modules (compare this with (6.16))

0 ⟶ M[z](−1, 1) ⟶^{z} M[z] ⟶^{e_{0}} M ⟶ 0.  (6.31)

Here M is a graded S-module which becomes a graded S[y]-module under the evaluation map e_{0} : S[y] → S, y ↦ 0. Again, similar to (6.25), defining

ϕ_{1}(M) = M[z](−1, 1)
ϕ_{2}(M) = M[z]
ϕ_{3}(M) = M,

the exact sequence 6.31 and the exact sequence of the functors Theorem (§6.2.1) immediately give,

\overline{ϕ}_{3} = \overline{ϕ}_{2} − \overline{ϕ}_{1},  (6.32)

and \overline{ϕ}_{3} = \overline{i}. One can then observe that \overline{i} is injective, so the exact sequence 6.30 reduces to

0 ⟶ ℤ[x, x^{−1}] ⊗ K_{n}^{gr}(A) ⟶^{\overline{i}} ℤ[x, x^{−1}] ⊗ K_{n}^{gr}(A) ⟶ K_{n}^{gr}(A[y]) ⟶ 0.

Finally, from this exact sequence it follows that

K_{n}^{gr}(A[y]) ≅ K_{n}^{gr}(A).  (6.33)

We are in a position to relate graded K-theory to non-graded K-theory.

Theorem 6.3.1. Let A be a right regular Noetherian ℤ-graded ring. Then there is a long exact sequence

⋯ ⟶ K_{n+1}(A) ⟶ K_{n}^{gr}(A) ⟶^{\overline{i}} K_{n}^{gr}(A) ⟶^{U} K_{n}(A) ⟶ · · · .  (6.34)

Here \overline{i} = \overline{s} − 1, where s : Gr-A → Gr-A, M ↦ M(1) and U is the forgetful functor.

Proof. By Example 6.2.2, we have a long exact sequence

⋯ ⟶ K_{n+1}^{gr}(A[y, y^{−1}]) ⟶ K_{n}^{gr}(A) ⟶^{i} K_{n}^{gr}(A[y]) ⟶ K_{n}^{gr}(A[y, y^{−1}]) ⟶ · · · .

By (6.33), K_{n}^{gr}(A[y]) ≅ K_{n}^{gr}(A). Since A[y, y^{−1}] is strongly graded, by Dade’s Theorem 1.5.1, K_{n}^{gr}(A[y, y^{−1}]) ≅ K_{n}(A). Replacing these into above long exact sequence, the theorem follows.

The following is an immediate corollary of the Theorem 6.3.1. It shows that for the graded regular Noetherian rings, the shifting of modules is all the difference between the graded Grothendieck group and the usual Grothendieck group.

Corollary 6.3.2. Let A be a right regular Noetherian ℤ-graded ring. Then

K_{0}^{gr}(A)/⟨[P (1)] − [P]⟩ ≅ K_{0}(A),

where P is a graded projective A-module.

Proof. The long exact sequence 6.3.1, for n = 0, reduces to

\xymatrix{K_{0}^{gr}(A) \ar[r]^{[P]↦[P (1)]−[P]} & K_{0}^{gr}(A) \ar[r]^{U} & K_{0}(A) \ar[r] & 0.}

The corollary is now immediate.

Remark 6.3.3. Both the fundamental theorem (§6.2.2) and Theorem 6.3.1 can be written for the case of graded coherent regular rings, as it was demonstrated by Gersten in [37] for the non-graded case.

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