Algebras and Representation Theory 5 (2002), 1 – 53

Dimension Theory and Nonstable K_{1}

Arun Mundkur Faculty of Mathematics, University of Bielefeld, P.O.Box 100131, 33501 Bielefeld, Germany.

Abstract: This article provides an introduction to A. Bak’s theory of group valued functors on categories with structure and dimension and applies this theory to the algebraic K-theory functors nonstable K_{1} and K_{2}.

Mathematics Subject Classifications (2000): 19Bxx, 19C09, 19C40, 19Cxx Keywords: category with structure, dimension, excision, linking diagram, nonstable K-group

1 Introduction

In [3] the foundations are laid for a theory of group valued functors on categories with dimension. A K-theory of natural transformations of group-valued functors on categories with dimension is developed and normal, solvable and nilpotent filtrations on group-valued functors are established.

In the current paper a self-contained development of a portion of this theory is provided and published for the first time. It is applied to give a new proof of the following result. Let R be a commutative ring. Let GL_{n}(R) denote the general linear group of rank n ≥ 5 over R and let E_{n}(R) denote its elementary subgroup. A main result in Bak [4] is that the dimension filtration GL_{n} = GL_{n}^{−1} ⊇ GL_{n}^{0} ⊇ ⋯ GL_{n}^{i} ⊇ ⋯ ⊇ E_{n} of the functor GL_{n} has the property that for any commutative ring R and any n ≥ 3 each GL_{n}^{i}(R) ⊲ GL_{n}(R) is a normal subgroup, GL_{n}(R)/GL_{n}^{0}(R) is Abelian and the filtration

GL_{n}^{0}(R) ⊇ ⋯ GL_{n}^{i}(R) ⊇ ⋯

is a descending, central series such that GL_{n}^{i}(R) = E_{n}(R) whenever i ≥ Krull − dimension(R). In the introduction to [4] the author writes that this result “will be extended to classical and Kac-Moody groups...using Mayer-Vietoris sequences for these functors to replace some of the computations in this article.” The purpose of the current paper is to develop the Mayer-Vietoris sequences referred to in [4] in the general framework of group valued functors on categories with dimension [3], and then show how to apply the sequences to prove the result above for n ≥ 5.

In [3], the concept of Mayer-Vietoris sequence is replaced by a more general noncommutative notion called a linking diagramme. A linking diagramme is a diagramme

\xymatrix{K′ \ar[dr]^{κ′} &  &  & L′ \\  & K \ar[r]^{∂} & L \ar[ur]^{λ′} \ar[dr]_{λ′′} \\ K′′ \ar[ur]_{κ′′} &  &  & L′′}

of groups such that Ker(λ′′) ∩ Ker(λ′) = Image(∂) and Image(κ′′) · Image(κ′) = Ker(∂). Such diagrammes arise in the study of natural transformations τ : S ⟶ G of group valued functors S and G defined on a category C with dimension. We explain how they arise and in particular, the relationship of the K’s and the L’s to τ.

The motivating case is when C is the category of commutative rings, S the Steinberg group functor St_{n}, G the general linear group functor GL_{n}, and τ : S ⟶ G the canonical natural transformation τ_{n} : St_{n} ⟶ GL_{n}. In the general situation one sets E = Image(τ) and assumes E is normal in G. In the motivating case, E corresponds to the elementary subgroup functor E_{n} of GL_{n} and by a theorem of [15], E_{n} is normal in GL_{n}. We define in the general situation

K = Ker(τ), L = Coker(τ) = G/E.

Thus in the motivating case K corresponds to the nonstable K-theory functor K_{2,n} and L to the nonstable K-theory functor K_{1,n}. A category C with dimension is by definition a category which is equipped with a class S(C) of commutative squares

\xymatrix{A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D}

in C called structure squares and a function d : Obj C ⟶ ℤ^{≥0} ∪ {∞} called a dimension function such that if A is any object of C whose dimension d(A) is finite and nonzero then there is a structure square as above such that d(C) < d(A). In the motivating case, the structure squares are so-called localization-completion squares of rings and d is Bass-Serre dimension for rings. An important part of the article is to establish in the general situation conditions concerning the behaviour of S and G on S(C) which allow us to associate functorially to each structure square a linking diagramme

\xymatrix{K(B) \ar[dr]^{κ′} &  &  & L(B) \\  & K(D) \ar[r]^{∂} & L(A) \ar[ur]^{λ′} \ar[dr]_{λ′′} \\ K(C) \ar[ur]_{κ′′} &  &  & L(C) & .}

Such a linking diagramme is called equivariant if the action by conjugation of certain elements of G(A) on L(A) can be lifted to an action of these elements on K(D) such that ∂ becomes equivariant. The key idea for constructing linking diagrammes is that of excision for τ. One shows that for any morphism A ⟶ B in C, there is 5-term functorial exact sequence

K(A) ⟶ K(B) ⟶ R(A, B) ⟶ L(A) ⟶ L(B).

The excision condition is that if

\xymatrix{A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D}

is a structure square then the functorially given homomorphism R(A, B) ⟶^{≅} R(C, D) is an isomorphism. This allows one to construct in a routine way the linking diagramme functor above. The main goal of the paper is to prove the following theorem. It is a special case of a more general result in [3]. One defines the dimension filtration G = G^{−1} ⊇ G^{0} ⊇ ⋯ G^{i} ⊇ ⋯ of τ on G by

G^{i}(A) = ⋂_{A⟶B_{d(B)≤i}} Ker(G(A) ⟶ L(B)).

It is obvious that G^{i} is normal in G.

Theorem Let C denote a category with dimension whose structure squares are denoted by S(C). Let τ : S ⟶ G be a natural transformation of group valued functors on C. If τ satisfies excision on S(C) and the “stable elementwise” action of G^{0} on K(D) is trivial for any structure square

\xymatrix{A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D}

then for any object A of C, the descending chain G^{0}(A) ⊇ ⋯ G^{n}(A) ⊇ ⋯ E(A) is a descending, central series of normal subgroups of G(A) such that G^{i}(A) = E(A) whenever i ≥ d(A).

The theorem is stated and proved under Theorem 3.2. The τ - induced linking diagram functors on S(C) are the keystone of the proof.

The rest of the paper is organized as follows.

Section 2 introduces tools which will be used throughout. We talk about a pair S and G of group valued functors on an arbitrary category C endowed with a fixed natural transformation τ : S ⟶ G between them. Section 2.1 discusses conditions for the existence and uniqueness of equivariant liftings of an action on G by conjugation of a subfunctor F ⊆ G to an action of F on S. Such liftings will be used later to obtain descending, central series in G. It is shown that the lifting conditions hold under the familiar group-theoretic concepts of perfect, central extension and universal, perfect, central extension. In Section 2.2 the notion of a Linking Diagramme in the category of groups is defined. The main tool of the paper will be functors on categories with dimension with values in linking diagrammes. Section 2.3 introduces the concept of categories with dimension. These categories form the general setting for this paper. One makes an arbitrary category C into a category with structure by equipping it with Infrastructure Functors and a class of commutative squares called Structure Squares. Provided with a Dimension Function, a category with structure becomes a Category with Dimension. Dimension functions are nonnegative integral or infinity valued functions with a certain behaviour on the structure squares. We prove a theorem which shows constructively how an arbitrary category with structure can be made into a category with dimension. Section 2.4 recalls the standard ring-theoretic constructions localization and completion. Localization-completion Squares are used to make categories of rings into categories with structure. These squares share, cf. Lemma 2.15, the property of being fibre squares which will be important in applications to GL_{n}. Then Krull dimension and Bass-Serre dimension are discussed briefly as examples of dimension functions. Bass-Serre dimension is used to make ring categories with structure into ring categories with dimension in Theorem 2.20.

A major goal of this article is to show that dimension filtrations on group valued functors G are descending central series under appropriate conditions. To do this, all one needs is that there are enough equivariant linking diagrammes. Section 3 makes this assumption and then proves that the dimension filtration G^{0} ⊇ ⋯ G^{i} ⊇ ⋯ is a descending central sequence such that G^{i}(A) = E(A) if i ≥ d(A).

Section 4 tackles the problem of obtaining “enough” equivariant linking diagramme functors from a natural transformation τ : S ⟶ G of group valued functors. We construct first a class E(C) of commutative squares in C called Excision Squares from which we obtain the linking diagramme functors (K, L, ∂). Then we make use of so-called infrastructure functors to transform the linking diagramme functors into equivariant ones.

Section 5 begins applying the results of Section 4 to categories of rings. In Section 2, we showed already that localization- completion squares and Bass-Serre dimension make commutative Noetherian rings into a category with dimension. Section 5 sets up the topological tools for rings and groups which are used to prove that localization-completion squares are excision squares for the natural transformation τ_{n} : St_{n} ⟶ GL_{n}.

In Section 6 we reap the fruits of karma. First the group valued functors GL_{n}, St_{n}, K_{1,n} and K_{2,n} are recalled. Then we quote well known results from Suslin, and Tulenbaev and results from the previous sections of the paper to show that all the conditions for the dimension filtration on GL_{n}^{0} to be a descending central series are satisfied, except possibly that localization-completion squares are excision squares for τ_{n} : St_{n} ⟶ GL_{n}. Then we use the tools developed in Section 5 to show that localization-completion squares are indeed excision squares for τ_{n}.

2 Instruments of Dimension Theory

2.1 Equivariant Liftings to Extensions of Group Valued Functors

Let C be an arbitrary category.

Let S, G : C ⟶ groups be group-valued functors and τ : S ⟶ G be a natural transformation of functors. If A is an object of C, define

K(A) = Ker(τ_{A} : S(A) ⟶ G(A))
E(A) = Im(τ_{A} : S(A) ⟶ G(A))

and

L(A) = G(A)/E(A)

Thus K and E are functors C ⟶ groups and L is a functor C ⟶ coset spaces. Throughout the article it will be assumed that E(A) is normal in G(A) for all A ∈ C ; thus each L(A) is also a group.

Recall the following definition from [14] p 43. Given any group E, a pair ( J, ϕ ), where J is a group and ϕ : J ⟶ E is a group homomorphism, is called a central extension of E if ϕ : J ↠ E is surjective and the kernel of the map ϕ : J ⟶ E is contained in Centre(J). A central extension is called a perfect, central extension if the group J is perfect. Given a pair of group-valued functors we can define one being a perfect, central extension of the other as follows. Let S and T be any group-valued functors and χ : S ⟶ T a natural transformation. The functor S is called a perfect central extension of T if

(1) for each object A of C the group S(A) is perfect,

(2) the map X_{A} : S(A) → T (A) is surjective, and

(3) for each object A of C the kernel of the map χ_{A} : S(A) ⟶ T (A) is contained in Centre(S(A)).

Since E(A) is normal in G(A), there is an action of G(A) on E(A) and on L(A) by conjugation. Sometimes there is an induced action of G(A) on S(A). These actions will play an important role in this article. We make a definition and establish certain results which will be needed later.

2.1 Definition Let F : C ⟶ groups be a subfunctor of G. Thus F normalises E , i.e. for any object A of C , F (A) normalises E(A). We say that the action of F on E by conjugation lifts equivariantly to S if for each object A of C , there is a group homomorphism F (A) ⟶ Aut(S(A)) such that the resulting action of F (A) on S(A) has the following properties :

(1) The homomorphism (τ_{A} : S(A) ⟶ G(A)) is F(A)-equivariant.

(2) If (A ⟶ B) ∈ C , g ∈ F (A) and x ∈ S(A) then S_{A⟶B}(gx) = F_{A⟶B}(g)S_{A⟶B}(x).

2.2 Lemma Suppose the action of G on E by conjugation lifts equivariantly to S . Suppose S is a perfect, central extension of E . Then the action of S on itself by conjugation is the same as the action of S on itself defined by the action of G on S .

Proof : If K(A) is central in S(A), then the action of S(A) on K(A) by conjugation is obviously trivial. If s ∈ S(A) let ρ_{s} denote the inner automorphism x ↦ sxs^{−1} of S defined by s. Let G(A) ⟶ Aut(S(A)), f ↦ σ_{f} be the homomorphism defined by the action of G(A) on S(A). We shall show that σ_{τ_{A}(s)} = ρ_{s}. Since τ_{A} is G(A)-equivariant, τ_{A}(σ_{τ_{A}(s)}(x)) = τ_{A}(ρ_{s}x) for all x ∈ S(A).

Thus σ_{τ_{A}(s)}(x)(ρ_{s}x)^{−1} ∈ K(A), i.e. there is an element c_{x} ∈ K(A) such that σ_{τ_{A}(s)}(x) = c_{x}ρ_{s}(x). Since S(A) is a perfect group, each x ∈ S(A) is a product of commutators [u, v] of elements u, v ∈ S(A). But for any commutator [u, v] of S(A) one has σ_{τ_{A}(s)}([u, v]) = [σ_{τ_{A}(s)}(u), σ_{τ_{A}(s)}(v)] = [c_{u}ρ_{s}(u), c_{v}ρ_{s}(v)] = ρ_{s}([u, v]), since c_{u}, c_{v} ∈ K(A) are central elements.

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2.3 Lemma Under the same assumptions as in the previous lemma, there is only one equivariant lifting of the action of G to S .

Proof : Denote by x ⟼ δ_{g}(x) another equivariant lifting of the action of G(A) to S(A). We see that δ_{g}(x) = σ_{g}(x) ∀x ∈ S(A) and ∀g ∈ G(A), as follows . Since τ_{A} is G(A)-equivariant under each of the liftings, τ_{A}(δ_{g}(x)) = τ_{A}(σ_{g}x) for all x ∈ S(A). Thus (δ_{g}(x))(σ_{g}x)^{−1} ∈ K(A), i.e. there is an element c_{x} ∈ K(A) such that δ_{g}(x) = c_{x}σ_{g}(x). It follows that for any commutator [u, v] in S(A) δ_{g}[u, v] = [δ_{g}(u), δ_{g}(v)] = [c_{u}σ_{g}(u), c_{v}σ_{g}(v)] = σ_{g}([u, v]), since c_{u}, c_{v} ∈ K(A) are central elements. But since S(A) is perfect, it follows from the above that δ_{g}(x) = σ_{g}(x) for any x ∈ S(A).

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The following corollary emphasizes an obvious but important fact about equivariant liftings of G to perfect, central extensions of E.

2.4 Corollary Under the same assumptions as in the previous lemma, the action of G on S induces an action of L on K , which is trivial.

Proof : Clearly the action of G(A) on S(A) leaves K(A) invariant. To show that L(A) acts on K(A), it suffices to prove that E(A) acts trivially on K(A). But this follows from Lemma 2.2 and the fact that K(A) ⊆ centre(S(A)).

□

Recall another concept of group theory also from [14] p 43. Given any group E a pair ( J, τ ), where J is a group and τ : J ⟶ E is a group homomorphism, is called a universal, central extension of E if

(1) the pair ( J, τ ) is a central extension of E and

(2) if the pair ( J′, τ′ ) is another central extension of E, there is a unique group homomorphism ϕ : J ⟶ J′ such that τ = τ′ ∘ ϕ.

Notice that a universal, central extension is always perfect, [14] Theorem 5.3.

We also formulate this concept in the language required in this article. The pair (S, τ) is called a universal, central extension of the functor E if for each object A of C , (S(A), τ(A)) is a universal, central extension of E(A). It is clear that if the functor (T, χ) is another central extension of E , then there is a transformation ϕ : S ⟶ T such that τ = χ ∘ ϕ , namely for each object A of C , the homomorphism ϕ(A) : S(A) ⟶ T (A) defined by the universality of (S(A), τ(A)) defines the natural transformation ϕ : S ⟶ T .

2.5 Lemma Suppose S is a universal, perfect, central extension of E . Then the action of G on E by conjugation lifts to S . Moreover there is only one lifting.

Proof : E(A) is assumed to be normal in G(A) . Let σ ∈ Inn(G(A)) be any inner automorphism of G(A). The pair (S(A), τ_{A}) is a universal, central extension of E(A). The pair (S(A), σ ∘ τ_{A}) is another perfect, central extension of E(A). Thus there is a group homomorphism κ : S(A) ⟶ S(A) such that τ_{A} = σ ∘ τ_{A} ∘ κ. Moreover the pair (S(A), σ^{−1} ∘ τ_{A}) is a further perfect, central extension of E(A). Thus there is also a group homomorphism κ′ : S(A) ⟶ S(A) such that τ_{A} = σ^{−1} ∘ τ_{A} ∘ κ′. This implies τ_{A} = σ ∘ (σ^{−1} ∘ τ_{A} ∘ κ′) ∘ κ = τ_{A} ∘ κ′ ∘ κ. Denote the group homomorphism κ′ ∘ κ by α. Then the equation just derived clearly shows that for each element s in S(A) there is an element c_{s} in Ker(τ_{A}) ⊆ Centre(S(A)) such that α(s) = c_{s} · s. The group S(A) is generated by commutators [s, t] of elements s and t in S(A). We have α([s, t]) = [ α(s), α(t) ] = [c_{s} · s, c_{t} · t] = [s, t]. Thus every element of S(A) is fixed by α i.e. α = κ′ ∘ κ is the identity on S(A). Exactly the same proof shows that κ ∘ κ′ is also the identity on S(A). Thus κ is a group isomorphism on S(A), i.e κ ∈ Aut(S(A)). If σ is conjugation by g ∈ G(A) then the assignment g ⟼ κ defines a group homomorphism G(A) ⟶ Aut(S(A)) which is the action we are after. Its uniqueness is given by Lemma 2.3.

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The strategy used above is based on properties of universal, central extensions. In the concrete case where C is the category of commutative rings, S the Steinberg group functor St_{n} , G the general linear group functor GL_{n}, and τ : S ⟶ G the usual natural transformation τ_{n} : St_{n} ⟶ GL_{n} , it is known by [16] that St_{n} ⟶ E_{n} is the universal, perfect, central extension whenever n ≥ 5, and so we can apply the strategy above. However for the case n = 4 it is still known thanks to [16] that St_{n} ⟶ E_{n} is a perfect, central extension and so we can apply the next lemma.

2.6 Lemma Suppose S is a perfect, central extension of E . Then the action of E on E by conjugation lifts to an action of E on S. Moreover there is only one lifting. Proof The action of S on itself by conjugation is clearly the lifting of the action of E on itself by conjugation. The last part of Lemma 2.2 shows that the lifting is unique.

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2.2 Linking Diagrammes

We now come to the definition of the main tool used in this article, the linking diagramme. Linking diagrammes are defined and used in [3] for coset spaces which are not neccessarily groups. Linking diagrammes associated to Excision Squares (see below) generalise the concept of Mayer-Vietoris sequences and will be crucial in proving our main results. However in this paper the coset spaces in linking diagrammes will be groups since E is always assumed to be normal in G i.e. E(A) is normal in G(A) for all A ∈ C. Hence one can always work with factor groups L(A) = G(A)/E(A) rather than just with coset spaces on the right-hand side of a linking diagramme. The following definition of linking diagrammes is a specialisation of the general definition in [3], to the case of groups.

2.7 Definition Let

\xymatrix{K′ \ar[dr]^{κ′} &  &  & L′ \\  & K \ar[r]^{∂} & L \ar[ur]^{λ′} \ar[dr]_{λ′′} \\ K′′ \ar[ur]_{κ′′} &  &  & L′′}

be a diagramme in groups such that all composite morphisms are zero morphisms. Such a diagramme is called a linking diagramme if

(1) Im(κ′′)Im(κ′) = Ker ∂.

(2) The canonical homomorphism Coim ∂ ⟶ Ker(λ′′)∩ Ker(λ′) is bijective.

Given a linking diagramme as above, the image of ∂ is a normal subgroup of L. Thus L acts on this image by conjugation. If this action lifts equivariantly to an action of L on K then the linking diagramme is called L-equivariant. More generally let J be any group. Then a J-equivariant linking diagramme is by definition a linking diagramme as above together with an action of J on K and L such that ∂ is J-equivariant.

A morphism of linking diagrammes is a commutative diagramme

\xymatrix{ & K′ \ar[dr] \ar[dddr] &  &  & L′ \ar[dddr] \\  &  & K \ar[r]^{∂} \ar[dddr] & L \ar[ur] \ar[dr] \ar[dddr] \\  & K′′ \ar[ur] \ar[dddr] &  &  & L′′ \ar[dddr] \\ (∗) &  & K′_{1} \ar[dr] &  &  & L′_{1} \\  &  &  & K_{1} \ar[r]^{∂} & L_{1} \ar[ur] \ar[dr] \\  &  & K′′_{1} \ar[ur] &  &  & L′′_{1}}

of groups.

A morphism of equivariant linking diagrammes is a morphism of linking diagrammes in the above sense together with a homomorphism J ⟶ J_{1} of groups such that the homomorphisms K ⟶ K_{1} and L ⟶ L_{1} above are equivariant.

2.3 Dimension Theory

The notion of a category with structure and dimension which is developed in this section is taken from [3].

Let C be a category. Let S(C) denote a category of commutative squares in C , which is closed under isomorphism of commutative squares in C . S(C) is called a category of structure squares, if for each object A of C , the square

\xymatrix{A \ar[r] \ar[d] & A \ar[d] \\ A \ar[r] & A}

with identity morphisms is a member of S(C) . Such squares are called constant or trivial squares. Let I(C) denote a category of functors F : I ⟶ C , which is closed under natural equivalence of functors, where I ranges over a class of directed partially ordered sets viewed as categories. I(C) is called a category of infrastructure functors , if for each object A of C, the functor F_{A} : {∗} ⟶ C, ∗ ↦ A, where {∗} denotes the partially ordered set with precisely one element ∗, is a member of I(C) and if for any (F : I ⟶ C) ∈ I(C) , the direct limit lim_{⟶}F exists in C. The functors F_{A} are called constant functors. A choice of appropriate structure squares and infrastructure functors endows an arbitrary category with the kind of structure one would like to have at his disposal in the context of a given problem. This will be illustrated later in this article, when a class of structure squares and an infrastructure are built up for a category C of commutative rings R in order to show that the dimension filtration on GL_{n}(R) is a descending central series.

2.8 Definition A category with structure is a triple ((C, S(C), I(C)) consisting of a category C , a category S(C) of structure squares in C , and a category I(C) of infrastructure functors in C. A morphism F : ((C, S(C), I(C)) ⟶ ((C′, S(C′), I(C′)) of categories with structure is a functor F : C ⟶ C′ which preserves structure squares and infrastructure functors: i.e. if

\xymatrix{A \ar[r] \ar[dd] & B \ar[dd] \\  &  & ∈ S(C) , \\ C \ar[r] & D}

then

\xymatrix{F(A) \ar[r] \ar[dd] & F(B) \ar[dd] \\  &  & ∈ S(C′), \\ F(C) \ar[r] & F(D)}

if

(F : I ⟶ C) ∈ I(C)

then lim_{→}FF exists in C and (FF : I ⟶ C′) ∈ I(C′) .

Proofs of statements referring to the objects of a category with stucture can be given a uniform format by applying induction over a suitable concept of dimension. It is possible to take any class of objects which is convenient for the purpose at hand to be the class of 0-dimensional objects. The definition below is followed by a theorem proving this.

2.9 Definition A category with dimension is a quadruple ((C, S(C), I(C)), d) where (C, S(C), I(C)) is a category with structure and d : ObjC ⟶ ℤ^{≥0} ∪ {∞} is a function which preserves isomorphism classes of objects and has the following properties : if A is an object of C of finite value d(A) > 0 then there is an infrastructure functor B : I ⟶ C , i ↦ B_{i} and a set of structure squares

\xymatrix{A \ar[r] \ar[dd] & B_{i} \ar[dd] \\  &  & (i ∈ I) \\ C_{i} \ar[r] & D_{i}}

(the rules i ↦ C_{i} and i ↦ D_{i} are not neccessarily functorial) satisfying the following:

(∗) F or all i ∈ I, d(C_{i}) < d(A).
(∗∗) F or any i ≤ j ∈ I, the diagramme
\xymatrix{ & B_{j} \\ A \ar[ur] \ar[dr] \\  & B_{i} \ar[uu]}

commutes,

in which case we say that B : I ⟶ C is compatible with the structure squares above.

(∗ ∗ ∗) d(lim_{⟶} B) = 0.

The function d is called a dimension function for (C, S(C), I(C)) and a family of structure squares

\xymatrix{A \ar[r] \ar[dd] & B_{i} \ar[dd] \\  &  & (i ∈ I) \\ C_{i} \ar[r] & D_{i}}

together with an infrastructure functor B : I ⟶ C , i ↦ B_{i}, satisfying (*), (**) and (***) is called a reduction of A.

A morphism (C, S(C), I(C), d) ⟶ (C′, S(C′), I(C′), d′) of categories with dimension is a morphism F : (C, S(C), I(C)) ⟶ (C′, S(C′), I(C′)) of categories with structure such that for all objects A of C , d′(F(A)) ≤ d(A).

2.10 Remark The kind of dimension function described above is of the sort needed to solve the problems treated in this article. Other problems lead naturally to other kinds of dimension functions. There are in fact 2^{6} − 1 useful kinds, each kind corresponding to a specific, nonempty subset of 6 possible conditions. It is helpful to recall the general definition, since it provides perspective for the special case above and many commonly encountered dimension functions will satisfy all 6 conditions. The general definition is taken from [3]. Let (C, S(C), I(C)) be a category with structure. Let d : Obj(C) ⟶ ℤ^{≥0} ∪ {∞} be a function. Then d is called a dimension function on (C, S(C), I(C)) if for each object A of C such that 0 < d(A) < ∞ there is a set

\xymatrix{A \ar[r] \ar[dd] & B_{i} \ar[dd] \\  &  & (i ∈ I) \\ C_{i} \ar[r] & D_{i}}

of structure squares in S(C) and an infrastructure functor B : I ⟶ C , which is compatible in the sense of (2.4)(**) with the structure squares above, such that a fixed subset of the following 6 conditions holds.

(i) d(C_{i}) < d(A) for all i ∈ I. (ii) d(D_{i}) < d(A) for all i ∈ I.

(iii) d(lim_{→} B) < d(A).

(iv) d(C_{i}) = 0 for all i ∈ I.

(v) d(lim_{→} B)) = 0.

(vi) d(B_{i}) ≤ d(A) for all i ∈ I.

The example below of a dimension function satisfies all 6 conditions above. Moreover it is the most efficient dimension function in the sense explained in the example. This sense will form the basis for the definition later of a universal dimension function. It is a theorem in [3] that for any kind of dimension function on a category with structure, there is a universal dimension function of that kind. We shall prove the result for the kind of dimension functions we are interested in.

2.11 Example Injective dimension of modules Let C denote the category of all (left) modules over an associative ring R with identity. The injective dimension injd(M) of an R-module M is by definition the smallest n ∈ ℤ^{≥0} ∪ {∞} such that there is an exact sequence 0 ⟶ M ⟶ Q_{0} ⟶ Q_{1} ⋯ ⟶ Q_{n} ⟶ 0 of R-modules with each Q_{i} injective. Let S(C) denote the smallest category of structure squares containing all commutative squares

\xymatrix{A \ar[r] \ar[d] & B \ar[d] \\ 0 \ar[r] & D}

where 0 is the trivial module and B is injective. Let I(C) denote the smallest possible category of infrasructure functors, namely the constant ones F_{A} : {∗} ⟶ C, ∗ ↦ A. Then ((C, S(C), I(C)), injd) is a category with dimension and if d is any dimension function on (C, S(C), I(C)) , whose 0-dimensional objects are contained in the class C^{0} of all 0-dimensional objects of injd, namely the class of all injective modules, then injd(A) ≤ d(A) for all R-modules A. Thus injd is the most efficient among all dimension functions on (C, S(C), I(C)) , whose 0-dimensional modules are injective (this follows from the theorem below).

2.12 THEOREM : Universal Dimension Function Let ((C, S(C), I(C)) be a category with structure. If C^{0} is a class of objects of C , closed under isomorphism, then there is dimension function δ on ((C, S(C), I(C)) (of the kind defined in Definition 2.9 whose 0-dimensional objects are precisely those in C^{0} such that for any dimension function d for ((C, S(C), I(C)) , whose 0-dimensional objects are contained in C^{0} , the identity functor C ⟶ C defines a morphism ((C, S(C), I(C)), d) ⟶ ((C, S(C), I(C)), δ) of categories with dimension.

Proof : Define disjoint classes C^{0}, C^{1}, ⋯ of objects of C as follows. Let C^{0} be as in the theorem and suppose C^{0}, ⋯, C^{n−1} have been defined.

Define an object A of C to be n-reductive if it satisfies the following conditions : there is an infrastructure functor (B : I ⟶ C) ∈ I(C) and a set

\xymatrix{A \ar[r] \ar[dd] & B_{i} \ar[dd] \\  &  & (i ∈ I) \\ C_{i} \ar[r] & D_{i}}

of structure squares in S(C) satisfying the following :

(*) For all i ∈ I, C_{i} ∈ ⋃_{k=0}^{n−1} C^{k}

(**) B is compatible with the structure squares and (lim_{⟶} B) ∈ C^{0}

Define an object of C to be n-good if it either is in ⋃_{k=0}^{n−1} C^{k} or is n-reductive. Let

C^{n−good} denote the class of all n-good objects of C and C^{n} = C^{n−good} \ ⋃_{k=0}^{n−1} C^{k}.

Using this construction, one may now define a dimension function by setting

δ(A) = \{ \begin{matrix}n, if & A ∈ C^{n} \\ ∞, if & A ∉ ⋃_{n≥0} C^{n}. \end{matrix}

Let C_{d}^{n} denote the class of all objects of C of d-dimension n. To prove that δ is uni-

versal, it suffices to show that for each n ∈ ℤ^{≥0}, ⋃_{k=0}^{n} C_{d}^{k} ⊆ ⋃_{k=0}^{n} C^{k}. For n = 0 the in-

clusion holds by definition. By induction on n, we can assume that ⋃_{k=0}^{n−1} C_{d}^{k} ⊆ ⋃_{k=0}^{n−1} C^{k}.

But this shows that ⋃_{k=0}^{n} C_{d}^{k} is an n-good class. Thus ⋃_{k=0}^{n} C_{d}^{k} ⊆ C^{n−good} = ⋃_{k=0}^{n} C^{k}.

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2.13 Definition The function δ in the theorem is called the universal dimension function for ((C, S(C), I(C)) and C^{0} .

2.4 Noetherian Rings as a Category with Dimension

The main application of dimension theory shown in this article will be to categories of commutative rings. In this section we shall provide such categories with a class of structure squares and infrastructure functors, which make them into categories with structure. Then we shall show that the notion of Bass-Serre dimension for rings is a dimension function on the category with structure of commutative Noetherian domains. We begin by reviewing quickly some standard ring-theoretic notions including localization, completion, spectrum and maximal spectrum. Our structure squares will be localization-completion squares, our infrastructure functors will be certain localizations and our dimension function will be the topological dimension of the maximal spectrum.

Let R denote a commutative ring. If S is a multiplicative set in R, let S^{−1}R denote the ring of S-fractions of R, called also the localization of R at S.

It is well-known, cf. [5] Ch III, §4, Prop. 4.1 that the functor S^{−1}R ⨂_{R} − is exact on the category of all R-modules. If 𝔮 is an ideal of R, let R̂_{𝔮} denote the completion lim_{⟵_{k≥0}} R/𝔮^{k} of R at 𝔮. It is well-known, cf. [8] Lemma 7.15, that the

functor R̂_{𝔮} ⨂_{R} − is exact on the category of all Noetherian R-modules.

¿From the exactness of S^{−1}R ⨂_{R} − it follows that if 𝔭 is a prime ideal of R such that 𝔭 ∩ S = ∅ then S^{−1}R𝔭 is a prime ideal of S^{−1}R. ( If 𝔭 ∩ S ≠ ∅ then S^{−1}R𝔭 = S^{−1}R ). In fact the correspondence 𝔭 ⟼ S^{−1}R𝔭 is a bijection of the prime ideals 𝔭 of R such that 𝔭 ∩ S = ∅ onto the set of prime ideals of S^{−1}R. If R is Noetherian then so is S^{−1}R, cf. [8] Cor. 2.3.

Suppose now that R is Noetherian. Then from the exactness of R̂_{𝔮} ⨂_{R} − on the category of finitely generated R-modules, it follows that if 𝔭 is a prime ideal of R such that 𝔭 + 𝔮 ≠ R then R̂_{𝔮}𝔭 = R̂_{𝔮} ⨂_{R} 𝔭 is a prime ideal of R̂_{𝔮}. Obviously if s is not a divisor of zero in R, then its image is also not a divisor of zero in R̂_{𝔮}. It is not difficult (relying on the Hilbert Basis Theorem) to show that R̂_{𝔮} is Noetherian cf. [8] Th. 7.1.

2.14 Definition Let s ≠ 0 be any element in the commutative ring R. Let < s > denote the multiplicative set {s^{k} | k ≥ 0} generated by s and (s) denote the principal ideal Rs generated by s. Then the commutative square

\xymatrix{R \ar[d] \ar[r] & < s >^{−1} R \ar[d] \\ R̂_{(s)} \ar[r] & < s >^{−1} R̂_{(s)}}

is called the localization-completion square of R at s.

2.15 Lemma If R is a Noetherian ring then each ring in a localization-completion square of R is Noetherian and such a square is a fibre product square. Moreover if s is not a divisor of zero then the horizontal maps are injective.

Proof : The first assertion recalls standard results we have already mentioned. The third statement is obviously true. We now prove the fibre product property. Let α ∈ R̂_{(s)} and \frac{a}{s^{k}} ∈< s >^{−1} R be such that α and \frac{a}{s^{k}} coincide in < s >^{−1} R̂_{(s)}. We must show that there is a unique element a ∈ R such that it is a common preimage of α and \frac{a}{s^{k}}. The equality α = \frac{a}{s^{k}} in < s >^{−1} R̂_{(s)} means that there is an m ≥ 0 such that s^{k+m}α = s^{m}a in R̂_{(s)}, hence s^{m}a ∈ s^{k+m}R̂_{(s)}. The canonical map R/s^{k+m}R ⟶ R̂_{(s)}/s^{k+m}R̂_{(s)} is an isomorphism, hence s^{m}a ∈ s^{k+m}R also. Thus there is an element b ∈ R such that s^{m}a = s^{k+m}b. But then \frac{a}{s^{k}} = \frac{s^{m}a}{s^{k+m}} = \frac{b}{1}. Let 𝔭 ⊆ R be the ideal annihilated by the element s^{k+m}, then elements of 𝔭̂_{(s)} are sequences of elements in R which are constant beyond the index k + m, so that 𝔭̂_{(s)} ≅ 𝔭. The sequence

0 ⟶ 𝔭 ⟶ R ⟶^{×s^{k+m}} R is exact. By exactness of the functor R̂_{s} ⨂_{R} − the

sequence 0 ⟶ 𝔭 ⟶ R̂_{(s)} ⟶^{×s^{k+m}} R̂_{(s)} is also exact. Since s^{k+m}α = s^{m}a = s^{k+m}b

in R̂_{(s)}, s^{k+m}(α − b) = 0 so that α − b ∈ 𝔭, i.e. there is an element p ∈ 𝔭 such that α = b + p. Then b + p is a common preimage for α and \frac{a}{s^{k}}. To prove that it is unique we recall Krull’s Theorem cf. [1] Th.10.17, which implies that the map R ⟶ R̂_{(s)} × < s >^{−1} R is injective. For clearly if a ∈ R vanishes in < s >^{−1} R then there is an n ≥ 0 such that s^{n}a = 0. If a vanishes in R̂_{(s)} then by Krull’s Theorem there is an element c ∈ R such that a · (1 − sc) = 0. Together these show that even s^{n−1}a = 0 holds. By induction a = 0.

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2.16 Definition Let S be a multiplicative set in a commutative ring R. Define a partial ordering on S by s ≤ t ⟺ ∃u ∈ A : su = t. Clearly the partial ordering is directed. If s ≤ t then the rule \frac{a}{s} ↦ \frac{au}{su} = \frac{au}{t} defines a ring homomorphism < s >^{−1} R ⟶< t >^{−1} R such that the function S ⟶ commutative rings , s ↦< s >^{−1} R is functorial. Such functors will be called infrastructure functors.

2.17 Lemma Let R be a commutative ring and S a multiplicative set in R. Then the set of localization-completion squares

\xymatrix{R \ar[dd] \ar[r] & < s >^{−1} R \ar[dd] \\  &  & (s ∈ S) \\ R̂_{(s)} \ar[r] & < s >^{−1} R̂_{(s)}}

is compatible in the sense of Def (2.4) with the infrastructure functor S ⟶

commutative rings , s ↦< s >^{−1} R. Moreover lim_{⟶_{S}} < s >^{−1} R = S^{−1}R.

Proof : The first assertion is trivial. There is a canonical homomorphism of each < s >^{−1} R ⟶ S^{−1}R such that the diagramme

\xymatrix{< s >^{−1} R \ar[dr] \ar[dd] \\  & S^{−1}R \\ < t >^{−1} R \ar[ur]}

commutes whenever s ≤ t. By the universality of direct limits there is a unique homomorphism lim_{⟶_{S}} < s >^{−1} R ⟶ S^{−1}R making the diagramme

\xymatrix{< t >^{−1} R \ar[dr] \ar[dd] \\  & S^{−1}R \\ lim_{⟶_{S}} < s >^{−1} R \ar@{-->}[ur]}

commute for any t ∈ S. Moreover for any t ∈ S, the image of t in lim_{⟶_{S}} < s >^{−1} R

is invertible. Thus by the universality of localization there is a unique homomorphism S^{−1}R ⟶ lim_{⟶_{S}} < s >^{−1} R such that the diagramme

\xymatrix{< t >^{−1} R \ar[dr] \ar[dd] \\  & lim_{⟶_{S}} < s >^{−1} R \\ S^{−1}R \ar[ur]}

commutes. The homomorphisms lim_{⟶_{S}} < s >^{−1} R ⟶ S^{−1}R and S^{−1}R ⟶ lim_{⟶_{S}} <

s >^{−1} R are obviously mutually inverse.

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On any category with structure, given a class of objects which is closed under isomorphisms there is a universal dimension function, say δ, for that class. In practice, to show that δ is a good dimension function, one shows that some well-known notion d of dimension on the category is a dimension function in the technical sense for a category with structure and that the 0-dimensional objects for d are also 0-dimensional for δ. This shows that δ ≤ d and therefore that δ is a good dimension function. For categories of Noetherian rings we shall show that Bass-Serre dimension is a dimension function in the technical sense required.

We recall next Krull dimension Kdim and then Bass-Serre dimension BSdim. Since BSdim ≤ Kdim and BSdim is a dimension function in the technical sense, it will follow that the universal dimension function δ ≤ BSdim ≤ Kdim, although Kdim itself will not be a dimension function in the technical sense we require.

Let R denote again an arbitrary commutative ring. Let Spec(R) denote the space of all prime ideals of R with the Zariski topology. This means that the subsets V (𝔞) = {𝔭 | 𝔭 ∈ Spec(R), 𝔭 ⊃ 𝔞} of Spec(R) where 𝔞 is an arbitrary ideal in R are precisely the closed subsets of Spec(R). If f : R ⟶ R′ is a ring homomorphism then the function f^{∗} : Spec(R′) ⟶ Spec(R), 𝔭′ ↦ f^{−1}(𝔭′) is a continuous map of topological spaces and Spec defines a contravariant functor Spec : commutative rings ⟶ topological spaces. Let Max(R) denote the subspace of Spec(R) containing all the maximal ideals of R. It is not true in general that if f : R ⟶ R′ is a ring homomorphism and 𝔪′ ∈ Max(R′) then f^{−1}(𝔪′) ∈ Max(R). Thus the canonical map Spec(R′) ⟶ Spec(R) does not take in general Max(R′) to Max(R), i.e. Max is not functorial as is Spec. Still, as we shall see Max will be better suited to our purposes.

We recall next a notion of dimension for topological spaces which will be applied to Spec(R) and Max(R) above, in the case of Noetherian rings R.A topological space is called irreducible if it is non-empty and not the union of two nonempty, proper closed subsets cf. [5] Ch III, §3, p 97. Thus irreducibility implies connectivity. Since the inverse image of a closed set under a continuous map is also closed, it is clear that continuous maps preserve irreducibility. A subspace is irreducible if and only if its closure is irreducible. Every irreducible subspace is contained in a maximal one and the latter is closed. The space is the union of its maximal irreducible subspaces which are called its irreducible components cf. [5] Ch III, §3, Prop 3.7. A topological space is called Noetherian if its closed subsets satisfy the descending chain condition ( d.c.c. for short). A Noetherian space has only finitely many irreducible components cf. [5] Ch III, §3, Prop 3.8. A subspace of a Noetherian space is also Noetherian. The irreducible components of the closure of a subspace are the closures of its components.

The dimension, dim(X) of a topological space X is the length m of the longest properly descending chain X_{0} ⊃ X_{1} ⊃ ⋯ X_{m} of nonempty, closed, irreducible subsets X_{i} of X. If there is no nonnegative integer m as above then the dimension of X is infinity. If Y ⊆ X is a subspace then dim(Y) ≤ dim(X), cf. [5] : p 98. For any commutative ring R the correspondence Spec(R) ⟶ closed, irreducible subsets of Spec(R), 𝔭 ↦ V (𝔭), is bijective cf. [5] Ch III, §3, Prop. 3.10. It follows that properly ascending chains 𝔭_{0} ⊂ 𝔭_{1} ⋯ ⊂ 𝔭_{l} of prime ideals of R of length l correspond 1 − 1 to properly descending chains V (𝔭_{l}) ⊃ ⋯ ⊃ V (𝔭_{1}) ⊃ V (𝔭_{0}) of closed, ireducible subsets of Spec(R) of length l. By definition the Krull dimension Kdim(R) of R is the length l of the longest properly ascending chain of prime ideals of R. If there is no such nonnegative integer then Kdim(R) is infinity. Clearly Kdim(R) = dim(Spec(R)). Define Bass-Serre dimension by setting BSdim(R) = dim(Max(R)).

Clearly if R is a commutative, Noetherian ring then from the fact that V is inclusion reversing it follows that Spec(R) is Noetherian. Thus for Noetherian rings R, Spec(R) and Max(R) have only a finite number of irreducible components.

2.18 Lemma Let R be a commutative Noetherian ring such that 0 < BSdim(R) < ∞. Let 𝔪_{1}, ⋯, 𝔪_{r} ∈ Max(R) be a finite set of maximal ideals of R such that each irreducible component of Max(R) contains some 𝔪_{i}. Let S = R\𝔪_{1}∪⋯∪𝔪_{r}. If s ∈ S then BSdim(R/sR) < BSdim(R).

Proof : Since the homomorphism R ⟶ R/sR is surjective, the map Spec(R/sR) ⟶ Spec(R) maps Max(R/sR) homeomorphically onto Max(R) ∩ V (sR). But the closed subspace Max(R) ∩ V (sR) of Max(R) is missing a point from each irreducible component of Max(R). Thus the irreducible components of Max(R) ∩ V (sR) are strictly smaller than those of Max(R). Thus BSdim(R/sR) = dim (Max(R/sR)) = dim(Max(R) ∩ V (sR)) < dim(Max(R)) = BSdim(R).

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2.19 Corollary Let R be a commutative Noetherian ring such that 0 < BSdim(R) < ∞. Let 𝔪_{1}, ⋯, 𝔪_{r} ∈ Max(R) be as in the previous lemma and let S = R\𝔪_{1} ∪ ⋯ ∪ 𝔪_{r}. If s ∈ S then BSdim(R̂_{(s)}) < BSdim(R).

Proof : By [8] Prop 7.10 and Ex 7.3, sR̂_{(s)} ⊆ Jac(R̂_{(s)}). Thus the canonical map Spec(R̂_{(s)}/sR̂_{(s)}) ⟶ Spec(R̂_{(s)}) maps Max(R̂_{(s)}/sR̂_{(s)}) homeomorphically onto Max(R̂_{(s)}). Thus BSdim(R̂_{(s)}/sR̂_{(s)}) = BSdim(R̂_{(s)}). But R̂_{(s)}/sR̂_{(s)} ≅ R/sR and by Lemma 2.18, BSdim(R/sR) < BSdim(R).

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2.20 THEOREM Let C be any category of commutative Noetherian rings which is closed under localization R ↦ S^{−1}R and completion R ↦ R̂_{(s)}. Let S(C) denote the category of all localization-completion squares, cf. Definition 2.16, in C where s is any non zero-divisor, and let I(C) denote the category of all infrastructure functors defined in Definition 2.7, where S is any set of non zero-divisors . Then (C, S(C), I(C)) is a category with structure and BSdim is a dimension function on (C, S(C), I(C)).

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Proof : It is obvious that (C, S(C), I(C)) is a category with structure. It remains to show that BSdim is a dimension function on (C, S(C), I(C)).

Let R be a ring in C such that 0 < BSdim(R) < ∞. Let S = R\𝔪_{1} ∪ ⋯ ∪ 𝔪_{r} be as in Lemma 2.17. By [11] Th. 2 and Th. 80, there is a finite number of maximal ideals of R which contain all zero-dvisors of R. It can be assumed that the set {𝔪_{1}, ⋯, 𝔪_{r}} contains all of these maximal ideals. Thus every element of S is a non zero-divisor. Consider the family of structure squares

\xymatrix{R \ar[dd] \ar[r] & < s >^{−1} R \ar[dd] \\  &  & (s ∈ S) \\ R̂_{(s)} \ar[r] & < s >^{−1} R̂_{(s)}}

and the infrastructure S ⟶ C, s ↦< s >^{−1} R. By Lemma 2.15, the infrastructure is compatible with the set of structure squares above. By Corollary 2.19, BSdim(R̂_{(s)}) < BSdim(R). Since Max(S^{−1}R) has a finite number of points, namely S^{−1}R𝔪_{1}, ⋯, S^{−1}R𝔪_{r} and each point is closed, it follows that each point is an irreducible component of Max(S^{−1}R). Thus BSdim(S^{−1}R) = 0.

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3 MAIN RESULT

The main theorem is a result on functors from a category with structure to equivariant linking diagrammes. We fix notation as follows.

Let (C, I(C), S(C)) be a category with structure. Let J, K and L be functors C :⟶ groups. We do not assume that K and L are derived from a natural transformation τ : S ⟶ G as in Section 2.

Instead we assume that for each structure square

\xymatrix{A \ar[r] \ar[dd] & B \ar[dd] \\  &  & in S(C) \\ C \ar[r] & D}

there is a group homomorphism

∂ : K(D) ⟶ L(A)

such that the rule which assigns to each structure square above a diagramme

\xymatrix{K(B) \ar[dr] &  &  & L(B) \\  & K(D) \ar[r]^{∂} & L(A) \ar[ur] \ar[dr] \\ K(C) \ar[ur] &  &  & L(C)}

is a functor from S(C) to equivariant linking diagrammes. We assume further that L preserves direct limits of infrastructure functors, i.e. if (S : I ⟶ C) ∈ I(C) then the canonical homomorphism lim_{⟶_{S}} L ⟶ L(lim_{⟶_{S}}) is an isomorphism.

We shall then call such a triple (K, L, ∂) a linking diagramme functor on (C, S(C), I(C)) .

Let J : C ⟶ groups be a group valued functor on C. Let (K, L, ∂) be a linking diagramme functor on (C, S(C), I(C)). We say that J acts on (K, L, ∂) if J acts functorially on L and for each object A of C there is a set

\xymatrix{A \ar[dd] \ar[r] & B_{i} \ar[dd] \\  &  & (i ∈ I) \\ C_{i} \ar[r] & D_{i}}

of structure squares and a compatible infrastructure functor

B : I ⟶ C , i ↦ B_{i}

with the following property. Let σ ∈ J(A). Then there is an i_{0} ∈ I such that for all i ≥ i_{0} the action of σ on L(A) lifts to an action on K(D_{i}) such that ∂ : K(D_{i}) ⟶ L(A) is σ-equivariant.

3.1 Remark : In the setting above let σ_{1}, σ_{2}, ⋯, σ_{n} ∈ J(A) be a finite set of elements. Then there is an i_{0} ∈ I such that for all i ≥ i_{0} each σ_{j} above lifts to an action on K(D_{i}) such that the connecting homomorphism ∂ : K(D_{i}) ⟶ L(A) is σ_{j}-equivariant. Let J′ =< σ_{1}, σ_{2}, ⋯, σ_{n} > denote the subgroup of J(A) generated by σ_{1}, σ_{2}, ⋯, σ_{n}. Let J̃′ denote the free group on the elements σ_{1}, ⋯, σ_{n}. Whereas J′ does not neccessarily act on K(D_{i}), J̃′ does and the homomorphism ∂ : K(D_{i}) ⟶ L(A) is J̃′-equivariant. In fact the action of J′ on L(A) lifts equivariantly to an action of J′ on K(D_{i}) if Ker(J̃′ ⟶ Aut(K(D_{i}))) ⊇ Ker(J̃′ ⟶ J′).

If d is any dimension function on (C, S(C), I(C)) define the dimension filtration of L with respect to d as follows:

L^{−1}(A) = L(A).
For k ≥ 0,

L^{k}(A) = ⋂_{f:A⟶B_{d(B)≤k}} Ker(L(f) : L(A) ⟶ L(B)).

The theorem below is a special case of results in [3].

3.2 MAIN THEOREM

Let (K, L, ∂) be an equivariant linking diagramme functor on (C, S(C), I(C)) and let J : C ⟶ groups be a functor which acts on (K, L, ∂) . Let C^{0} denote a class of objects B′ of C which is closed under isomorphisms and has the following property: Let

\xymatrix{A \ar[r] \ar[dd] & B_{i} \ar[dd] \\  &  & (i ∈ I) \\ C_{i} \ar[r] & D_{i}}

be any set of structure squares and

B : I ⟶ C , i ↦ B_{i},

be any infrastructure functor which is compatible with the set of squares above and lim_{⟶_{I}} B = B′. The property is: given σ ∈ J(A) there is an i_{0} ∈ I such that for all

i ≥ i_{0} the action of σ on K(D_{i}) is trivial. Clearly the class of all B′ satisfying the above property is closed under isomorphism. Let d be a dimension function on (C, S(C), I(C)) whose 0-dimensional objects are precisely those of C^{0} , e.g. the universal dimension function for the data above. Let

L = L^{−1} ⊇ L^{0} ⊇ ⋯ ⊇ L^{k} ⊇ ⋯

be the dimension filtration on L defined by d. Then this filtration is a descending J -central series, i.e. the action of J on each L^{k}/L^{k+1} is trivial.

Proof:

Let k ≥ 0.

Since there is a set M of morphisms f : A ⟶ B such that B is (k + 1)- dimensional and the induced morphism L^{k}(A)/L^{k+1}(A) ↣ ∏_{(f:A⟶B)∈M} L^{k}(B) is

injective, it suffices to show that the action of J(B) on L^{k}(B) is trivial. Thus we can assume above that d(A) = k + 1 and therefore L^{k+1}(A) = 1. Let

\xymatrix{A \ar[r] \ar[dd] & B_{i} \ar[dd] \\  &  & (i ∈ I) \\ C_{i} \ar[r] & D_{i}}

be a reduction of A in the sense of Def 2.9. Let σ ∈ G_{F}^{0} (A) and x ∈ L^{k}(A). Since d(C_{i}) < k, L^{k}(C_{i}) is trivial (for any i). Choose i_{0} such that the action of σ on K(D_{i_{0}}) is trivial and the image of x in L(B_{i_{0}}) is trivial. Consider the linking diagramme

\xymatrix{K(B_{i_{0}}) \ar[dr] &  &  & L(B_{i_{0}}) \\  & K(D_{i_{0}}) \ar[r]^{∂} & L(A_{i_{0}}) \ar[ur] \ar[dr] \\ K(C_{i_{0}}) \ar[ur] &  &  & L(C_{i_{0}})}

Since x vanishes in L(B_{i_{0}}) and in L(C_{i_{0}}), there is an element y ∈ K(D_{i_{0}}) such that ∂(y) = x. By definition the action of σ on y is trivial and ∂ is equivariant. Thus σxσ^{−1} = x.

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The theorem above is easy to prove but there is honest work left to do. This article presents a proof by new methods of the fact that for any commutative Noetherian ring R the filtration

GL_{n}(R) = GL_{n}^{−1}(R) ⊇ GL_{n}^{0}(R) ⊇ ⋯ GL_{n}^{i}(R) ⊇ ⋯ E_{n}(R)

is a descending, central series. The effect of the Main Theorem is to focus work on showing that (K_{2,n}, GL_{n}/E_{n}, ∂) where n ≥ 5, really is a functor localization− completion squares ⟶ linking diagrammes, and that the group valued functor GL_{n} acts upon it. The next section prepares the ground for this by analyzing in general terms interrelated structure squares S(C) and linking diagramme functors (K, L, ∂). A group valued functor called J_{τ} ⊆ G, the largest possible with respect to acting on a given linking diagramme functor, is identified, where a priori J_{τ} ≠ G is possible. Additional flexibility is thus acheived, which though not in fact needed for the retricted aim above does serve the purpose of showing the logical structure of the methods used. Moreover the flexibility may be required in other contexts.

4 Excision Let τ : S ⟶ G denote a natural transformation of group valued functors on an arbitrary category C such that E = image τ is a normal subgroup functor of G . Define as in Section 2 K = Ker τ and L = Coker τ . In this situation we shall construct a class E(C) of commutative squares

\xymatrix{A \ar[d] \ar[r] & B \ar[d] \\ C \ar[r] & D}

in C called excision squares and for each excision square a group homomorphism ∂ : K(D) ⟶ L(A) such that the triple (K, L, ∂) is a linking diagramme functor on the category with structure (C, E(C), I(C)) where I(C) are all possible infrastructure functors F : I ⟶ C such that S and G commute with direct limits of F , i.e. the canonical homomorphisms lim_{⟶_{I}}SF = S(lim_{⟶_{I}}F) and

lim_{⟶_{I}}GF = G(lim_{⟶_{I}}F)are isomorphisms. Then we shall construct a subgroup

functor J_{τ} ↣ G of G such that the quadruple (J, K, L, ∂) is an equivariant linking diagramme functor on (C, E(C), I(C)) . Finally we apply the conclusions of the Main Theorem to construct a descending J -central series on L for any subgroup functor J ↣ J_{τ} .

4.1 Relative Functors and the 5-term Exact Sequence

4.1 Definition Let A ⟶ B denote a morphism in C . Define

τ (A, B) = τ(A ⟶ B)

by a pullback diagramme

\xymatrix{τ(A, B) \ar[r] \ar[d] & S(B) \ar[d] \\ G(A) \ar[r] & G(B)}

The pullback τ(A, B) will frequently be identified with its explicit construction as the fibre product G(A)×_{G(B)}S(B) = {(σ, s) ∈ G(A)×S(B) | images of σ and s in G(B) coincide}.

¿From the universality of the pullback construction or from its explicit construction as a fibre square, one obtains for each commutative square

\xymatrix{A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D}

in C a canonically defined group homomorphism τ(A, B) ⟶ τ(C, D).

Let C^{2} denote the morphism category of C . By definition the objects of C^{2} are all morphisms A ⟶ B in C and the morphisms of C^{2} are all commutative squares as above. Obviously the assignment (A ⟶ B) ⟼ τ(A, B) defines a functor which we also denote by τ : C^{2} ⟶ groups. The universal property of the pullback τ(A, B) gives a canonical group homomorphism S(A) ⟶ τ(A, B), whose image we denote by ′S(A)′. Setting R(A, B) = τ(A, B)/′S(A)′ we obtain another functor R : C^{2} ⟶ cosets called the relative functor defined by τ. The universal property of τ(A, B) gives a group homomorphism K(B) ⟶ τ(A, B), x ⟼ ( 1, x ) which induces a map K(B) ⟶ R(A, B), x ⟼ ( 1, x ) · ′S(A)′ of coset spaces. The group homomorphism τ(A, B) ⟶ G(A), ( g, s ) ⟼ g appearing in the definition of τ(A, B) induces also a homomorphism R(A, B) ⟶ L(A), ( g, s ) · ′S(A)′ ⟼ g · E(A) of coset spaces. Thus there is associated to each object A ⟶ B of C^{2} a 5-term sequence

(4.1.1) K(A) ⟶ K(B) ⟶ R(A, B) ⟶ L(A) ⟶ L(B)

of coset spaces and it is routine to check that this association defines a functor

C^{2} ⟶ 5 − term sequences in cosets,

i.e., if

\xymatrix{A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D}

is a morphism in C^{2} then the diagramme

\xymatrix{ & K(A) \ar[r] \ar[dd] & K(B) \ar[r] \ar[dd] & R(A, B) \ar[r] \ar[dd] & L(A) \ar[r] \ar[dd] & L(B) \ar[dd] \\ (4.1.2) \\  & K(C) \ar[r] & K(D) \ar[r] & R(C, D) \ar[r] & L(C,) \ar[r] & L(D)}

commutes. Moreover if R(A, B) is a group then the sequence (4.1.1) is obviously a sequence of groups and group homomorphisms and if R(A, B) is a group for any object A ⟶ B in C^{2} then (4.1.2) is a commutative diagramme of groups and group homomorphisms, which implies that we have a functor

C^{2} ⟶ 5 − term sequences in groups

.

4.2 THEOREM The sequence (4.1.1) is exact in the category of pointed sets. In particular, if R(A, B) is a group then (4.1.1) is an exact sequence of groups.

Proof : Exactness at K(A) : Let σ ∈ G(A) whose image in L(B) is trivial. Let ϵ denote the image of σ in E(B). Let t be a lifting to S(B). Then the class of (σ, t) in R(A, B) is a preimage for the class of σ in L(A).

Exactness at R(A, B) : Let (σ, t) ∈ τ(A, B) whose image in L(A) is trivial. Then σ ∈ E(A). Let s be preimage of σ in S(A) and let s_{1} denote its image in S(B). Then (σ, t) and (1, s_{1}^{−1}t) determine the same element of R(A, B). But (1, s_{1}^{−1}t) ∈ image(K(B) ⟶ τ(A, B)). Thus the element s_{1}^{−1}t is a preimage in K(B) for the class of (σ, t) in R(A, B).

Exactness at K(B) : Let κ ∈ K(B) whose image in R(A, B) is trivial. Thus there is an element s ∈ S(A) whose image (ϵ, t) ∈ τ(A, B) is equal to (i, κ). Thus ϵ = 1. Thus s ∈ S(A) is a preimage for κ.

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The next lemma provides a general condition under which R(A, B) is a group for any morphism A ⟶ B in C.

4.3 Lemma It τ : S ⟶ G is central and S is perfect then for any morphism A ⟶ B, R(A, B) is a group.

Proof : Let τ̄(A, B) be defined by the pullback diagramme

\xymatrix{τ̄(A, B) \ar[r] \ar[d] & E(B) \ar[d] \\ G(A) \ar[r] & G(B).}

Let Ḡ(A, B) = Ker(G(A) ⟶ G(B)/E(B)). Obviously the canonical homomorphism Ḡ(A, B) ⟶ τ̄(A, B), σ ↦ ( σ, ϵ(σ) ), where ϵ(σ) denotes the image of σ in G(B), is an isomorphism. Using this isomorphism we identify Ḡ(A, B) with τ̄(A, B). Consider the short exact sequence K(B) ↣ τ(A, B) ↠ τ̄(A, B). Let x ∈ τ(A, B). We must show that x normalizes ′S(A)′. Let s ∈′ S(A)′ ⊆ τ(A, B). Since E(A) is normal in τ̄(A, B), xsx^{−1} = s′κ_{s} for some s′ ∈′ S(A)′ and κ_{s} ∈ K(B). If t ∈′ S(A)′ then x[s, t]x^{−1} = [xsx^{−1}, xtx^{−1}] = [s′κ_{s}, t′κ_{t}] = [s′, t′] since K(B) is also contained in the centre(τ(A, B)). But [s′, t′] ∈′ S(A)′ and ′S(A)′ is perfect. Thus conjugation by x normalizes ′S(A)′.

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4.2 Excision Squares and Linking Diagrammes

The results of this section are special cases of more general ones in [3].

4.4 Definition A commutative square

\xymatrix{A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D}

in C is called an excision square for τ if the following conditions are satisfied. (4.4.1) The canonical map

R(A, B) ⟶ R(C, D)

is bijective. (This does not imply that there is a group homomorphism τ(C, D) ⟶ τ(A, B) which induces an inverse R(C, D) ⟶ R(A, B) to the bijection above.)

(4.4.2) The composite function

\xymatrix{ & R(A, B) \ar[r] & L(A) \\ ∂ : K(D) \ar[r] & R(C, D) \ar[u]^{≅}}

is a group homomorphism. (This is obviously guaranteed by (4.2.1) whenever the bijection in (4.2.1) is a group homomorphism.)

4.5 THEOREM If

\xymatrix{A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D}

is an excision square for τ then

\xymatrix{K(B) \ar[dr] &  &  & L(B) \\  & K(D) \ar[r]^{∂} & L(A) \ar@{-->}[ur] \ar[dr] \\ K(C) \ar[ur] &  &  & L(C)}

is a linking diagramme.

Proof : It is very easy to check that the composition of any two consecutive hommorphisms in the diagramme is trivial. We skip this part of the verification.

Exactness at L(A) : Let x ∈ L(A) whose image in L(B) and L(C) is trivial. From the exactness Th 4.2, of the sequence R(A, B) ⟶ L(A) ⟶ L(B), x has a preimage x̃ in R(A, B). Let ỹ denote the preimage of x̃ in R(C, D). From the commutative square

\xymatrix{R(A, B) \ar[r] \ar[d] & L(A) \ar[d] \\ R(C, D) \ar[r] & L(C)}

we see that ỹ vanishes in L(C). From the exactness Th 4.2, of the sequence K(D) ⟶ R(C, D) ⟶ L(C), it follows that ỹ has a preimage z in K(D). Clearly ∂(z) = x.

Exactness at K(D) : Let x ∈ K(D) such that ∂(x) = 1. Let x′ denote the image of x in R(A, B). From the exactness Th 4.2, of the sequence K(B) ⟶ R(A, B) ⟶ L(A), it follows that x′ has a preimage y′ in K(B). Let y denote the image of y′ in K(D). From the commutativity of the diagramme

\xymatrix{K(B) \ar[r] \ar[d] & R(A, B) \ar[d] \\ K(D) \ar[r] & R(C, D)}

it follows that x and y have the same image in R(C, D). Let (1, x) and (1, y) denote respectively the images of x and y in τ(C, D). Since the classes of (1, x) and (1, y) in R(C, D) are equal, there is an element s ∈ S(C) such that if (ϵ, t) denotes its image in τ(C, D) then (1, x) = (ϵ, t)(1, y). Thus ϵ = 1 and x = ty. The latter equation proves exactness at K(D).

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4.6 THEOREM Let E_{τ}(C) denote the category of all excision squares for τ. Let I_{τ}(C) denote the category of all infrastructure functors F : I ⟶ C such that lim_{⟶_{I}}SF ⟶^{≅} S(lim_{⟶_{I}}F) and lim_{⟶_{I}}GF ⟶^{≅} G(lim_{⟶_{I}}F). Then (K, L, ∂) is a

linking diagamme functor on the category (C, E_{τ}(C), I_{τ}(C)) .

Proof : It is obvious that (C, E_{τ}(C), I_{τ}(C)) satisfies the requirements for a category with structure, namely E_{τ}(C) contains all constant squares and I_{τ}(C) all constant infrastructure functors.

By the theorem above, the rule

\xymatrix{A \ar[r] \ar[dd] & B \ar[dd] \\  &  & ⟼ \\ C \ar[r] & D}

\xymatrix{K(B) \ar[dr] &  &  & L(B) \\  & K(D) \ar[r]^{∂} & L(A) \ar@{-->}[ur] \ar[dr] \\ K(C) \ar[ur] &  &  & L(C)}

associates to each excision square a linking diagramme. The fact that this process is functorial in morphisms of excision squares follows by a routine diagramme chasing argument from the constructions in Th 4.2 and the definition of ∂.

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We fix now an infrastructure subcategory I(C) ⊆ I_{τ}(C) . One reason for doing this is that I_{τ}(C) might be too big for purposes of getting a large subgroup functor J ⊆ G to act on (K, L, ∂) .

4.7 Definition Let A be an arbitrary object of C . Call an element σ ∈ G(A) a special element if the following property is satisfied. Let A ⟶ A′ be an arbitrary morphism and let

\xymatrix{A′ \ar[r] \ar[dd] & B_{i} \ar[dd] \\  &  & (i ∈ I) \\ C_{i} \ar[r] & D_{i}}

be any set of τ -excision squares such that there is an infrastructure functor B : I ⟶ C , i ⟼ B_{i} in I(C) which is compatible with the above squares. Then there is an element i_{0} ∈ I such that for all i ≥ i_{0} the following holds. Let σ′, σ_{B_{i}}, σ_{C_{i}} or σ_{D_{i}} denote the image of σ in G(A), G(B_{i}), G(C_{i}) and G(D_{i}) respectively. Then the action by conjugation of each of the elements above on its G-group lifts to some action on its S-group such that the diagrammes

\xymatrix{S(B_{i}) \ar[dr] \\  & S(D_{i}) & and & ∂ : K(D_{i}) \ar[r] & L(A) \\ S(C_{i}) \ar[ur]}

are σ-equivariant. It follows that the linking diagrammes

\xymatrix{K(B_{i}) \ar[dr] &  &  & L(B_{i}) \\  & K(D_{i}) \ar[r]^{∂} & L(A) \ar[ur] \ar[dr] \\ K(C_{i}) \ar@{-->}[ur] &  &  & L(C_{i})}

must be also σ-equivariant, and the induced action of J(D_{i}) on K(D_{i}) is trivial. Obviously 1 ∈ G(A) is a special element. Let J_{τ}(A) denote the set of all special elements of G(A).

The above refinement J_{τ} is frankly superfluous for the application we give below in Section 6, to “Filtrations of K_{1,n}”. However this is only so because we shall assume n ≥ 5 and it is known that St_{n} of commutative rings is the universal, central extension of E_{n} if n ≥ 5. Then the the results proved in Section 2.1 on “Equivariant Liftings to Extensions of Group Valued Functors” imply that J_{τ} = GL_{n}. When things are less simple the extra machinery might be one useful technique. Unitary groups over rings are a case in point. It is at present still not clear whether the quadratic K_{2,n} of commutative rings is a central subgroup of quadratic St_{n}. Moreover even in the linear case, if n = 3 one could try to find an explicit description of J_{τ} and thus obtain a J_{τ}-nilpotent filtration on GL_{3}.

4.8 Lemma J_{τ} (A) is a subgroup of G(A) and J_{τ} is a subgroup functor of G.

Proof : Clearly 1 is a special element and if σ is a special element then so is σ^{−1}. Suppose σ and ρ are special elements. In the Def 4.7 of special elements let i_{0} be an index for σ and j_{0} an index for ρ. If k_{0} ≥ i_{0} and k_{0} ≥ j_{0} then k_{0} is an index for σρ, i.e. if i ≥ k_{0} and (σρ)′, (σρ)_{B_{i}}, (σρ)_{C_{i}} or (σρ)_{D_{i}} denote the image of σρ in G(A), G(B_{i}), G(C_{i}) or G(D_{i}) respectively then each action by conjugation of an element above on its G-group lifts to an action on its S-group such that the homomorphisms

\xymatrix{S(B_{i}) \ar[dr] \\  & S(D_{i}) & and & ∂ : K(D_{i}) \ar[r] & L(A′) \\ S(C_{i}) \ar[ur]}

are σρ-equivariant. Thus σρ is a special element. This shows that J_{τ} (A) is a group.

To show that J_{τ} is a subgroup functor of G we must show that if A ⟶ B is a morphism in C then the image of J_{τ} (A) in G(B) lands in J_{τ} (B). But if σ ∈ J_{τ} (A) then its image σ_{B} in G(B) is obviously a special element of G(B).

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4.9 Corollary View (K, L, ∂) as a linking diagramme functor on (C, E_{τ}(C), I(C)) . Then the functor J_{τ} : C ⟶ groups acts on (K, L, ∂) .

Proof : This follows trivially from the definition of a special element.

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4.10 THEOREM Define C_{τ}^{0} as in Theorem 2.1 for the category with structure (C, E_{τ}(C), I(C)) . Let d be any dimension function on (C, E_{τ}(C), I(C)) whose 0-dimensional objects are precisely the objects of C_{τ}^{0}. Then the dimension filtration on L

L^{0} ⊇ ⋯ L^{k} ⊇ ⋯

is a descending J_{τ} -central filtration on L for d, i.e. [J(A), L^{k}(A)] ⊆ L^{k+1}(A) for all A ∈ C_{τ}^{0} and all k ≥ 0. Furthermore if for each A ∈ C_{τ}^{0} the action of J_{τ} (A) on L(A) is trivial then

L = L^{−1} ⊇ L^{0} ⊇ ⋯ L^{k} ⊇ ⋯

is a descending J_{τ} -central filtration on L for d.

Proof : This follows immediately from the Corollary above and the Main Theorem.

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4.11 Corollary Let E ⊆ E_{τ}(C) be any category of structure squares contained in E_{τ}(C) . Let C^{0} ⊆ C_{τ}^{0} be any class of objects of C which is closed under isomorphism and contained in C_{τ}^{0} . Let d be any dimension function on (C, E(C), I(C)) whose 0-dimensional objects are precisely the objects of C^{0} . Let J ⊆ J_{τ} be any subfunctor of J_{τ} . Then the dimension filtration

L^{0} ⊇ ⋯ L^{k} ⊇ ⋯

on L for d is is a descending J -central filtration. Furthermore if for each A ∈ C^{0} the action of J(A) on L(A) is trivial then

L = L^{−1} ⊇ L^{0} ⊇ ⋯ L^{k} ⊇ ⋯

is a descending J -central filtration.

Proof : Same as above.

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4.12 Corollary Suppose τ : S ⟶ G has the property that τ : S ⟶ E is a universal, perfect, central extension. Then J_{τ} = G for any choice of E(C) and I(C) .

Proof : This follows from Lemma 2.5, which says that under the assumptions of the corollary, the action of G(A), where A is an arbitrary object of C, lifts equivariantly to an action on S(A). Moreover the uniqueness of this lifting which is also asserted by Lemma 2.5, shows that it is functorial in A. This gives the σ-equivariance of the S-triangle in Def 4.7 for any element σ of G(A). Let any morphism A ⟶ A′ in C be given, then Th 4.5 implies that for any square

\xymatrix{A′ \ar[r] \ar[dd] & B_{i} \ar[dd] \\  &  & (i ∈ I) \\ C_{i} \ar[r] & D_{i}}

in E_{τ} (C), the map ∂ : K(D_{i}) ⟶ L(A′) defines a linking diagramme. Functoriality of the lifting now means that ∂ : K(D_{i}) ⟶ L(A′) is σ-equivariant. Thus every element σ of G(A) is a special element in the sense of Def 4.7, i.e. G ⊆ J_{τ}.

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4.13 Lemma Let \underline{τ} (A, B) denote the image of τ(A, B) ⟶ S(B). If the homomorphism G(A) ⟶ G(B) is injective then the homomorphism τ(A, B) ⟶ \underline{τ}(A, B) is an isomorphism. In particular if τ : S ⟶ G is central and S is perfect then a commutative square

\xymatrix{A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D}

in C such that G(A) ↣ G(B) and G(C) ↣ G(D) are injective is a τ-excision square ⟺ the group homomorphism \underline{τ}(A, B)/S(A) ⟶ \underline{τ}(C, D)/S(C) is an isomorphism. Furthermore if in addition

\xymatrix{G(A) \ar[r] \ar[d] & G(B) \ar[d] \\ G(C) \ar[r] & G(D)}

is a pullback square and S(D) = S(C)S(B) then the homomorphism above of \underline{τ}- groups is surjective; it is an isomorphism ⟺ the map S(B)/S(A) ⟶ S(D)/S(C) of coset spaces is bijective.

Proof : To obtain the isomorphism τ(A, B) ⟶ \underline{τ}(A, B) we only need to check injectivity. But an element (g, s) of τ(A, B) is in the kernel of this map only if it is the common preimage of 1 ∈ G(A) and 1 ∈ S(B), since G(A) ⟶ G(B) is injective by assumption. Then (g, s) = (1, 1), from the definition of fibre squares.

Assume that τ : S ⟶ G is central and S is perfect. By Lemma 4.1 R(A, B) and R(C, D) are groups. Suppose G(A) ⟶ G(B) and G(C) ⟶ G(D) are injective for the given square. By the pararagraph above τ(A, B) ⟶^{≅} \underline{τ}(A, B),

τ(C, D) ⟶^{≅} \underline{τ}(C, D), R(A, B) ⟶^{≅} \underline{τ}(A, B)/S(A) and R(C, D) ⟶^{≅} \underline{τ}(C, D)/S(C).

But then the second assertion is obviously true.

Suppose now that additionally the G-square is a pullback square and S(D) = S(C)S(B). Then clearly any element in S(D)/S(C) has a preimage in S(B)/S(A). Diagramme chasing in

\xymatrix{τ(A, B) \ar[rr] \ar[dd] \ar[dr] &  & S(B) \ar[dd] \ar[dr] \\  & τ(C, D) \ar[rr] \ar[dd] &  & S(D) \ar[dd] \\ G(A) \ar[rr] \ar[dr] &  & G(B) \ar[dr] \\  & G(C) \ar[rr] &  & G(D)}

shows that if the element of S(D)/S(C), which we started with, lies in \underline{τ}(C, D)/S(C) then its preimage in S(B)/S(A) lies in \underline{τ}(A, B)/S(A). The remaining assertion follows also by diagramme chasing in the diagramme above.

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5 Topological Tools

In this section we summarize in a convenient form standard materials from [6] which will be used in the following sections.

A topological group is an abstract group G with a topology on G such that multiplication and taking inverses are continuous. Let G be an abstract group. A basis of open neighbourhoods of the identity in G is a set B of subsets of G each containing 1 ∈ G and satisfying the following conditions: [6] Vol I, Ch III, §1.2, p 222 ,

∀ V ∈ B ∃ U ∈ B : U · U ⊆ V.
∀ V ∈ B ∃ U ∈ B : U^{−1} ⊆ V.
∀ g ∈ G ∀ V ∈ B ∃ U ∈ B : g · U · g^{−1} ⊆ V.

If G is a topological group then it is easy to check that the set of all open subsets of G containing 1 is a basis of open neighbourhoods of the identity.

Given a basis B of open neighbourhoods of an abstract group G one makes G into a topological group by observing that the set of all subsets σU where σ ∈ G and U ∈ B is a basis for a topology on G and then showing that G under this topology is a topological group.

Let G be an abstract group and let B and B′ be bases of neighbourhoods of 1 for G. B is called finer than B′ if given any U′ ∈ B′ and any element σ ∈ U′ there is a U ∈ B such that σU ⊆ U′. B and B′ are called equally fine if each is finer than the other. Suppose now that B is finer than B′ and let \underline{G} denote the topological group defined by the basis B and \underline{G′} that defined by the basis B′. Then the identity map on G defines a continuous homomorphism \underline{G} ⟶ \underline{G′} of topological groups. In particular if B and B′ are equally fine then they define the same topology on G.

Let G be a topological group and let B be the basis of neighbourhoods of 1 consisting of all open subsets of G containing 1. Then a basis B′ of open neighbourhoods of 1 of the abstract group G defines the same topology on G ⟺ B and B′ are equally fine.

Let G and G′ be topological groups and let B be a basis of open neighbourhoods of 1 for the topology on G. Then an abstract group homomorphism f : G ⟶ G′ is continuous ⟺ given an open subset U′ of G′ and an element σ′ ∈ U′ there is an element U ∈ B and a σ ∈ G such that f(σ) = σ′ and f(σU) ⊆ U′.

Let G be an abstract group and let B and B′ be bases of neighbourhoods of 1 in G consisting of subgroups of G. Then B is called cofinal in B′ if given any subgroup H′ ∈ B′ there is a subgroup H ∈ B such that H ⊆ H′. One shows easily that B is cofinal in B′ ⟺ B is finer than B′.

Suppose now that G and G′ are topological groups with bases B and B′ respectively of open neighbourhoods of 1 consisting of subgroups. Let f : G ⟶ G′ be an abstract group homomorphism. Then f is continuous ⟺ given H′ ∈ B′ there is an H ∈ B such that f(H) ⊆ H′.

Let G be a topological group. Let X ⊆ G be a dense subset and let U ⊆ G be an open subset. Then G = UX = XU. On the other hand suppose X ⊆ G is subset such that its closure X̄ is a subgroup of G. Let Z be a set of generators of G. Then X is dense in G ⟺ Z is contained in the closure of X. Moreover an element σ ∈ G is contained in the closure of X ⟺ for any U in a basis B of open neighbourhoods of 1 for the topology on G, σU ∩ X ≠ ∅.

In the next Section 6, the category of rings with dimension discussed in Section 2 will be taken up and a theorem for descending, central series on the group valued functors GL_{n}, St_{n} : rings ⟶ groups will be proved. We shall need to prove that localization-completion squares of rings are excision squares. In order to do so it is helpful to introduce topologies on the rings in localization-completion squares.

Let A be any commutative ring and s ∈ A any element. Then

A = s^{0}A ⊇ s^{1}A ⊇ ⋯ s^{k}A ⊇ ⋯

is a descending filtration of ideals, called the sA-adic filtration of A. It can obviously be chosen as a base for a neighbourhood filter of 0 ∈ A. Then A carries a topology [7] pp 164, 173 with a basis consisting of all sets a + s^{k}A where k ranges over all nonnegative integers and a ranges over A. It is well-known and easy to verify that this topology makes A into a topological ring, i.e. multiplication in A is also continuous.

We obtain a corresponding topology on the localization < s >^{−1} A of A at s as follows. For m ≥ 0 fixed, let ′′s^{m}A′′ denote the image \{\frac{s^{m}a}{1} | a ∈ A\} ⊆< s >^{−1} A, of s^{m}A ⊆ A. Each such subset is a subgroup of the additive group (< s >^{−1} A, +) of < s >^{−1} A, and the image of a subgroup of the additive group (A, +) of A. The descending filtration

′′s^{0}A′′ ⊇ ′′s^{1}A′′ ⊇ ⋯ ′′s^{k}A′′ ⊇ ⋯

of subgroups forms a basis of open neighbourhoods of 0 for a topology on < s >^{−1} A compatible with the additive group structure on (< s >^{−1} A, +).

The filtration is also compatible with the ring structure of < s >^{−1} A. We show that the multiplication map

< s >^{−1} A × < s >^{−1} A ⟶< s >^{−1} A

is continuous, where the product topology is used on the left. It suffices to show that given any m ≥ 0 and (\frac{a}{s^{i}} , \frac{b}{s^{k}}) ∈< s >^{−1} A × < s >^{−1} A there is an open neighbourhood of (\frac{a}{s^{i}} , \frac{b}{s^{k}}) such that its image under the multiplication map lies in \frac{ab}{s^{i+k}} + ′′s^{n}A′′. Consider a neighbourhood

(\frac{a}{s^{i}} + ′′s^{j}A′′) × (\frac{b}{s^{k}} + ′′s^{l}A′′)

of the element

(\frac{a}{s^{i}} , \frac{b}{s^{k}}) ∈ < s >^{−1} A× < s >^{−1} A,

chosen such that

\{ \begin{matrix}l > & i + m, \\ j > & k + m. \end{matrix}

Clearly the product set satisfies

\frac{ab}{s^{i+k}} + a · ′′s^{l−i}A′′ + b · s^{j−k}A + ′′s^{j+l}A′′ ⊆ \frac{ab}{s^{i+k}} + ′′s^{n}A′′,

where n = min{ l − i, j − k }. But then by the choice above of l and j, n > m.

The s-completion is a subring of a product ring: Â_{s} ⊆ ∏_{k≥1} A/s^{k}A. Elements

of this subring are [8] p 183 sequences

(a_{1}, a_{2}, ⋯) ∈ ∏_{k≥0} A/s^{k}A such that a_{j} ≡ a_{i} mod s^{k}A.

The sequence of ideals

Â = s^{0}Â_{(s)} ⊇ s^{1}Â_{(s)} ⊇ ⋯ s^{k}Â_{(s)} ⊇ ⋯

is a basis of neighbourhoods for a topology compatible with the ring structure. The filtration

< s >^{−1} Â_{(s)} ⊇ ′′s^{0}Â′′_{(s)} ⊇ ′′s^{1}Â′′_{(s)} ⊇ ⋯ ′′s^{k}Â′′_{(s)} ⊇ ⋯

by subgroups of the additive group (< s >^{−1} Â_{(s)}, +) is a basis of neighbourhoods of 0 for a topological group structure on (< s >^{−1} Â_{(s)}, +). The discussion above for the localisation < s >^{−1} A (of any commutative ring) shows that multiplication in < s >^{−1} Â_{(s)} is continuous, i.e. that < s >^{−1} Â_{(s)} is a topological ring.

5.1 Proposition Let R be a commutative ring. Let s ≠ 0 be a non zero-divisor in R. Topologize as above the rings in a localization-completion square

\xymatrix{R \ar[d] \ar[r] & < s >^{−1} R \ar[d] \\ R̂_{(s)} \ar[r] & < s >^{−1} R̂_{(s)}}

starting with the sR-adic topology on R. Then the horizontal maps are continuous, open and injective. The vertical maps are continuous and dense.

Proof : By the choice of s above it is clear that R̂_{(s)} is s-torsion free and that the top horizontal map is injective. Since R̂_{(s)} is s-torsion free, it follows that the bottom horizontal map is injective.

The openness assertions for the horizontal maps are obvious, since each takes a basis of open neighbourhoods of 0 onto a basis of open neighbourhoods of 0.

The vertical maps are canonical completion maps. Thus the assertions about them are standard results.

□

The topologies introduced on the rings in localization-completion squares define topologies on the corresponding St_{n}-groups. We discuss these now.

Let R be any ring and n ≥ 1 be a fixed number. Let symbols x_{ij}(a) be given for all a ∈ R and all (i, j) ∈ ℕ_{n} × ℕ_{n}, i ≠ j.

Then the Steinberg group St_{n}(R) is defined to be the free group on the symbols x_{ij}(a) as generators and satisfying the Steinberg relations

x_{ij}(a + b) = x_{ij}(a) · x_{ij}(b),
[x_{ij}(a), x_{kl}(b)] = 1 if i ≠ l and j ≠ k,

and

[x_{ij}(a), x_{jl}(b)] = x_{il}(ab) if i ≠ l.

The third relation implies for n ≥ 3 that St_{n}(R) is a perfect group. There is a canonical map St_{n}(R) ⟶ GL_{n}(R) where GL_{n}(R) is the general linear group of invertible n×n matrices over R. The image of this map is the elementary subgroup E_{n}(R) ⊂ GL_{n}(R) generated by elementary matrices ϵ_{ij}(r) = 1+e_{ij}(r) where 1 denotes the identity matrix in GL_{n}(R) and e_{ij}(r) is a n × n matrix with r in the ij’th place and 0’s elsewhere.

Now suppose R is commutative and consider the sR-adic filtration

R = s^{0}R ⊇ s^{1}R ⊇ ⋯ s^{k}R ⊇ ⋯

Let St_{n}(s^{k}R) denote for all k ≥ 0 the subgroup of St_{n}(R) generated by Steinberg generators x_{ij}(s^{k}a) with coefficients s^{k}a in the ideal s^{k}R. For the localization < s >^{−1} R of R define corresponding subgroups St_{n}(′′s^{k}R′′). In the case of St_{n}(R), we get a filtration

St_{n}(R) = St_{n}(s^{0}R) ⊇ St_{n}(s^{1}R) ⋯ ⊇ St_{n}(s^{k}R) ⊇ ⋯

of St_{n}(R) by subgroups of St_{n}(R) and in the case of St_{n}(< s >^{−1} R), we get a filtration

St_{n}(′′R′′) = St_{n}(′′s^{0}R′′) ⊇ St_{n}(′′s^{1}R′′) ⋯ ⊇ St_{n}(′′s^{k}R′′) ⊇ ⋯

of St_{n}(< s >^{−1} R) by subgroups of St_{n}(< s >^{−1} R).

Clearly the canonical homomorphism St_{n}(R) ⟶ St_{n}(< s >^{−1} R) takes St_{n}(s^{k}R) onto St_{n}(′′s^{k}R′′).

5.2 Lemma Let n be an integer, n ≥ 3. The filtrations above of St_{n}(R) and St_{n}(< s >^{−1} R) are bases of open neighbourhoods of the identity in their respective groups.

Proof : Since the sets in each filtration are groups, the first two properties of bases of open neighbourhoods of the identity are clear. We check the third property only for the case of St_{n}(< s >^{−1} R). The proof for St_{n}(R) is similiar and a little easier.

We must show that, given any element x = ∏_{μ=0}^{N} x_{i(μ)j(μ)}(\frac{r_{μ}}{s^{pμ}}) ∈ St_{n}(< s >^{−1} R)

and any subgroup St_{n}(′′s^{m}R′′) in the basis of open neighbourhoods of the identity in St_{n}(< s >^{−1} R) there is another subgroup St_{n}(′′s^{m′}R′′) in the basis of open neighbourhoods of the identity in St_{n}(< s >^{−1} R), such that for all x_{kl}(\frac{s^{m′}a}{1}) ∈ St_{n}(′′s^{m′}R′′),

xx_{kl}(\frac{s^{m′}a}{1}) = x · x_{kl}(\frac{s^{m′}a}{1}) · x^{−1} ∈ St_{n}(′′s^{m}R′′) ⋯ (∗)

A proof by induction on N requires only proving that given x_{ij}(\frac{r}{s^{p}}) there is an m′ such that

x_{ij}( \frac{r}{s^{p}})x_{kl}(\frac{s^{m′}a}{1}) ∈ St_{n}(′′s^{m}R′′).

Set q = m + p and m′ = 2q. If j ≠ k and also i ≠ l then the Steinberg relations above imply that

x_{ij}( \frac{r}{s^{p}})x_{kl}(\frac{s^{m′}a}{1}) = x_{kl}(\frac{s^{m′}a}{1}) ∈ St_{n}(′′s^{2m+2p}R′′) ⊆ St_{n}(′′s^{2m+p}R′′) ⊆ St_{n}(′′s^{m}R′′).

If i ≠ l but j = k

x_{ij}( \frac{r}{s^{p}})x_{kl}(\frac{s^{m′}a}{1}) = [x_{ij}(\frac{r}{s^{p}}), x_{kl}(\frac{s^{m′}a}{1})] · x_{kl}(\frac{s^{m′}a}{1}) ∈ St_{n}(′′s^{2m+p}R′′),

since here [x_{ij}(\frac{r}{s^{p}}), x_{kl}(\frac{s^{m′}a}{1})] = x_{il}(\frac{r}{s^{p}} · \frac{s^{m′}a}{1}) by the Steinberg relations. Finally if

j = k and i = l choose h ≠ i, j and write x_{ji}(\frac{s^{m′}a}{1}) = [x_{jh}(\frac{s^{q}}{1} ), x_{hi}(\frac{s^{q}a}{1})]. Then

x_{ij}( \frac{r}{s^{p}})x_{kl}(\frac{s^{m′}a}{1}) = [^{x_{ij}( \frac{r}{s^{p}})}x_{jh}(\frac{s^{q}}{1} ),^{x_{ij}( \frac{r}{s^{p}})} x_{hi}(\frac{s^{q}a}{1})] ∈ St_{n}(′′s^{m}R′′).

□

Insert We provide now a description of the induction process in the proof of the lemma. Later on, see the proof of Lemma 6.2, it will be convenient to have

this available. Consider as in the lemma any product x = ∏_{μ=0}^{N} x_{i(μ)j(μ)}(\frac{r_{μ}}{s^{pμ}}) ∈

St_{n}(< s >^{−1} R) and any subgroup St_{n}(′′s^{m}R′′). The process then looks in detail as follows. Conjugating by x is the same as conjugating successively by its factors, beginning with x_{i(N)j(N)}(\frac{r_{N}}{s^{p_{N}}} ) then conjugating by x_{i(N−1)j(N−1)}(\frac{r_{N−1}}{s^{p_{N−1}}} ) and so on through x_{i(N−k)j(N−k)}(\frac{r_{N−k}}{s^{p_{N−k}}}), N > k > 0 and ending with x_{i(0)j(0)}(\frac{r_{0}}{s^{p_{0}}} ). At each step the conjugate is an element of a certain subgroup whose generators are known, thus it can be rewritten as a product of such. The next conjugation can be distributed over this product of generators. Then the conjugate of each generator lies, as has been shown in the proof of the lemma, in the next subgroup we need. The final result is to lie in St_{n}(′′s^{m}R′′). Thus after the N + 1’th step in the process, which is conjugation by the last factor applied, x_{i(0)j(0)}(\frac{r_{0}}{s^{p_{0}}} ), the result will lie in St_{n}(′′s^{m}R′′) provided the previous, or N’th step has got us into St_{n}(′′s^{2m+2p_{0}}R′′). This in turn requires that we have achieved St_{n}(′′s^{4m+4p_{0}+2p_{1}}R′′) at the N − 1’th step. Thus at the N + 1 − M’th step, we must have got into St_{n}(′′s^{2^{M} m+2^{M} p_{0}+2^{M−1}p_{1}+2^{M−2}p_{2}+2^{M−3}p_{3}+···+2p_{M−1}}R′′). Taking M = N, we must have at the first step have achieved St_{n}(′′s^{2^{N} m+2^{N} p_{0}+2^{N−1}p_{1}+2^{N−2}p_{2}+2^{N−3}p_{3}+···+2p_{N−1}}R′′). Any generator of the subgroup St_{n}(′′s^{2^{N+1}m+2^{N+1}p_{0}+2^{N} p_{1}+···+2p_{N}} R′′) when conjugated by the element x_{i(N)j(N)}(\frac{r_{N}}{s^{p_{N}}} ) lies in St_{n}(′′s^{2^{N} m+2^{N} p_{0}+2^{N−1}p_{1}+2^{N−2}p_{2}+2^{N−3}p_{3}+···+2p_{N−1}}R′′), i.e.

x_{i(N)j(N)}( \frac{r_{N}}{s^{p_{N}}} )x_{kl}(\frac{s^{2^{N+1}m+2^{N+1}p_{0}+2^{N} p_{1}+···+2p_{N}} a}{1}) ∈ St_{n}(′′s^{2^{N} m+2^{N} p_{0}+2^{N−1}p_{1}+···+2p_{N−1}}R′′),

x_{i(M)j(M)}( \frac{r_{M}}{s^{p_{M}}} )x_{kl}(\frac{s^{2^{M+1}m+2^{M+1}p_{0}+2^{M} p_{1}+···+2p_{M}} a}{1}) ∈ St_{n}(′′s^{2^{M} m+2^{M} p_{0}+2^{M−1}p_{1}+···+2p_{M−1}}R′′),

x_{i(0)j(0)}( \frac{r_{0}}{s^{p_{0}}} )x_{kl}(\frac{s^{2^{1}m+2^{1}p_{0}}a}{1}) ∈ St_{n}(′′s^{2^{0}m}R′′) = St_{n}(′′s^{m}R′′).

We next prove a lemma from which topological properties of St_{n}-squares can be derived.

5.3 Lemma Let G be a topological group generated by the subgroups {H_{i} | i ∈ I} and let G′ be a topological group generated by the subgroups {H′_{i} | i ∈ I}. Let G ⟶ G′ be a continuous group homomorphism which maps H_{i} to H′_{i} for each i ∈ I. If each map H_{i} ⟶ H′_{i} is a dense map of topological groups, then the map G ⟶ G′ is also dense.

Proof : The subset X′ = ⋃_{i∈I} H′_{i} is a set of generators for G′. Thus it suffices to show that this set is contained in the closure Ḡ of the image of G in G′, because Ḡ is a subgroup of G′. (This last assertion is clear because a homomorphic image of group is a subgroup of the target group and the closure of any subgroup of a topological group is also a topological subgroup, [6] Vol I, Ch III, §2.1, Prop 1.) The image of G clearly contains the image of the set X = ⋃_{i∈I} H_{i}, so that Ḡ contains the closure Ȳ of the image Y of X. But the latter set contains X′ by assumption, and since Ḡ is a subgroup of G′, it follows that Ḡ = G′.

□

5.4 Proposition Let

\xymatrix{A \ar[d] \ar[r] & < s >^{−1} A \ar[d] \\ Â_{(s)} \ar[r] & < s >^{−1} Â_{(s)}}

be a localization-completion square in commutative rings. Let n ≥ 1. Then the square

\xymatrix{St_{n}(A) \ar[d] \ar[r] & St_{n}(< s >^{−1} A) \ar[d] \\ St_{n}(Â_{(s)}) \ar[r] & St_{n}(< s >^{−1} Â_{(s)})}

has continuous, open horizontal maps and dense, continuous, open vertical maps.

Proof : Each of the maps in the St_{n}-square obviously maps any subgroup in the basis of open neighbourhoods of the identity onto a subgroup in the basis of open neighbourhoods of the identity. This implies the map is continuous and open.

We now prove that the vertical maps are dense. For any ring R the group St_{n}(R) is generated by its subgroups x_{ij}(R), 1 ≤ i, j ≤ n. By Lemma 5.3 it suffices to show that for R = A or R = < s >^{−1} A and any (i, j) ∈ ℕ_{n} × ℕ_{n}, i ≠ j the natural map

x_{ij}(R) ⟶ x_{ij}(R̂_{(s)}), x_{ij}(r) ⟼ x_{ij}(∏_{k≥0} r + s^{k}R) ⋯ (∗)

is dense, i.e. that the vertical homomorphisms are dense on the generating subgroups. By Prop 5.1 the maps below are dense

A ⟶ Â_{(s)}, < s >^{−1} A ⟶< s >^{−1} Â_{(s)}.

We prove that the maps x_{ij}(< s >^{−1} A) ⟶ x_{ij}(< s >^{−1} Â_{(s)}) are dense. Clearly the maps

< s >^{−1} A ⟶ x_{ij}(< s >^{−1} A), \frac{a}{s^{ν}} ⟼ x_{ij}(\frac{a}{s^{ν}}),

and

< s >^{−1} Â ⟶ x_{ij}(< s >^{−1} Â), \frac{â}{s^{ν}} ⟼ x_{ij}(\frac{â}{s^{ν}}),

are isomorphisms of the additive group of < s >^{−1} A with the (multiplicative) subgroups x_{ij}(< s >^{−1} A) of St_{n}(< s >^{−1} A) respectively of the additive group of < s >^{−1} Â with the (multiplicative) subgroups x_{ij}(< s >^{−1} Â) of St_{n}(< s >^{−1} Â). If these maps are also homeomorphisms, then obviously the commutative squares

\xymatrix{< s >^{−1} A \ar[r]^{≅} \ar[dd] & x_{ij}(< s >^{−1} A) \ar[dd] \\  &  & (i, j) ∈ ℕ_{n} × ℕ_{n}, i ≠ j \\ < s >^{−1} Â_{(s)} \ar[r]_{≅} & x_{ij}(< s >^{−1} Â_{(s)})}

prove that the vertical maps on the right in each such square is dense. Thus it will be clear that (∗) is dense for R =< s >^{−1} A once we have shown that x_{ij}(< s >^{−1} A) carries the subspace topology induced by the topology on St_{n}(< s >^{−1} A), for then an ideal s^{k}A in the filtration defining the topology on R maps onto a subgroup x_{ij}(′′s^{k}A′′) defining the topology on x_{ij}(R) and the analogue holds for the inverse map. Clearly it suffices to show that St_{n}(′′s^{k}A′′) ∩ x_{ij}(< s >^{−1} A) = x_{ij}(′′s^{k}A′′), i.e. for each subgroup St_{n}(′′s^{k}A′′) in the basis of open neighbourhoods for the topology on St_{n}(< s >^{−1} A), the intersection St_{n}(′′s^{k}A′′) ∩ x_{ij}(< s >^{−1} A) is a subgroup in the basis of open neighbourhoods for the topology on x_{ij}(< s >^{−1} A).

Let Q = A or Q = Â_{(s)}, and consider the image of St_{n}(′′s^{k}Q′′) ∩ x_{ij}(< s >^{−1} Q) in GL_{n}(< s >^{−1} Q). By definition of Steinberg groups and of elementary subgroups of GL_{n}, the map x_{ij}(r) ↦ ϵ_{ij}(r), where ϵ_{ij}(r) is the elementary matrix with its sole nonzero off-diagonal entry = r at the ij-th coordinate, is bijective.

An element of St_{n}(′′s^{k}Q′′) is a product ∏_{μ=0}^{N} x_{i(μ),j(μ)}(\frac{s^{k}r_{μ}}{1} ), where for all μ, r_{μ} ∈ Q.

Thus its image in GL_{n}(< s >^{−1} Q) is a product of elementary matrices

∏_{μ=0}^{N} ϵ_{i(μ),j(μ)}(\frac{s^{k}r_{μ}}{1} ) = ∏_{μ=0}^{N} 1 + e_{i(μ),j(μ)}(\frac{s^{k}r_{μ}}{1} ) = 1 + ∑_{ν=0}^{M} e_{i(ν),j(ν)}(a_{ν}).

The second equality is obtained by multiplying out and we clearly have an invertible matrix which is the sum of the identity matrix 1 ∈ GL_{n}(Q) and a matrix ∑_{ν=0}^{M} e_{i(ν),j(ν)}(a_{ν}) all of whose entries a_{ν} are sums of products of some of the \frac{s^{k}r_{μ}}{1} and thus come from the subgroup ′′s^{k}Q′′ ⊆< s >^{−1} Q. Hence

ϵ_{ij}(r) = 1 + e_{ij}(r) = 1 + ∑_{ν=0}^{M} e_{i(ν),j(ν)}(a_{ν})

which implies r ∈ ′′s^{k}Q′′, i.e. GL_{n}(′′s^{k}Q′′) ∩ ϵ_{ij}(Q) = ϵ_{ij}((′′s^{k}Q′′). Thus by the bijectivity of x_{ij}(Q) ⟶ ϵ_{ij}(Q), St_{n}(′′s^{k}Q′′) ∩ x_{ij}(Q) = x_{ij}(′′s^{k}Q′′).

The proof for the case R = A is similar but simpler.

□
6 Filtrations of K_{1,n}

The results obtained in Section 2 to Section 5 can now be applied to categories of rings and to the groups St_{n}(A) and GL_{n}(A) . We first recall the non-stable functors of lower K-theory. For any category C of rings there are for all n ≥ 1 functors St_{n}, GL_{n} : C ⟶ groups and a canonical natural transformation τ_{n} : St_{n} ⟶ GL_{n}. Moreover there are K-theoretic groups

K_{2,n}(A) = Ker(τ_{n,A} : St_{n}(A) ⟶ GL_{n}(A)),
E_{n}(A) = Im(τ_{n,A} : St_{n}(A) ⟶ GL_{n}(A)),

and a K-theoretic coset space

K_{1,n}(A) = GL_{n}(A)/E_{n}(A).

Thus K_{2,n} is a functor C ⟶ groups and K_{1,n} is a functor C ⟶ coset spaces. The category coset spaces is defined in [3]. Moreover it is known by results of [15] that if A is commutative and n ≥ 3 then E_{n}(A) is normal in GL_{n}(A), so K_{1,n}(A) is a group as well. Furthermore by results of [10] and [16] if A is commutative and n ≥ 4, K_{2,n}(A) is Abelian, and if in addition n ≥ 5 then St_{n}(A) is the universal, perfect central extension of E_{n}(A). For the rest of this section let ((C, S(C), I(C)), BSdim) denote the category with dimension appearing in Th 2.20, where C is the category of all commutative Noetherian rings. By Th 2.20 ((C, S(C), I(C)), BSdim) is a category with dimension.

Restrict for the rest of this section the natural transformation τ_{n} : St_{n} ⟶ GL_{n} and the functors K_{1,n} and K_{2,n} to C and assume that n ≥ 5. Define the dimension filtration

K_{1,n} = K_{1,n}^{−1} ⊇ K_{1,n}^{0} ⊇ K_{1,n}^{1} ⊇ ⋯ K_{1,n}^{i} ⊇ ⋯

on K_{1,n} such that for i ≥ 0

K_{1,n}^{i}(A) = ⋂_{f:A⟶B_{BSdim(B)≤k}} Ker(K_{1,n}(f) : K_{1,n}(A) ⟶ K_{1,n}(B)).

The goal of this section is to prove the following theorem.

6.1 THEOREM The dimension filtration on K_{1,n} is a descending central series such that if d = BSdim(A) is finite then K_{1,n}^{d}(A) = 0.

Proof We want to apply Cor 4.11. Let C^{0} denote the class of rings A in C such that BSdim(A) = 0. ¿From the definition of BSdim, A ∈ C_{τ_{n}}^{0} ⟺ A has only a finite number of maximal ideals. Let us assume for the moment that each localization-completion square in C is an excision square for τ_{n}. It is routine to check that the direct limit of any infrastructure functor in I(C) commutes with τ_{n} : St_{n} ⟶ GL_{n} and therefore with K_{1,n} and K_{2,n}. Since St_{n}(A) is the universal, perfect central extension of E_{n}(A), it follows from Lemma 2.5 that each element of GL_{n}(A) is a special element in the sense of Def 4.7. Thus the data ((C, S(C), I(C)), BSdim), C^{0}, J where J = GL_{n} satisfies the conditions of Cor 4.11, except possibly the condition that GL_{n}(A) acts trivially on K_{1,n}(A) whenever BSdim(A) = 0. But this follows from [5] Ch V, Th 4.1(b) combined with [5] Ch V, Th 3.5. Thus the assertion of the theorem follows from the conclusions of Cor 4.11.

It remains to show that a localization-completion square is an excision square for τ_{n}. Let

\xymatrix{A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D}

be a localization-completion square. From Lemma 2.6, it follows that the square is a pullback square. To prove that it is an excision square we shall need injectivity for its horizontal maps. This again is guaranteed by Lemma 2.6. It is obvious that the commutative square of matrix rings of n × n matrices defined by the square above is a pullback square whose horizontal maps are injective. Thus the same holds also for the square of units of these rings. But this is the square

\xymatrix{GL_{n}(A) \ar[r] \ar[d] & GL_{n}(B) \ar[d] \\ GL_{n}(C) \ar[r] & GL_{n}(D).}

Lemma 4.13 reduces now the burden of proof to showing that the homomorphism St_{n}(B)/St_{n}(A) ⟶ St_{n}(D)/St_{n}(C) of coset spaces is bijective.

Recall that in Section 5 we put a topology on the square

\xymatrix{St_{n}(A) \ar[r] \ar[d] & St_{n}(B) \ar[d] \\ St_{n}(C) \ar[r] & St_{n}(D).}

such that the horizontal maps are open and the vertical maps are dense. Thus the image of St_{n}(C) in St_{n}(D) is open and the image of St_{n}(B) in St_{n}(D) is dense. Thus St_{n}(D) is the product of these images in any order; i.e. if U = image(St_{n}(C)) and V = image(St_{n}(B)) then St_{n}(D) = UV = V U. It follows obviously that the map St_{n}(B)/St_{n}(A) ⟶ St_{n}(D)/St_{n}(C) is surjective. Thus to complete the proof of the theorem, it suffices to show that this map is injective. This will occupy the rest of this section.

□

Let

\xymatrix{A \ar[d] \ar[r] & < s >^{−1} A \ar[d] \\ Â_{(s)} \ar[r] & < s >^{−1} Â_{(s)}}

be our localization-completion square. The injectivity proof is taken from [2], appropriately modifying concepts developed there, Def 7.13, p 122 to handle this problem.

We must prove that the set map below is injective.

St_{n}(< s >^{−1} A)/′′St_{n}(A)′′ ⟶ St_{n}(< s >^{−1} Â_{(s)})/′′St_{n}(Â_{(s)})′′.

This map is induced by the set map

St_{n}(< s >^{−1} A) ⟶ St_{n}(< s >^{−1} Â_{(s)}).

An element in St_{n}(< s >^{−1} Â_{(s)}) can be written in many different ways as a product of generators. But it turns out that given an arbitrary element x̂ of St_{n}(< s >^{−1} Â_{(s)}) any two elements x and x′ in St_{n}(< s >^{−1} A) whose images are close to x̂ in the topology we have put on St_{n}(< s >^{−1} Â_{(s)}) are close to each other in St_{n}(< s >^{−1} A). We set out to prove this, beginning with a lemma which proves that two elements in St_{n}(< s >^{−1} A) both of whose images are close to the same product of generators of St_{n}(< s >^{−1} Â_{(s)}) are the same modulo ′′St_{n}(A)′′. The lemma will be used to motivate in Definition 6.3 the notion of good approximation. Later we shall show that the same is also true no matter which product of generators is used to represent the element in St_{n}(< s >^{−1} Â_{(s)}). The proof of the lemma is taken from [2] Th. 7.16, p 124, with the neccessary changes.

6.2 Lemma Let x(t̂) = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(\frac{t̂_{μ}}{s^{kμ}}) be any given product of generators of

St_{n}(< s >^{−1} Â_{(s)}). Let x(b) = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(\frac{b_{μ}}{s^{kμ}}) and x(c) = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(\frac{c_{μ}}{s^{kμ}}) be

two elements in St_{n}(< s >^{−1} A) satisfying the following condition. Denote by b̂_{μ} and ĉ_{μ} in < s >^{−1} Â_{(s)} the images of elements \frac{b_{μ}}{s^{kμ}} and \frac{c_{μ}}{s^{kμ}} in < s >^{−1} A. The condition on x(b) and x(c) is the following.

There are nonnegative integers m′ ≥ m ≥ 0 such that

1) for all μ, 0 ≤ μ ≤ p, t̂_{μ} − b̂_{μ} ∈ ′′s^{(2^{p−μ}m+2^{p−μ}k_{p}+2^{p−μ−1}k_{p−1}+···+2k_{μ+1})}Â′′_{(s)}, and such that

2) for all μ, 0 ≤ μ ≤ p, t̂_{μ} − ĉ_{μ} ∈ ′′s^{(2^{p−μ}m′+2^{p−μ}k_{p}+2^{p−μ−1}k_{p−1}+···+2k_{μ+1})}Â′′_{(s)}.

Then x(b) and x(c) have the property that x(b)· ′′St_{n}(s^{m}A)′′ = x(c)· ′′St_{n}(s^{m}A)′′.

Proof : Let t̂∈< s >^{−1} Â_{(s)} and \frac{b}{s^{k_{b}}} , \frac{c}{s^{kc}} ∈< s >^{−1} A with images b̂, ĉ ∈< s >^{−1} Â_{(s)} be any elements. Suppose that b̂ − t̂ ∈ ′′s^{m}Â′′_{(s)} ⊆< s >^{−1} Â_{(s)} and that ĉ − t̂ ∈′′ s^{m′} Â′′_{(s)}, where m′ ≥ m. Then b̂ − ĉ = b̂ − t̂− (ĉ − t̂) ∈ ′′s^{m}Â′′_{(s)}. Thus there is an element s^{m}α in s^{m}Â_{(s)} which coincides with \frac{b}{s^{k_{b}}} − \frac{c}{s^{kc}} in < s >^{−1} Â_{(s)}. By Lemma 2.15 there is a unique common preimage a ∈ A for s^{m}α and \frac{b}{s^{k_{b}}} − \frac{c}{s^{kc}}. Since s^{m}α ∈ s^{m}Â_{(s)}, clearly there is an a′ ∈ A such that a = s^{m}a′ ∈ s^{m}A. Thus \frac{b}{s^{k_{b}}} − \frac{c}{s^{kc}} = \frac{a}{1} = \frac{s^{m}a′}{1} ∈ ′′s^{m}A′′ ⊆< s >^{−1} A.

Suppose x(b) = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(\frac{b_{μ}}{s^{kμ}}), x(c) = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(\frac{c_{μ}}{s^{kμ}}) ∈ St_{n}(< s >^{−1} A)

and x(t̂) = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(\frac{t̂_{μ}}{s^{kμ}}) ∈ St_{n}(< s >^{−1} Â_{(s)}) satisfy the assumptions of the

Lemma. Then the foregoing discussion shows that for each μ, 0 ≤ μ ≤ p, there is an element

a_{μ} = s^{(2^{p−μ}m+2^{p−μ}k_{p}+2^{p−μ−1}k_{p−1}+···+2k_{μ+1})}a′_{μ} ∈ s^{(2^{p−μ}m+2^{p−μ}k_{p}+2^{p−μ−1}k_{p−1}+···+2k_{μ+1})}A

such that

\frac{c_{μ}}{s^{k_{μ}}} − \frac{b_{μ}}{s^{k_{μ}}} = \frac{a_{μ}}{1} = \frac{s^{(2^{p−μ}m+2^{p−μ}k_{p}+2^{p−μ−1}k_{p−1}+···+2k_{μ+1})}a′_{μ}}{1}

∈ ′′s^{(2^{p−μ}m+2^{p−μ}k_{p}+2^{p−μ−1}k_{p−1}+···+2k_{μ+1})} < s >^{−1} A′′.

Let x(a) be the element

x_{i(0)j(0)}(\frac{a_{0}}{1} )^{x_{i(1)j(1)}( \frac{b_{1}}{s^{k_{1}}} )·x_{i(2)j(2)}( \frac{b_{2}}{s^{k_{2}}} )···x_{i(p)j(p)}( \frac{bp}{s^{kp}} )} · · · · · · · · · · · · · · · · · ·

· · · · · · x_{i(p−1)j(p−1)}(\frac{a_{p−1}}{1} )^{x_{i(p)j(p)}( \frac{bp}{s^{kp}} )} · x_{i(p)j(p)}(\frac{a_{p}}{1} )

then clearly x(c) = x(b) · x(a). Moreover, see the Insert after Lemma 5.2, for each μ, 0 ≤ μ ≤ p,

x_{i(μ)j(μ)}(\frac{a_{μ}}{1} ) )^{x_{i(μ+1)j(μ+1)}( \frac{b_{μ+1}}{s^{k_{μ+1}}} )···x_{i(p)j(p)}( \frac{bp}{s^{kp}} )} ∈ ′′St_{n}(s^{m}A)′′.

The differences are due to our conjugating on the right now, whereas conjugation was on the left in the proof of Lemma 5.2.

□

We now have to show that given an arbitrary element x̂ of St_{n}(< s >^{−1} Â_{(s)}) any two elements x and x′ in St_{n}(< s >^{−1} A) whose images are close in the topology we have put on St_{n}(< s >^{−1} Â_{(s)}), each to some formal product of generators representing x̂, but not both neccessarily to the same such formal product, are still close to each other in St_{n}(< s >^{−1} A). A convenient tool to use here is words in the sense of combinatorial group theory. We recall the general idea behind this from [13] p 5.

A word is used to represent a formal product of elements in a group in the same way that a polynomial with integer coefficients in several variables is used to represent a formal algebraic combination of elements in an integral domain. Just as we can assign values to the variables in a given polynomial from the elements in any integral domain, and compute the value of the polynomial in this domain, so we can assign values to the symbols in a given word from the elements in any group, and compute the value of the word in this group.

We use the following notation. Let

Y(< s >^{−1} Â_{(s)})

be the set of all words in the symbols y_{i,j}( t̂), cf. [13] p 4, where n is a natural number, 0 ≤ i, j ≤ n, i ≠ j are natural numbers and t̂ ∈ < s >^{−1} Â_{(s)}. A typical

element of Y(< s >^{−1} Â_{(s)}) is of the form ∏_{μ=1}^{p}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)}, where η(μ) = ±1.

Let ∅ denote the empty word, cf. [13] p 4, and let Y = Y(< s >^{−1} Â_{(s)}) ∪ ∅. Let further

Y_{0}(< s >^{−1} Â_{(s)}) =
{ y_{ij}(t̂+ û)y_{ij}(−t̂)y_{ij}(−û), y_{ij}(t̂)y_{kl}(û)y_{ij}(−t̂)y_{kl}(−û),

y_{ij}(t̂)y_{jl}(û)y_{ij}(−t̂)y_{jl}(−û)y_{il}(−t̂û) | i ≠ l, j ≠ k; t̂ and û ∈ < s >^{−1} Â_{(s)} }.

There are two equivalence relations on the set Y, such that the respective equivalence classes form a group in each case. One of these is the free group F(Y) with generators the elements of Y. The other is St_{n}(< s >^{−1} Â_{(s)}). We develop now a convenient language for handling the above. Recall first that a word in the set { y_{ij}(t̂)^{η}y_{ij}(t̂)^{−η} | t̂ ∈ < s >^{−1} Â_{(s)}, η = ±1} is called a trivial relator, cf. [13] p 6. Define two elements of Y to be freely equivalent if one can be obtained from the other by the insertion or deletion of trivial relators. Denote by Ȳ the set of classes of freely equivalent elements of Y and denote by [y]_{F} the free equivalence class of the element y ∈ Y. Define a composition on Y by means of the juxtaposed product of words, cf. [13] p 5, as follows

Y × Y ⟶ Y,

(∏_{μ=1}^{p}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)}, ∏_{μ=1}^{p′}(y′_{i(μ),j(μ)}(t̂′_{μ}))^{η′(μ)}) ↦

∏_{μ=1}^{p}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)} ∏_{μ=1}^{p′}(y′_{i(μ),j(μ)}(t̂′_{μ}))^{η′(μ)}.

It is understood in this definition that if ∅ denotes the empty word and z is an element of Y then

∅z = z∅ = z

Define now a composition on Ȳ as follows.

Ȳ × Ȳ ⟶ Ȳ ,

([∏_{μ=1}^{p}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)}]_{F} , [∏_{μ=1}^{p′}(y′_{i(μ),j(μ)}(t̂′_{μ}))^{η′(μ)}]_{F}) ↦

[∏_{μ=1}^{p}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)} ∏_{μ=1}^{p′}(y′_{i(μ),j(μ)}(t̂′_{μ}))^{η′(μ)}]_{F} .

This composition is a group law on Ȳ and the group it defines is the free group on the symbols y_{i,j}( t̂), where n is a natural number, 0 ≤ i, j ≤ n, i ≠ j are natural numbers and t̂ ∈ < s >^{−1} Â_{(s)}, cf. [13] p 36.

By definition of the Steinberg group, a word in Y_{0}(< s >^{−1} Â_{(s)}) “defines the identity element” cf. [13] p 6, in St_{n}(< s >^{−1} Â_{(s)}). Thus it is a relator for St_{n}(< s >^{−1} Â_{(s)}). In fact these words form a set of defining relators, cf. [13] p 7, for the Steinberg group, by definition. We introduce now a convenient way of dealing with defining relators and trivial relators. Let z be an element of Y_{0}(< s >^{−1} Â_{(s)}) or a trivial relator and let r and p be numbers, 1 ≤ r ≤ p − 1. An elementary move is a change of the following kind on formal products, i.e. elements of Y.

∏_{μ=1}^{p}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)} ↦ ∏_{μ=1}^{r}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)} z ∏_{μ=r+1}^{p}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)}

∏_{μ=1}^{r}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)} z ∏_{μ=r+1}^{p}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)} ↦ ∏_{μ=1}^{p}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)}

The former kind is called insertion and the latter deletion. We call two elements of Y equivalent to each other if one can be obtained from the other by a finite sequence of elementary moves, i.e. by inserting or deleting a finite number of trivial relators or relators for St_{n}(< s >^{−1} Â_{(s)}). Denote by [y] the class of elements equivalent to y ∈ Y and denote by Ỹ the set of equivalence classes. Define now a composition on Ỹ as follows.

Ỹ × Ỹ ⟶ Ỹ ,

([∏_{μ=1}^{p}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)}], [∏_{μ=1}^{p′}(y′_{i(μ),j(μ)}(t̂′_{μ}))^{η′(μ)}]) ↦

[∏_{μ=1}^{p}(y_{i(μ),j(μ)}(t̂_{μ}))^{η(μ)} ∏_{μ=1}^{p′}(y′_{i(μ),j(μ)}(t̂′_{μ}))^{η′(μ)}].

This composition is a group law on Ỹ and by definition Ỹ = St_{n}(< s >^{−1} Â_{(s)}). These observations yield a well-defined map

χ : Y ⟶ St_{n}(< s >^{−1} Â_{(s)}) , y ⟼ [y].

It is clear that

< Y ; Y_{0}(< s >^{−1} Â_{(s)}) >

is a presentation, cf. [13] p 7, for St_{n}(< s >^{−1} Â_{(s)}) under the map χ which sends each word to the element “defined” by it.

The following definition is the key to applying the topological tools of Section 5 to the problem of proving the injectivity of the map.

6.3 Definition Let m ≥ 0 be an integer. Denote by b̂_{μ} and ĉ_{μ} the images in < s >^{−1} Â_{(s)} of elements \frac{b_{μ}}{s^{kμ}} and \frac{c_{μ}}{s^{kμ}} belonging to < s >^{−1} A.

Let b = ∏_{μ=0}^{p}(x_{i(μ),j(μ)}(\frac{b_{μ}}{s^{kμ}}))^{η(μ)} be a formal product representing an element

of St_{n}(< s >^{−1} A). The element b is called an m−good approximation

for ŷ = ∏_{μ=0}^{p} y_{i(μ),j(μ)}(ŷ_{μ})^{η(μ)} ∈ Y if for all μ, 0 ≤ μ ≤ p, η(μ)ŷ_{μ} − b̂_{μ} ∈

′′s^{(2^{p−μ}m+2^{p−μ}k_{p}+2^{p−μ−1}k_{p−1}+···+2k_{μ+1})}Â′′_{(s)}. We also call an m-good approximation, m ≥ 0, just a good approximation.

If b and c both approximate the same formal product of generators of St_{n}(< s >^{−1} Â_{(s)}), then we have shown in Lemma 6.2 that b ·′′ St_{n}(A)′′ = c ·′′ St_{n}(A)′′. The next lemma proves that good approximations to equivalent elements of Y, i.e. to formal products which may be distinct from each other but are equivalent, are the same modulo ′′St_{n}(A)′′. Thus good approximations to an element of Ỹ = St_{n}(< s >^{−1} Â_{(s)}) are well-defined modulo ′′St_{n}(A)′′, i.e. as elements of St_{n}(< s >^{−1} A) / ′′St_{n}(A)′′.

6.4 Lemma Define a map

ϕ : Y ⟶ St_{n}(< s >^{−1} A) / ′′St_{n}(A)′′

by

y ↦ \{b = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(\frac{b_{μ}}{s^{k_{μ}}})^{η(μ)} | b good approximation to y \} · ′′St_{n}(A)′′

Let y_{1}, y_{2} ∈ Y and let z ∈ Y_{0}(< s >^{−1} Â_{(s)}), or let z be an element of the set \{ x_{ij}(\frac{t̂}{s^{k_{p+1}}} )^{ϵ}x_{ij}(\frac{t̂}{s^{k_{p+1}}} )^{−ϵ} | t̂ ∈ Â_{(s)}, ϵ = ±1\}.

Then

ϕ(y_{1}zy_{2}) = ϕ(y_{1}y_{2}).

Proof : Two good approximations to y represent the same coset of ′′St_{n}(A)′′, by Lemma 6.2, hence the map ϕ is well-defined. We must now show the following. Given a good aproximation b to y_{1}zy_{2}, there is a good approximation c to y_{1}y_{2} such that b and c determine the same coset of ′′St_{n}(A)′′. Clearly it suffices to show that b = c in St_{n}(< s >^{−1} A). We now proceed to show this.

Suppose y_{1} = ∏_{μ=0}^{p} y_{i(μ),j(μ)}(\frac{ŷ_{μ}}{s^{kμ}})^{η(μ)} and y_{2} = ∏_{ν=0}^{p′} y_{k(ν),l(μ)}(\frac{ŷ′_{ν}}{s^{iν}})^{η(ν)}. There are four

types of elements z to consider, three coming from Y_{0}(< s >^{−1} Â_{(s)}) and then the extra one in the other set of the lemma.

Suppose first z = x_{ij}(\frac{t̂}{s^{k_{p+1}}} )^{ϵ}x_{ij}(\frac{t̂}{s^{k_{p+1}}} )^{−ϵ}, ϵ = ±1. Then let b = ∏_{μ=0}^{p}(x_{i(μ),j(μ)}(\frac{b_{μ}}{s^{kμ}}))^{η(μ)}x_{ij}(\frac{ϵr}{s^{k_{p+1}}} )

x_{ij}(−\frac{ϵr}{s^{k_{p+1}}} ) ∏_{ν=0}^{p′}(x_{k(ν),l(ν)}(\frac{b′_{ν}}{s^{iν}}))^{η(ν)} be a good approximation for y_{1}zy_{2}. Let further

c = ∏_{μ=0}^{p}(x_{i(μ),j(μ)}(\frac{b_{μ}}{s^{kμ}}))^{η(μ)} ∏_{ν=0}^{p′}(x_{i(ν),j(ν)}(\frac{b′_{ν}}{s^{iν}}))^{η(μ)}. There is some m ≥ 0 such that

for all μ, 0 ≤ μ ≤ p, \frac{η(μ)ŷ_{μ}}{s^{kμ}} − \frac{b̂_{μ}}{s^{kμ}} ∈ ′′s^{(2^{p−μ}m+2^{p−μ}k_{p}+2^{p−μ−1}k_{p−1}+···+2k_{μ+1})}Â′′_{(s)};

moreover \frac{ϵt̂}{s^{k_{p+1}}} − \frac{r̂}{s^{k_{p+1}}} ∈ ′′s^{2^{p+1}m+2k_{p+1}} Â′′_{(s)} ; \frac{−ϵt̂}{s^{k_{p+1}}} + \frac{r̂}{s^{k_{p+1}}} ∈ ′′s^{2^{p+1}m+2k_{p+1}} Â′′_{(s)};

and for all ν, 0 ≤ ν ≤ p′ : \frac{η(ν)ŷ′_{ν}}{s^{iν}} − \frac{b̂′_{ν}}{s^{iν}} ∈ ′′s^{(2^{p−ν}m+2^{p−ν}k_{p}+2^{p−ν−1}k_{p−1}+···+2k_{ν+1})}Â′′_{(s)}.

The Steinberg relations hold in St_{n}(< s >^{−1} A), so that x_{ij}(\frac{ϵr}{s^{k_{p+1}}} )x_{ij}(−\frac{ϵr}{s^{k_{p+1}}} ) =

x_{ij}(\frac{ϵr}{s^{k_{p+1}}} )x_{ij}(\frac{ϵr}{s^{k_{p+1}}} )^{−1} = 1, giving b = ∏_{μ=0}^{p}(x_{i(μ),j(μ)}(\frac{b_{μ}}{s^{kμ}}))^{η(μ)} ∏_{ν=0}^{p′}(x_{i(ν),j(ν)}(\frac{b′_{ν}}{s^{iν}}))^{η(μ)},

i.e. b = c. But the above conditions on \frac{η(μ)ŷ_{μ}}{s^{kμ}} − \frac{b̂_{μ}}{s^{kμ}} respectively on \frac{η(ν)ŷ′_{ν}}{s^{iν}} − \frac{b̂′_{ν}}{s^{iν}}

imply that c = ∏_{μ=0}^{p}(x_{i(μ),j(μ)}(\frac{b_{μ}}{s^{kμ}}))^{η(μ)} ∏_{ν=0}^{p′}(x_{i(ν),j(ν)}(\frac{b′_{ν}}{s^{iν}}))^{η(μ)} is a good approxima-

tion to y_{1}y_{2}. For it suffices here that, as is clearly true, for all μ, 0 ≤ μ ≤ p : \frac{η(μ)ŷ_{μ}}{s^{kμ}} − \frac{b̂_{μ}}{s^{kμ}} ∈ ′′s^{(2^{p−μ}m+2^{p−μ}k_{p}+2^{p−μ−1}k_{p−1}+···+2k_{μ+1})}Â′′_{(s)};

and all ν, 0 ≤ ν ≤ p′ : \frac{η(ν)ŷ′_{ν}}{s^{iν}} − \frac{b̂′_{ν}}{s^{iν}} ∈ ′′s^{(2^{p−ν}m+2^{p−ν}k_{p}+2^{p−ν−1}k_{p−1}+···+2k_{ν+1})}Â′′_{(s)}.

In what follows we suppress for better legibility the inverted powers of s ∈ A.

Let z = x_{ij}(t̂+v̂)x_{ij}(−t̂)x_{ij}(−v̂) and b = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(b_{μ})x_{ij}(r +u)x_{ij}(−r)x_{ij}(−u)

∏_{ν=0}^{p′} x_{k(ν),l(ν)}(b′_{ν}) be a good approximation for y_{1}zy_{2} and let c = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(b_{μ}) ∏_{ν=0}^{p′} x_{k(ν),l(ν)}(b′_{ν}).

The Steinberg relations give x_{ij}(r + u)x_{ij}(−r)x_{ij}(−u) = x_{ij}(r + u)x_{ij}(−r − u) = 1,

so that b = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(b_{μ}) ∏_{ν=0}^{p′} x_{k(ν),l(ν)}(b′_{ν}). But c is a good approximation for

y_{1}y_{2} and we have just shown that b = c in St_{n}(< s >^{−1} A).

Let z = x_{ij}(t̂)x_{kl}(v̂)x_{ij}(−t̂)x_{kl}(−v̂). If b = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(b_{μ})x_{ij}(r)x_{kl}(u)x_{ij}(−r)

x_{kl}(−u) ∏_{ν=0}^{p′} x_{k(ν),l(ν)}(b′_{ν}) is a good approximation for y_{1}zy_{2}, then by the Steinberg

relations x_{ij}(r)x_{kl}(u)x_{ij}(−r)x_{kl}(−u) = [x_{ij}(r), x_{kl}(u)] = 1 in St_{n}(< s >^{−1} A), so

we get b = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(b_{μ}) ∏_{ν=0}^{p′} x_{k(ν),l(ν)}(b′_{ν}). But c = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(b_{μ}) ∏_{ν=0}^{p′} x_{k(ν),l(ν)}(b′_{ν})

is a good approximation for y_{1}y_{2} here also and we have just shown that b = c in the Steinberg group St_{n}(< s >^{−1} A).

Finally let z = x_{ij}(t̂)x_{jl}(v̂)x_{ij}(−t̂)x_{jl}(−v̂)x_{il}(−t̂v̂). Let b = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(b_{μ})x_{ij}(r)x_{jl}(u)

x_{ij}(−r)x_{jl}(−u)x_{il}(−ru) ∏_{ν=0}^{p′} x_{k(ν),l(ν)}(b′_{ν}) be a good approximation for y_{1}zy_{2} and

let c = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(b_{μ}) ∏_{ν=0}^{p′} x_{k(ν),l(ν)}(b′_{ν}). By the Steinberg relations x_{ij}(r)x_{jl}(u)x_{ij}(−r)x_{jl}(−u)x_{il}(−ru) =

[x_{ij}(r), x_{jl}(u)]·x_{il}(−ru) = x_{il}(ru)x_{il}(−ru) = 1 and hence b = ∏_{μ=0}^{p} x_{i(μ),j(μ)}(b_{μ}) ∏_{ν=0}^{p′} x_{k(ν),l(ν)}(b′_{ν}).

Again c is a good approximation for y_{1}y_{2} and we have just shown that b = c in the Steinberg group St_{n}(< s >^{−1} A).

□

6.5 Proposition The set map below

St_{n}(< s >^{−1} A)/′′St_{n}(A)′′ ⟶ St_{n}(< s >^{−1} Â_{(s)})/′′St_{n}(Â_{(s)})′′

is bijective.

Proof We already saw in the last paragraph of the proof of Theorem 6.1 that the map above is surjective. So it suffices to construct a retract of this map. But the map ϕ : Y ⟶ St_{n}(< s >^{−1} A) / ′′St_{n}(A)′′, y ↦ { good approximation to y } ·′′ St_{n}(A)′′ factors thanks to the conclusions of the Lemmas 6.2 and 6.4 through St_{n}(< s >^{−1} Â_{(s)}) by a diagramme

\xymatrix{Y \ar[rr]^{ϕ} \ar[rr] \ar[dr]_{χ} &  & St_{n}(< s >^{−1} A) / ′′St_{n}(A)′′ \\  & St_{n}(< s >^{−1} Â_{(s)}) . \ar[ur]^{ϕ′}}

The resulting map St_{n}(< s >^{−1} Â_{(s)})/′′St_{n}(Â_{(s)})′′ ⟶ St_{n}(< s >^{−1} A)/′′St_{n}(A)′′ is obviously a retract of the map in the lemma.

□

This completes the proof of Theorem 6.1. Acknowledgement I would like to thank Anthony Bak for introducing me to the topic and for his inestimable advice and the enormous amounts of time and aid of every kind he has given me so generously.

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e-mail : mundkur@mathematik.uni-bielefeld.de