A NOTE ON K-THEORY OF AZUMAYA ALGEBRAS

ROOZBEH HAZRAT AND JUDITH R. MILLAR

Abstract. For an Azumaya algebra A which is free over its centre R, we prove that Ktheory of A is isomorphic to K-theory of R up to its rank torsions. We conclude that K_{i}(A, ℤ/m) = K_{i}(R, ℤ/m) for any m relatively prime to the rank and i ≥ 0. This covers, for example, K-theory of division algebras, K-theory of Azumaya algebras over semi-local rings and K-theory of graded central simple algebras indexed by a totally ordered abelian group.

1. introduction

Let R be a ring and K_{i}(R), i ≥ 0, be Quillen’s K-groups. The construction of K-groups is functorial. Furthermore, K_{i} functors induce identity maps on inner-automorphisms of a ring and K_{i}(R) → K_{i}(M_{t}R) → K_{i}(R) is multiplication by t, where R → M_{t}R is the diagonal homomorphism, r ↦ rI_{t}.

For the class of division rings finite dimensional over their centres (which are fields), Green et. al. [8] proved that K-theory of a division algebra is essentially the same as K-theory of its centre, i.e., for a division algebra D over its centre F of index n,

(1.1)  K_{i}(D) ⊗ ℤ[1/n] ≅ K_{i}(F) ⊗ ℤ[1/n].

Their proof combines the above observations with the Skolem-Noether theorem which guarantees that algebra homomorphisms in the setting of central simple algebras are inner, and then uses the main result of [7] which states that

lim_{i→∞} M_{n^{2i}}F ≅ lim_{i→∞} M_{n^{2(i+1)}}D.

This note is a continuation of [10] where it was observed that one can naturally deduce (1.1) by using the fact that an F -central simple algebra A is a twisted form of a matrix algebra, i.e., there is a finite field extension L/F such that A ⊗_{F} L = M_{k}L. However, using this takes us out of the category of F -algebras to the category of field extensions of F.

Another equivalent defining property for a central simple algebra A is that the opposite algebra is the inverse element in the Brauer group, i.e., A ⊗_{F} A^{op} ≅ M_{m}F, where m is the dimension of A over F as a vector space. In this note we will use this to prove (1.1) type properties (see also Remark 2). This enables us to extend our category from that of central simple algebras to the category of Azumaya algebras free over their centres. In particular we will prove that K-theory of an Azumaya algebra over a semi-local ring is essentially the same as K-theory of its centre. Along the same lines, another corollary of this approach is to conclude that K-theory of graded central simple algebras indexed by a totally ordered abelian group has (1.1) type property. However, it remains as a question whether the same can be said about any Azumaya algebra of constant rank:

The first author acknowledges the support of EPSRC first grant scheme EP/D03695X/1. Part of this work has been done in the Winter of 2007 at KIAS, Seoul, Korea and the Summer of 2008 at ICTP, Trieste, Italy. The authors would like to thank Raymond Hoobler for several discussions regarding the subject.

Question 1. Let A be an Azumaya algebra over its centre R of constant rank n. Then is it true that for any i ≥ 0, K_{i}(A) ⊗ ℤ[1/n] ≅ K_{i}(R) ⊗ ℤ[1/n]?

Remark 2. The term Azumaya algebra originates from the work done by Azumaya in his 1951 paper [2]. The definition has developed since then, and an Azumaya algebra is now defined to be an R-algebra A such that A is faithfully projective as an R-module and the natural homomorphism A ⊗_{R} A^{op}→ End_{R}(A) is an isomorphism (see [16], Chapter III, §5.1). An equivalent condition is that (i) A is a finitely generated R-module and (ii) for every maximal ideal m of R, A/mA is a central simple algebra over R/m (see [16], Chapter III, Theorem 5.1.1). This equivalence has been proven by Azumaya ([2], Theorem 15, page 130) under the additional assumption that A is free over R, and by Bass ([3], Chapter III, Theorem 4.1) under the assumption that A is projective over R, thanks to the work of Auslander and Goldman [1]. So for an Azumaya algebra A as originally defined by Azumaya with A free over its centre R, our Theorem 6 covers its K-theory.

2. 𝒟-functors

In this section we set up an abstract functor, called a 𝒟-functor, which will handle several quotient groups coming from K-groups. Some of the ideas behind this definition have already been employed in [9] to define a functor in the category of central simple algebras (compare this definition with the one in [9] and [10]). One main distinction of this definition, besides being defined over the category of Azumaya algebras, is that the domain is restricted to a fixed base ring. The aim is to show that such functors have bounded torsion abelian groups as their range.

Let Az(R) be the category of Azumaya algebras over the commutative ring R, C(R) the category of all commutative R-algebras and 𝒜b the category of abelian groups. Consider a functor ℱ : Az(R) → ℱunc(C(R), 𝒜b). Thus for any Azumaya R-algebra A, ℱ_{A} : C(R) → 𝒜b is a functor from C(R) to the category of abelian groups. Consider the following property for ℱ:

(1) ℱ_{R} is a trivial functor, i.e., ℱ_{R}(S) = 1 for any R-algebra S. Consider also the following properties at the base point R:

(2) For any faithfully projective R-module P, there is a homomorphism

d : ℱ_{A⊗End_{R}(P)}(R) → ℱ_{A}(R)

such that the composition

ℱ_{A}(R) → ℱ_{A⊗End_{R}(P)}(R) → ℱ_{A}(R)

is η_{[P:R]} if P has a constant rank [P : R]. Here η_{k}(x) = x^{k}.

(3) If P has a constant rank over R then ker(d) is [P : R]-torsion.

We call a functor ℱ with the above three properties a 𝒟-functor with respect to the class of constant faithfully projective modules.

Remark 3. When evaluating ℱ at the base point, i.e., ℱ_{A}(R), one can drop R and simply write ℱ(A). With this simplification, when P is a free module of rank n, from the condition (2) it follows that the composition of ℱ(A) → ℱ(M_{n}A) → ℱ(A) is η_{n}.

Remark 4. The goal is to show that ℱ_{A}(R) is a torsion abelian group. However, one would like to prove the same statement for any ℱ_{A}(S) where S is an R-algebra. This could be done if one further assumes:

(4) For any S ∈ C(R), ℱ_{A}(S) = ℱ_{A⊗S}(S). Here ℱ_{A⊗S} is a functor from C(S) to 𝒜b. There is no need to introduce an extra notation, say, ℱ_{A}^{R} and ℱ_{A⊗S}^{S} here. The condition (4) simply states that evaluating ℱ_{A} at S is the same as evaluating the functor ℱ_{A⊗S} at the base point.

Theorem 5. Let A be an Azumaya algebra over R of constant rank n. Then ℱ_{A}(R) is n^{2}-torsion, where ℱ is a 𝒟-functor.

Proof. Since we will be working with the functor ℱ on the base point, for an R-Azumaya algebra A, we simply write ℱ(A) instead of ℱ_{A}(R) (see Remark 3 above). The two R-algebra homomorphisms, i : A → A ⊗_{R} A^{op} and r : A^{op} → End_{R}(A^{op}) induce the homomorphisms ℱ(A) → ℱ(A ⊗_{R} A^{op}) and ℱ(A^{op}) → ℱ(End_{R}(A^{op})). Since A is an Azumaya algebra over R, we have A ⊗_{R} A^{op} = End_{R}(A) and A is faithfully projective of rank n (see [16], Chapter III, §5.1 and Theorem 5.1.1). By (2) in the definition of a 𝒟-functor, we have a homomorphism d : ℱ(End_{R}(A)) → ℱ(R). But ℱ(R) is trivial by (1) and thus the kernel of d is ℱ(End_{R}(A)) which is, by (3), n-torsion. Consider the following diagram

\xymatrix{ & ℱ(A) \ar[dl]^{i} \ar[ddr]^{η_{n}} \\ ℱ(End_{R}(A)) ≅ ℱ(A ⊗_{R} A^{op}) \ar[dr]^{r} \\  & ℱ(A ⊗_{R} End_{R}(A^{op})) \ar[r]^{d} & ℱ(A)}

(2.1)

By (2), d ∘ r ∘ i = η_{n}. But from the commutativity of the diagram and the fact that ℱ(End_{R}(A)) is n-torsion, it follows that ℱ(A) is n^{2}-torsion. □

For a ring A with centre R, consider the inclusion R → A. This induces the map K_{i}(R) → K_{i}(A) for i ≥ 0. Consider the exact sequence

(2.2)  1 → ZK_{i}(A) → K_{i}(R) → K_{i}(A) → CK_{i}(A) → 1

where ZK_{i} and CK_{i} are the kernel and cokernel of the map K_{i}(R) → K_{i}(A) respectively. It is not difficult to see that one can consider CK_{i} as the following functor:

CK_{i} : Az(R) → ℱunc(C(R), 𝒜b)
A ↦ CK_{i}(A ⊗_{R} −).

Thus for any morphism S → T in C(R), we have CK_{i}(A ⊗_{R} S) → CK_{i}(A ⊗_{R} T). We will check that CK_{i} (and in a similar manner ZK_{i}) is a 𝒟-functor. Condition (1) in the definition of a 𝒟-functor is straightforward to check. To prove the second condition, since A ⊗_{R} End_{R}(P) ≅ End_{A}(P ⊗_{R} A) (see [5], Chapter II, §5.3, Proposition 7) and P ⊗_{R} A is faithfully projective over A, it is enough to work with faithfully projective modules over A. Let P be a faithfully projective A-(bi-)module. Let 𝒫(A) and 𝒫(End_{A}(P)) denote the categories of finitely generated projective (left) modules over A and End_{A}(P) respectively.

Consider the functors

(2.3)  𝒫(A) ⟶^{ϕ} 𝒫(End_{A}(P))
X ⟼ End_{A}(P) ⊗_{A} X

and

(2.4)  𝒫(End_{A}(P)) ⟶^{ψ} 𝒫(A)
Y ⟼ Hom_{A}(P, A) ⊗_{End_{A}(P)} Y

By the Morita theory, ψ is a natural equivalence (see [16], Chapter I, Theorem 9.1.3), thus it gives rise to the isomorphism from K_{i}(End_{A}(P)) to K_{i}(A). Furthermore, ψϕ(X) = Hom_{A}(P, X). Clearly if P = A^{k}, then ψϕ(X) = X^{k}, i.e., k copies of X. Since K_{i} are functors which respect direct sums ([17], §2), this composition induces a multiplication by k in the level of K-groups. This implies that Condition (2) holds for K-functors with respect to free modules of finite ranks.

If we could prove that ZK_{i} and CK_{i} are 𝒟-functors in the generality that we set up our definition of a 𝒟-functor, i.e., for all faithfully projective modules of constant rank, then we would be able to answer Question 1 positively. However, at this stage, we can show that CK_{i} and ZK_{i} are in fact 𝒟-functors with respect to free modules of finite ranks and, consequently, for Theorem 5 to follow, we need to have Azumaya algebras free over their centres. Examples of Azumaya algebras free over their centres, apart from division algebras, are Azumaya algebras over semi-local rings and graded central simple algebras. Thus restricting to free modules, (2.3) and (2.4) give rise to the homomorphisms ϕ and ψ on the level of K-groups and we have the following commutative diagram,

\begin{matrix}K_{i}(R) & [figure] & = & [figure] & K_{i}(A) & [figure] & ϕ & [figure] & CK_{i}(A) & [figure] & [figure] & 1 & K_{i}(R) & [figure] & η_{k} & [figure] & K_{i}(M_{k}A) & [figure] & ψ & ≅ & [figure] & CK_{i}(M_{k}A) & [figure] & d & [figure] & 1 & K_{i}(R) & [figure] & K_{i}(A) & [figure] & CK_{i}(A) & [figure] & 1 \\ (2.5) \end{matrix}

where compositions of columns are η_{k}, and thus Condition (2) holds. Now if x ∈ CK_{i}(M_{k}A) such that d(x) = 1, then a diagram chase shows that x^{k} = 1, satisfying Condition (3) of a 𝒟-functor. In the same manner one can show that ZK_{i} is a 𝒟-functor with respect to free modules of finite ranks.

We are now in a position to prove that the K-theory of Azumaya algebras over semi-local rings and K-theory of graded central simple algebras are isomorphic to K-theory of their centres up to (their ranks) torsions.

Theorem 6. Let A be an Azumaya algebra free over its centre R of rank n. Then for any i ≥ 0,

K_{i}(A) ⊗ ℤ[1/n] ≅ K_{i}(R) ⊗ ℤ[1/n].

Proof. The argument before the theorem shows that CK_{i} (and in the same manner ZK_{i}) is a 𝒟-functor with respect to free modules of finite ranks, and thus by Theorem 5, CK_{i}(A) and ZK_{i}(A) are n^{2}-torsion abelian groups. Tensoring the exact sequence (2.2) by ℤ[1/n], since CK_{i}(A) ⊗ ℤ[1/n] and ZK_{i}(A) ⊗ ℤ[1/n] vanish, the result follows. □

Recall that K-theory with coefficients is related to absolute K-theory by the following exact sequence (see [20])

(2.6)  1 ⟶ K_{i}(A) ⊗ ℤ[1/m] ⟶ K_{i}(A, ℤ/m) ⟶ _{m}K_{i−1}(A) ⟶ 1

for any ring A and m ≥ 0. The following corollary is now immediate.

Corollary 7. Let A be an Azumaya algebra free over its centre R of rank n. Then for any m relatively prime to n, and for any i ≥ 0,

K_{i}(A, ℤ/m) ≅ K_{i}(R, ℤ/m).

Proof. Consider the diagram

\begin{matrix}1 & [figure] & K_{i}(R) ⊗ ℤ[1/m] & [figure] & [figure] & K_{i}(R, ℤ/m) & [figure] & [figure] & _{m}K_{i−1}(R) & [figure] & [figure] & 1 & 1 & [figure] & K_{i}(A) ⊗ ℤ[1/m] & [figure] & K_{i}(A, ℤ/m) & [figure] & _{m}K_{i−1}(A) & [figure] & 1 \\ (2.7) \end{matrix}

Since ZK_{i}(A) and CK_{i}(A) are n^{2}-torsion and n and m are relatively prime, it is an easy exercise to show that the outer vertical maps are isomorphisms. Thus the middle vertical map is an isomorphism. □

Corollary 8. Let A be an Azumaya algebra over its centre R, a semi-local ring, of rank n. Then for any i ≥ 0, K_{i}(A) ⊗ ℤ[1/n] ≅ K_{i}(R) ⊗ ℤ[1/n].

Proof. Since A is finitely generated projective of constant rank and R is a semi-local ring, it follows that A is a free module over R (see [6], Chapter II, §5.3, Proposition 5), and thus the corollary follows from Theorem 6. □

Let us note that, for an F -central division algebra D, and i = 1, CK_{1}(D) is the group D^{∗}/F^{∗}D′ where D^{∗} is the multiplicative group of D and D′ is its commutator subgroup. This group has a direct application in solving the open problem whether a multiplicative group of a division algebra has a maximal subgroup [12]. Indeed, since CK_{1}(D) is torsion of bounded exponent (by Theorem 5), if it is not trivial, it has maximal subgroups and therefore D^{∗} has (normal) maximal subgroups. Thus finding the maximal subgroups in D^{∗} reduces to the case that CK_{1}(D) is trivial. It is a conjecture that CK_{1}(D) is trivial if and only if D is an ordinary quaternion division algebra over a Pythagorean field. It was shown in [12] that such division algebras do have (non-normal) maximal subgroups. Thus if the above conjecture is settled positively, one concludes that the multiplicative group of a division algebra does have a maximal subgroup.

The theory of graded division algebras, indexed by a totally ordered abelian group, has recently been given considerable attention as the theory is closely related to the theory of division algebras equipped with a valuation. Let D be a division algebra with a valuation. To this one associates a graded division algebra grD = ⨁_{γ∈Γ_{D}} gD_{γ} where Γ_{D} is the value group of D and gD_{γ} comes from the principal fractional ideals of the valuation ring (see §4 in [15] for details). As it is mentioned in [15], even though computations in the graded setting are easier (and discrete), it seems not so much is lost in passage from D to its corresponding graded division algebra grD. This was a motivation to the systematic study of graded central simple algebras and their correspondences, notably by Boulagouaz [4], Hwang, Tignol and Wadsworth [13, 14, 15, 19] and to the comparison of certain functors defined on these objects, notably the Brauer group and the reduced Whitehead group.

Recall that a unital ring R = ⨁_{γ∈Γ} R_{γ} is called a graded ring if Γ is a group, each R_{γ} is a subgroup of (R, +) and R_{γ} · R_{δ} ⊆ R_{γ+δ} for all γ, δ ∈ Γ. Throughout this paper, we will assume that Γ is a totally ordered abelian group (thus it is a torsion free abelian group). The elements of R_{γ} are called homogeneous of degree γ. A graded ring E = ⨁_{γ∈Γ_{E}} E_{γ} is called a graded division ring if every non-zero homogeneous element has a multiplicative inverse. Note that Γ_{E}, the support of E, is a group and that E_{0} is a division ring. Since Γ_{E} is totally ordered, it follows that E has no zero divisors and that E^{∗}, the multiplicative group of E, is the set E^{h} \ {0}. Note that every graded E-module M is a graded free E-module, i.e., M is a graded E-module which is free as an E-module with a homogeneous basis. A graded field is a commutative graded division ring. A graded algebra A over a graded field R is said to be a graded central simple algebra over R if A is a simple graded ring (i.e., no non-trivial graded two sided ideals), finite dimensional over R with centre R.

Corollary 9. Let A be a graded central simple algebra over its graded centre R of degree n. Then for any i ≥ 0,

K_{i}(A) ⊗ ℤ[1/n] ≅ K_{i}(R) ⊗ ℤ[1/n].

Proof. By [4], Proposition 5.1 (see also [15], Corollary 1.2), a graded central simple algebra A over R is an Azumaya algebra. Since R is a graded field (indexed by a torsion free group), A is a free R-module. The corollary now follows from Theorem 6. □

The above result seems also to follow (in a hard way) by using Corollary 8, in combination with the fundamental theorem of K-theory, i.e., Theorem 8, its corollary and the exercise on p. 38 of Quillen’s paper [17].

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Dept. of Pure Mathematics, Queen’s University, Belfast BT7 1NN, United Kingdom

   E-mail address: r.hazrat@qub.ac.uk

Dept. of Pure Mathematics, Queen’s University, Belfast BT7 1NN, United Kingdom

   E-mail address: jmillar12@qub.ac.uk