DIMENSION THEORY AND NONSTABLE

K-THEORY FOR NET GROUPS

Anthony Bak, Alexei Stepanov

Fakultät für Mathematik Universität Bielefeld 33615 Bielefeld, Deutschland and Chair of Mathematics II Sankt-Petersburg Electrotechnical University 194356, Sankt-Petersburg, Russia

Abstract. The article applies concepts of structure and dimension in arbitrary categories to establish that nonstable net K_{1} of a net of finite Bass-Serre dimension is a nilpotent by Abelian group.

Introduction

This article together with the articles Mundkur [Mu] and Hazrat [H] apply concepts of structure and dimension in arbitrary categories to prove structural results for classical-like groups defined over rings and related objects. The current paper provides applications to net groups associated to the general linear group, the paper [Mu] to the general linear group, and the paper [H] to the general quadratic group. Each article including the current one has a short, self-contained introduction to a different aspect of the general theory of structure and dimension in categories and of group valued functors on categories with structure and dimension, which is developed in Bak [Bk4] and Bak [Bk5]. The current articles are intended to illustrate this theory. The main application in the present article is to prove that nonstable K_{1} of a major net of rank ⩾ 4 and finite Bass-Serre dimension is a nilpotent by Abelian group.

We describe the general theory as it pertains to the current paper. An arbitrary category 𝒞 is structured by fixing a class of commutative diagrams called structure diagrams and a class of functors called infrastructure functors on directed quasiordered sets with values in 𝒞, whose direct limits exist in 𝒞. In the current article

Typeset by AMS-TEX

Key words and phrases. General linear group, Elementary block diagonal subgroup, Normal subgroups, Steinberg group, Net subgroup.

The work on this article was partially supported by the INTAS 93–436 and DFG–RFBR 96-01-00092G. The second named author gratefully acknowledges the support of the SFB 343 an der Universität Bielefeld.

the diagrams are commutative squares. A function d : 𝒪bj(𝒞) → ℤ^{⩾0}∪{∞} is called a dimension function, if it satisfies a certain property called reduction relating it to the structure on 𝒞. Let τ : 𝒮 → 𝒢 denote a natural transformation of group valued functors 𝒮 and 𝒢 on 𝒞. The dimension filtration of τ on 𝒢 is defined by

𝒢^{i}(A) = ⋂_{A→B_{d(B)⩽i}} Ker(𝒢(A) → 𝒢(B)/ Im(𝒮(B) → 𝒢(B)))

for any integer i ⩾ 0 and object A of 𝒞. Let ℰ = Im(τ). The main result used in the current paper for group valued functors on a category with structure and dimension is the following: If τ is good with respect to the structure on 𝒞 then 𝒢^{0} ⩾ 𝒢^{1} ⩾ ⋯ ⩾ ℰ is a descending central series such that 𝒢^{i}(A) = ℰ(A) for all i ⩾ d(A). Moreover, if the coset space 𝒢(A)/ℰ(A) is an Abelian group for all 0dimensional objects A then the quotient functor 𝒢/𝒢^{0} takes its values in Abelian groups. In this case, the quotient functor 𝒢/ℰ is a nilpotent by Abelian group on all finite dimensional objects of 𝒞.

Let ((ν-nets)) denote the category of all nets corresponding to an equivalence relation ν on a subset of the natural numbers. Let ((major ν-nets)) denote the full subcategory of all major ν-nets. Let G : ((ν − nets)) → ((groups)), σ ↦ G(σ), denote the usual group valued functor associating to each ν-net σ, its group G(σ). Let E(ν, σ) denote the elementary subgroup of G(σ) and St(ν, σ) the Steinberg group of σ. The main results concerning ν-nets are obtained by exhibiting ((νnets)) as a category with structure and dimension whose dimension function is the Bass-Serre dimension of a net. It is then shown that the natural transformation π : St(ν,_{−}) → G(_{−}) is good on the category ((major ν-nets)) and that for any 0-dimensional ν-net σ, G(σ)/E(ν, σ) is an Abelian group. It follows now from the general theory above that the dimension filtration G ⩾ G^{0} ⩾ G^{1} ⩾ ⋯ ⩾ E(ν,_{−}) on G makes the quotient functor K_{1}G := G /E(ν,_{−}) into a nilpotent by Abelian group valued functor on finite dimensional major ν-nets. This allows one to deduce that each sandwich E(ν, σ) ⩽ H ⩽ C(ν, σ) in the sandwich classification theorem for subgroups H of the general linear group normalized by a block diagonal elementary subgroup, has the property that the quotient C(ν, σ)/E(ν, σ) is a G^{0}([ν] + σ)- nilpotent group for any finite dimensional ν-net σ.

The rest of the paper is organized as follows. In section 1, a certain group homomorphism χ is constructed, which plays a crucial role in section 2 in showing that the filtration 𝒢^{0} ⩾ 𝒢^{1} ⩾ ⋯ is a descending central series. When χ is bijective, it turns out that its inverse is the connecting map in a certain nonabelian Mayer-Vietoris sequence. This result is included for completeness, but is not necessary for proving the main results. Section 2 develops notions of structure and dimension in arbitrary categories and proves the main results for dimension filtrations on group valued functors. Section 3 recalls nets and net groups and establishes stable properties of these groups. Section 4 proves the injective stability theorem for the functor K_{1} of nets. The surjective stability theorem is known by work of Vavilov [Vv]. Section 5 studies nets over quasi-finite rings. In particular, it shows that elementary subgroups of net groups are normal and that the so-called standard commutator formula holds. Golubchik’s sandwich classification theorem is recalled. Section 6 exhibits the categories ((ν-nets)) and ((major ν-nets)) as categories with structure and dimension and shows that the natural transformation π : St(ν,_{−}) → G(_{−}) is good on the category ((major ν-nets)). A nonstable K_{2}–K_{1} Mayer-Vietoris sequence for nets is also proved. Section 7 puts together the results of sections 2, 5, and 6 to prove our main results concerning net groups.

1. The map χ and a nonabelian Mayer-Vietoris sequence

The materials in this section are taken in a selfcontained way from Bak [Bk2] and [Bk3] (unpublished). We begin by sharpening the Mayer-Vietoris sequences in[Bk2], following the exposition given in [Bk3]. In the second half of the section, we investigate the equivariant properties of the connecting map in the Mayer-Vietoris sequence. 1.1. Let ℛ be a category and let 𝒢, ℰ, 𝒮 be functors from ℛ to ((groups)). Suppose that for any object A ∈ ℛ, these functors satisfy the following conditions.

(i) ℰ(A) is a normal subgroup of 𝒢(A) and the inclusion map defines a natural transformation of functors.

(ii) 𝒮(A) is a central extension of ℰ(A) and the covering 𝒮(A) →^{π} ℰ(A) defines a

natural transformation of functors. 1.2. For each object A in ℛ, define the groups

K_{1}(A) = 𝒢(A)/ℰ(A) and K_{2}(A) = Ker(𝒮(A) → ℰ(A)).

Clearly K_{1} and K_{2} are functors from ℛ to ((groups)).

For a commutative square

(∗)  \begin{matrix}A & ⟶^{φ} & B \\ ^{ψ}↓ & & ↓^{ψ′} \\ C & ⟶^{φ′} & D \end{matrix}

in ℛ consider the corresponding squares of groups

(𝒢 − square) \begin{matrix}𝒢(A) & ⟶^{φ} & 𝒢(B) \\ ^{ψ}↓ & & ↓^{ψ′} \\ 𝒢(C) & ⟶^{φ′} & 𝒢(D) \end{matrix} (𝒮 − square) \begin{matrix}𝒮(A) & ⟶^{φ} & 𝒮(B) \\ ^{ψ}↓ & & ↓^{ψ′} \\ 𝒮(C) & ⟶^{φ′} & 𝒮(D). \end{matrix}

Factoring K_{2}(A), K_{2}(B) and K_{2}(C) out of the right hand square, one obtains the commutative square

(factor-square)  \begin{matrix}ℰ(A) & ⟶^{φ} & ℰ(B) \\ ^{ψ}↓ & & ↓^{ρ} \\ ℰ(C) & ⟶^{ω} & 𝒮(D)/(K_{2}(B) K_{2}(C)) . \end{matrix}

Note that we denote functorial group homomorphisms in the diagrams above with the same letters used for the corresponding morphisms in the category ℛ, e.g. we write φ instead of 𝒢(φ), 𝒮(φ) and ℰ(φ), but we introduce into the factor-square the new letters ω and ρ to denote the group homomorphisms induced from the functorial ones on the 𝒮-level. We shall use such notation when it cannot cause confusion. 1.3. A commutative diagram

K_{2}(B)  K_{1}(B)
↘ ↗ ↘
K_{2}(D) ⟶^{∂} K_{1}(A)  K_{1}(D)
↗ ↘ ↗
K_{2}(C)  K_{1}(C)

will be called a Mayer–Vietoris sequence if the following conditions are satisfied.

(i) The square is exact, i.e. the canonical homomorphism from K_{1}(A) to the pullback of the diagram K_{1}(C) → K_{1}(D) ← K_{1}(B) is surjective.

(ii) The intersection of the kernels of the maps K_{1}(A) → K_{1}(B) and K_{1}(A) →

K_{1}(C) coincides with Im(∂). (iii) The set {bc | b ∈ Im(K_{2}(B) → K_{2}(D)), c ∈ Im(K_{2}(C) → K_{2}(D))} formed

by the product of the images of K_{2}(B) and K_{2}(C) in K_{2}(D) coincides with Ker(∂).

The purpose of this section is to construct for certain commutative squares (∗) in ℛ, a Mayer–Vietoris sequence which is functorial over these squares. 1.4. Following [Bk2] we define the kinds of squares that will be needed. The square (∗) will be called:

(i) weak ℰ-fibred, if factor-square is a fibre square.

(ii) ℰ-surjective, if given b ∈ 𝒢(B) and c ∈ 𝒢(C) such that ψ′(b)φ′(c) ∈ ℰ(D), there are elements b′ ∈ ℰ(B) and c′ ∈ ℰ(C) such that

ψ′(b′)φ′(c′) = ψ′(b)φ′(c).

(iii) 𝒮-surjective, if given d ∈ K_{2}(D), there are elements b ∈ 𝒮(B) and c ∈ 𝒮(C)

such that d = ψ′(b)φ′(c). (iv) 𝒢-fibred, if 𝒢-square is a fibre square. The next lemma provides conditions which are easier to check in practice than

those in (1.4)(i)-(iii). It will be used in the Goodness Lemma 6.9.

1.5. Excision Lemma. Let (∗) be a 𝒢-fibred square and θ : 𝒮(B)/ Im 𝒮(φ) → 𝒮(D)/ Im 𝒮(φ′) the map of coset spaces induced by 𝒮-square.

If θ is injective and

Ker ℰ(φ′) ⩽ ℰ(ψ)( Ker ℰ(φ))

then the square (∗) is weak ℰ-fibred.

If θ is surjective then the square (∗) is ℰ-surjective and 𝒮-surjective.

Proof. Let b ∈ ℰ(B) and c ∈ ℰ(C) be such that ω(c) = ρ(b) and let b′ and c′ be preimages in 𝒮(B) and 𝒮(C), respectively. Obviously we can choose b′ and c′ such that 𝒮(φ′)(c′) = 𝒮(ψ′)(b′). We have θ(b′ Im 𝒮(φ)) = 1 · Im 𝒮(φ′) and since θ is injective, there is an element a′ ∈ 𝒮(A) such that 𝒮(φ)(a′) = b′. Let ā denote the image of a′ in ℰ(A). Since the element ℰ(ψ)(ā)^{−1}c belongs to Ker ℰ(φ′), it follows from the hypotheses in the lemma that it has a preimage ã ∈ Ker ℰ(φ). If a = āã then it is easy to check that ℰ(φ)(a) = b and ℰ(ψ)(a) = c. The uniqueness of such an element a follows immediately from condition (1.4)(iv). Thus we have shown that factor-square is fibred.

The surjectivity of θ immediately implies that each element from 𝒮(D) can be written as product 𝒮(φ)(b′)𝒮(ψ)(c′) for some b′ ∈ 𝒮(B), c′ ∈ 𝒮(C). The rest of the proof is very easy and will be left to the reader.

1.6. Let

ℰ̃ = 𝒢(φ)^{−1}(ℰ(B)) ∩ 𝒢(ψ)^{−1}(ℰ(C)).

Obviously ℰ(A) ⩽ ℰ̃. Consider the commutative diagram

ℰ(B) ⟶^{ρ} 𝒮(D)/(K_{2}(B) K_{2}(C))
φ↗  ↘
ℰ̃  ℰ(D) .
↘^{ψ}  ↗
ℰ(C) ⟶^{ω} 𝒮(D)/(K_{2}(B) K_{2}(C))

Define a function χ : ℰ̃ → 𝒮(D)/(K_{2}(B) K_{2}(C)) by the formula

χ(a) = ρφ(a^{−1})ωψ(a).

1.7. Lemma. χ is a homomorphism from ℰ̃ to K_{2}(D)/(K_{2}(B) K_{2}(C)). Suppose that (∗) is a 𝒢-fibred square.

If (∗) is 𝒮-surjective then χ is an epimorphism.

If (∗) is weak ℰ-fibred then Ker χ = ℰ(A).

Proof. Since 𝒢-square is commutative, the image of χ lies in the subgroup K_{2}(D)/(K_{2}(B) K_{2}(C)) which is central in 𝒮(D)/(K_{2}(B) K_{2}(C)) by (1.1)(ii). Hence

χ(a′)χ(a) = ρφ(a^{−1})χ(a′)ωψ(a) = ρφ(a^{−1})ρφ(a′^{−1})ωψ(a′)ωψ(a) = χ(a′a)

for any a, a′ ∈ ℰ̃. Thus χ is a homomorphism.

Let d̄∈ K_{2}(D)/(K_{2}(B) K_{2}(C)) and d be a preimage of d̄ in K_{2}(D). By (1.4)(iii) there are elements b ∈ 𝒮(B) and c ∈ 𝒮(C) such that d = ψ′(b)φ′(c). Hence, d̄ = ρπ(b)ωπ(c) (where π denotes as in (1.1) the natural transformation 𝒮 → ℰ). Since the image of d̄ in ℰ(D) is trivial, it follows that ℰ(ψ′)π(b)^{−1} = ℰ(φ′)π(c). By (1.4)(iv), there is an element a ∈ 𝒢(A) such that φ(a) = π(b)^{−1} and ψ(a) = π(c). Obviously a ∈ ℰ̃, since π(b)^{−1} ∈ ℰ(B) and π(c) ∈ ℰ(C). But χ(a) = ρφ(a^{−1})ωψ(a) = ρπ(b)ωπ(c) = d̄. Thus χ is surjective.

Clearly ℰ(A) ⩽ Ker(χ) because factor-square is commutative. Suppose ωψ(a) = ρφ(a) for some a ∈ ℰ̃. Since (∗) is weak ℰ-fibred, there is by definition (1.4)(i) an element a′ ∈ ℰ(A) such that φ(a′) = φ(a) and ψ(a′) = ψ(a). But, since 𝒢-square is a fibre square, a = a′ which proves the last assertion of the lemma.

Suppose that (∗) is 𝒢-fibred, weak ℰ-fibred, and 𝒮-surjective. Define the homomorphism ∂ : K_{2}(D) → K_{1}(A) as the composite map

K_{2}(D) ↠ K_{2}(D)/(K_{2}(B) K_{2}(C)) ⟶^{χ^{−1}} ℰ̃/ℰ(A) ↣ K_{1}(A).

1.8. Theorem. There is a functorial Mayer–Vietoris sequence (1.3) for 𝒢-fibred, weak ℰ-fibred, 𝒮-surjective, ℰ-surjective squares (∗).

Proof. The proof is absolutely the same as that of Theorem 5.31 of [Bk2] (using (1.7) instead of [Bk2, Lemma 5.29]).

In fact, exactness of the K_{1}-square follows from (1.4)(iv) and ℰ-surjectivity; exactness at the term K_{2}(D) follows from 𝒮-surjectivity; and exactness at K_{1}(A) follows from the condition that (∗) is weak ℰ-fibred. Of course, in the last two cases we must change the definition of χ′ in the obvious way if χ is not an isomorphism.

In subsequent sections, the Mayer-Vietoris sequence in (1.8) will not be fully used, but the homomorphism χ which is used to construct it, will play a crucial role. Furthermore it will be important that χ is equivariant in a certain sense. For the sake of precision and clarity, we define and establish the equivariance of χ in a purely group theoretic context. The arguments refine those given already above and are taken from [Bk3].

1.9. For the rest of this section, let

S_{A}⟶ S_{B}
↙  ↙
S_{C} ⟶ S_{D}
↓ ↓
G_{A}− ; ⟶ G_{B}
↓ ↙ ↓ ↙
G_{C}⟶ G_{D}

be a commutative cube of groups. For X = A, B, C or D, let E_{X} = Im(S_{X} → G_{X}) and K_{X} = Ker(S_{X} → G_{X}). Assume that K_{D} ⩽ center(S_{D}). Define the group Ẽ_{A} = {a ∈ G_{A} | image of a in G_{X} (X = B, C) lies in E_{X}}. Define the homomorphisms φ, ψ, ρ, and ω as in the commutative diagram

E_{B} ⟶^{ρ} S_{D}/(K_{B}K_{C})
φ↗  ↘
Ẽ_{A}  E_{D}
↘^{ψ}  ↗
E_{C} ⟶^{ω} S_{D}/(K_{B}K_{C})

and set

\begin{matrix}χ_{A} : Ẽ_{A} & → K_{D}/K_{B}K_{C} \\ a & ↦ (ρφ(a))^{−1}(ωψ(a)). \end{matrix}

Since K_{D} ⩽ center(S_{D}), it follows that χ_{A} is a group homomorphism.

1.10. Key Lemma. Suppose that the cube in (1.9) satisfies the following conditions.

(i) The group K_{D} is central in S_{D}.

(ii) The groups E_{X} are perfect for X = B, C.

(iii) The maps S_{B}/S_{A} → S_{D}/S_{C} and E_{C} → E_{D} are injective.

(iv) The G-square is fibred.

Let J_{A} ⩽ G_{A} be a subgroup satisfying the following conditions.

(v) Im(J_{A} → G_{X}) (X = B, C) normalizes E_{X}.

(vi) Im(J_{A} → G_{D}) ⩽ E_{D}. Lift in the obvious way the action of J_{A} on E_{D} by conjugation to an action of J_{A} on S_{D}.

Then J_{A} normalizes Ẽ_{A} and leaves each element of K_{D} fixed, the homomorphism χ_{A} : Ẽ_{A} → K_{D}/K_{B}K_{C} is J_{A}-equivariant, and Ker χ_{A} = E_{A}. In particular the action of J_{A} on Ẽ_{A}/E_{A} is trivial.

Proof. Since J_{A} normalizes E_{B} and E_{C}, it follows from the definition of Ẽ_{A} that J_{A} normalizes Ẽ_{A}. Since K_{D} ⩽ center(S_{D}), the obvious lifting to S_{D} of the conjugation action of J_{A} on E_{D} leaves each element of K_{D} fixed. Let J_{A} act on E_{B} and E_{C} by conjugation. To show that χ_{A} is J_{A}-equivariant, it suffices to show that all the homomorphisms in the second diagram in (1.9) are J_{A}-equivariant. The homomorphisms E_{B} → E_{D} and E_{C} → E_{D} are clearly J_{A}-equivariant. Since E_{B} and E_{C} are perfect and K_{D}/K_{B}K_{C} ⩽ center(S_{D}/K_{B}K_{C}), it follows that the homomorphisms ρ : E_{B} → S_{D}/K_{B}K_{C} and ω : E_{C} → S_{D}/K_{B}K_{C} are J_{A}-equivariant (cf. [M, lemma 5.4]). The remaining homomorphisms in the second diagram in (1.9) are obviously J_{A}-equivariant and thus χ_{A} is J_{A}-equivariant. From the J_{A}- equivariance of χ_{A} and the triviality of the J_{A}-action on K_{D}/K_{B}K_{C}, it follows that the J_{A}-action on Ẽ_{A}/ Ker χ_{A} is trivial. But the third and fourth conditions in the lemma show as in the proof of (1.7) that Ker χ_{A} = E_{A}.

There are frequently situations in which all of the hypotheses of (1.10) hold except the one that the map E_{C} → E_{D} is injective. In order to get around this problem and still have the final conclusion of (1.10) hold, one develops a relative version of the above.

Let G and Ḡ be groups with an action of G on Ḡ by automorphisms of Ḡ. The semidirect or smash product of this action is denoted by Ḡ ⋊ G. By definition, Ḡ ⋊ G is a group whose underlying set is the Cartesian product Ḡ × G and whose multiplication is given by (σ̄, σ)(ρ̄, ρ) = (σ̄(σ ∘ ρ̄), σρ). A precrossed module of the action is a group homomorphism f : Ḡ → G which is G-equivariant under the action of G on itself by conjugation. This implies of course that Im(f) is normal in G. The group Ḡ ⋊ G is called the smash product group associated to f. A precrossed module f : Ḡ → G is called a crossed module if the action of Ḡ on itself induced from that of G on Ḡ is the conjugation action. This implies of course that Ker(f) ⩽ center (Ḡ). A homomorphism of precrossed modules is a commutative diagram

Ḡ ⟶ G
f̄↓  ↓^{f}
H̄ ⟶ H

of groups such that the group homomorphism f̄ is G-equivariant. A homomorphism of crossed modules defines a homomorphism Ḡ⋊G → H̄ ⋊H, (σ̄, σ) ↦ (f̄(σ̄), f(σ)),

of groups. Let G ⟶^{1} G denote the crossed module defined by the action of G on itself by conjugation and the identity map 1 : G → G.

1.11. Corollary. [Bk3] Let

(SĀ → S_{A}) ⟶ (S_{B} ⟶^{1} S_{B})
↙  ↙
(SC̄ → S_{C}) ⟶ (S_{D} ⟶^{1} S_{D})
↓  ↓
(GĀ → G_{A})− ⟶(G_{B} ⟶^{1} G_{B})
↓  ↙  ↓  ↙
(GC̄ → G_{C})⟶(G_{D} ⟶^{1} G_{D})

be a commutative cube of precrossed modules such that the associated cube of groups

SĀ ⋊ S_{A} ⟶ S_{B} ⋊ S_{B}
↙  ↙
SC̄ ⋊ S_{C} ⟶ S_{D} ⋊ S_{D}
↓ ↓
GĀ ⋊ G_{A}− ⟶G_{B} ⋊ G_{B}
↓ ↙ ↓ ↙
GC̄ ⋊ G_{C}⟶G_{D} ⋊ G_{D}

and the subgroup J_{A} ⩽ G_{A} ⩽ GĀ ⋊ G_{A} of GĀ ⋊ G_{A} satisfy the assumptions of (1.10), except possibly the third and fourth assumptions. Let EX̄ = Im(SX̄ → GX̄) (X = A, C) and let ẼĀ = {ā ∈ GĀ | image of ā in G_{B} (resp. GC̄) lies in E_{B} (resp. EC̄)}. Assume the following.

(i) The maps S_{B}/SĀ → S_{D}/SC̄ and EC̄ → E_{D} are injective.

(ii) \begin{matrix}GĀ & ⟶ & G_{B} \\ ↓ & & ↓ \\ GC̄ & ⟶ & G_{D} \end{matrix} is a fibred subsquare of \begin{matrix}G_{A} & ⟶ & G_{B} \\ ↓ & & ↓ \\ G_{C} & ⟶ & G_{D} \end{matrix}

of normal subgroups and the action of G_{X} on GX̄ (X = A, C) is by conjugation.

(iii) The map ẼĀ/EĀ → Ẽ_{A}/E_{A} is surjective.

(iv) E_{A} is J_{A}-invariant.

Then the action of J_{A} on the coset space Ẽ_{A}/E_{A} is trivial. (No assertion is being made concerning the normality of E_{A} in Ẽ_{A}.)

Proof. Obviously ẼĀ is J_{A}-invariant. We show next that EĀ is J_{A}-invariant and normal in ẼĀ and that the action of J_{A} on ẼĀ/EĀ is trivial. Let KX̄ = Ker(SX̄ → GX̄) (X = A, C). Define χĀ : ẼĀ → K_{D}/K_{B}KC̄ and χ : ẼĀ ⋊ Ẽ_{A} → K_{D} ⋊ K_{D}/(K_{B} ⋊ K_{B})(KC̄ ⋊ K_{C}) as in (1.9). Then χ = χĀ ⋊ χ_{A}. The proof of (1.10) shows that χ is J_{A}-equivariant. Since the action of of J_{A} on K_{D} ⋊ K_{D} is trivial, it follows that the action of J_{A} on ẼĀ ⋊ Ẽ_{A}/ Ker χĀ ⋊ Ker χ_{A} is trivial. But the first and second assumptions in the corollary imply as in the proof of (1.7) that Ker χĀ = EĀ. Thus EĀ is J_{A}-invariant and normal in ẼĀ and the action of J_{A} on ẼĀ/EĀ is trivial. The map ẼĀ/EĀ → Ẽ_{A}/E_{A} of coset spaces is obviously J_{A}- equivariant and by the third assumption in the corollary, it is surjective. Thus the action of J_{A} on Ẽ_{A}/E_{A} is trivial.

2. Dimension Theory and group valued functors

The materials in this section are taken from Bak [Bk3] (unpublished). We define the notion of a category with structure and introduce one kind of dimension function on a category with structure. A category with structure equipped with a dimension function is called a category with dimension. Let τ : 𝒮 → 𝒢 be a natural transformation of group valued functors 𝒮, 𝒢 : 𝒞 → ((groups)) on a category 𝒞 with dimension. We define the dimension filtration 𝒢 = 𝒢^{−1} ⩾ 𝒢^{0} ⩾ 𝒢^{1} ⩾ ⋯ ⩾ 𝒢^{i} ⩾ ⋯ ⩾ Im(𝒮 → 𝒢) of τ on 𝒢 and show under certain conditions that 𝒢^{0} ⩾ 𝒢^{1} ⩾ ⋯ ⩾ 𝒢^{i} ⩾ ⋯ is a descending central series such that 𝒢^{i}(A) = Im(𝒮(A) → 𝒢(A)) whenever i ⩾ dimension(A).

Recall that a quasi-ordered set is a set I together with a reflexive, transitive relation ⩽ on I. A quasi-ordered set (I, ⩽) is called directed if given i, j ∈ I, there is a k ∈ I such that i ⩽ k and j ⩽ k. An equivalent definition of a quasi-ordered set is a category whose objects form a set and for any pair i, j of objects, Mor(i, j) has at most one element.

2.1. Definition. A category with structure is a category 𝒞 together with a class 𝒮(𝒞) of commutative squares in 𝒞 called structure squares and a class ℐ(𝒞) of functors called infrastructure functors whose domain categories are directed quasi-ordered sets and whose target category is 𝒞, satisfying the following conditions.

(i) 𝒮(𝒞) is closed under isomorphism of commutative squares. For each object A of 𝒞, the constant or trivial square

A ⟶^{1} A
1↓  ↓^{1}
A ⟶_{1} A

is in 𝒮(𝒞). (ii) ℐ(𝒞) is closed under isomorphism of functors. For each object A of 𝒞, the

constant or trivial functor F_{A} : {∗} → 𝒞, ∗ ↦ A, is in ℐ(𝒞), where {∗} denotes the directed quasi-ordered set with precisely one element ∗. For each (F : I → 𝒞) in ℐ(𝒞), the direct limit \underrightarrow{lim}_{I} F exists in 𝒞.

2.2. Definition. Let (𝒞, 𝒮(𝒞), ℐ(𝒞)) be a category with structure. Let d : 𝒪bj(𝒞) → ℤ^{⩾0} ∪ {∞} be a function which is constant on isomorphism classes of objects. Let A ∈ 𝒪bj(𝒞) such that 0 < d(A) < ∞. A d-reduction of A is a set

\begin{matrix}A & ⟶ & B_{i} \\ ↓ & & ↓ \\ C_{i} & ⟶ & D_{i} \end{matrix} (i ∈ I)

of structure squares where I is a directed quasi-ordered set and B : I → 𝒞, i ↦ B_{i}, is an infrastructure functor such that the following holds.

(i) If i ⩽ j ∈ I then the triangle

\begin{aligned}& & B_{j} \\ & ↗ \\ A & & ↑ \\ & ↘ \\ & & B_{i} \end{aligned}

commutes.

(ii) d(\underrightarrow{lim}_{I} B) = 0.

(iii) d(C_{i}) < d(A) for all i ∈ I. The function d is called a dimension function on (𝒞, 𝒮(𝒞), ℐ(𝒞)) if each object A of 𝒞 such that 0 < d(A) < ∞ has a d-reduction. In this case, the quadruple (𝒞, 𝒮(𝒞), ℐ(𝒞), d) is called a category with dimension.

In [Bk3], a more general concept of dimension function is developed, which allows conditions other than those above to be placed on d(\underrightarrow{lim}_{I} B_{i}) and the d(X_{i})’s (X =

B, C, D), e.g. d(\underrightarrow{lim}_{I} B_{i}) < d(A), d(C_{i}) < d(A), and d(D_{i}) < d(A).

A dimension function d on (𝒞, 𝒮(𝒞), ℐ(𝒞)) is called tame or universal, if the existence of a d-reduction

\begin{matrix}A & ⟶ & B_{i} \\ ↓ & & ↓ \\ C_{i} & ⟶ & D_{i} \end{matrix} (i ∈ I)

for A such that d(C_{i}) ⩽ n for all i ∈ I implies that d(A) ⩽ n + 1.

The next result is fundamental for appreciating the concept of a dimension function, but will not be applied in the sequel.

2.3. Theorem. [Bk4] Let (𝒞, 𝒮(𝒞), ℐ(𝒞)) be a category with structure and 𝒞^{0} a nonempty class of objects of 𝒞, closed under isomorphism. Then there is a universal dimension function δ on (𝒞, 𝒮(𝒞), ℐ(𝒞)) such that 𝒞^{0} is the class of 0-dimensional objects of δ and such that if d is any other dimension function on (𝒞, 𝒮(𝒞), ℐ(𝒞)) whose 0-dimensional objects are contained in 𝒞^{0} then δ ⩽ d, i.e. δ(A) ⩽ d(A) for all A ∈ 𝒪bj(𝒞).

The example below of a category with dimension will play a role in the current paper. 2.4. Example. Let 𝒞 denote the category of all algebras A_{R} over Noetherian commutative rings R such that A_{R} is module finite over R. A morphism A_{R} → A′_{R′} is a pair of ring homomorphisms A_{R} → A′_{R′} and R → R′ such that the diagram

R ⟶ R′
↓  ↓
A_{R} ⟶ A′_{R′}

commutes. If s ∈ R, let ⟨s⟩ denote the multiplicative set generated by s. For any Noetherian R-module M, let ⟨s⟩^{−1}M denote the module of ⟨s⟩-fractions of M and let M̂_{(s)} denote the completion \underleftarrow{lim}_{n∈ℕ} M/s^{n}M of M. Let ⟨s⟩^{−1}A_{R} denote the ⟨s⟩^{−1}R algebra (⟨s⟩^{−1}A_{R})_{⟨s⟩^{−1}R} and (Â_{R})_{(s)} the R̂_{(s)}-algebra ((Â_{R})_{(s)})R̂_{(s)}. Let 𝒮(𝒞) denote the class of commutative squares in 𝒞 isomorphic to a square of the kind

A_{R} ⟶  ⟨s⟩^{−1}A_{R}
↓  ↓
(Â_{R})_{(s)} ⟶ ⟨s⟩^{−1}(Â_{R})_{(s)} .

Such squares are called localization-completion squares. The Noetherian and finiteness conditions guarantee that they are pullback diagrams in 𝒞. If S ⩽ R is a multiplicative set, let S have the directed quasi-ordering defined by s ⩽ t ⇔ there is a u ∈ S such that su = t. Let F_{S,A_{R}} : S → 𝒞 denote the functor s ↦ ⟨s⟩^{−1}A_{R}. Clearly \underrightarrow{lim}_{S} F_{S,A_{R}} exists and is the S^{−1}R algebra (S^{−1}A_{R})_{S^{−1}R}. Let ℐ(𝒞) be the

class of all functors isomorphic to some functor F_{S,A_{R}}. Then Bass-Serre dimension BS and Jacobson-Krull dimension JK are dimension functions on (𝒞, 𝒮(𝒞), ℐ(𝒞)). A definition of Bass-Serre dimension is found in [Bk1] where it is shown that BS is a dimension function on (𝒞, 𝒮(𝒞), ℐ(𝒞)). Jacobson-Krull dimension is defined as follows. An ideal of R is called Jacobson if it is the intersection of the maximal ideals containing it. JK(A_{R}) is by definition the largest nonnegative integer n such that there is a chain 𝔓_{0} ⫋ 𝔓_{1} ⫋ ⋯ ⫋ 𝔓_{n} of prime Jacobson ideals 𝔓_{i} in R. It is not difficult to show that JK(A_{R}) = dimension (maxspec (R)) where maxspec (R) is the space of all maximal ideals of R under the Zariski topology. Using this fact, one can show easily that JK(A_{R}) ⩾ BS(A_{R}) and can adapt to JK the proof in [Bk1] that BS is a dimension function on (𝒞, 𝒮(𝒞), ℐ(𝒞)). It can also be shown, but is beyond the scope of the current article, that neither JK nor BS is the universal dimension function on (𝒞, 𝒮(𝒞), ℐ(𝒞)) for their class of 0-dimensional algebras, namely all A_{R} such that R is semilocal.

2.5. Definition-Lemma. Let (𝒞, 𝒮(𝒞), ℐ(𝒞), d) be a category with dimension. Let τ : 𝒮 → 𝒢 be a natural transformation of functors 𝒮, 𝒢 : 𝒞 → ((groups)). Let ℰ = Im(τ). Define the dimension filtration or d-filtration 𝒢 = 𝒢^{−1} ⩾ 𝒢^{0} ⩾ 𝒢^{1} ⩾ ⋯ ⩾ 𝒢^{i} ⩾ ⋯ ⩾ ℰ of τ on 𝒢 by

𝒢^{i}(A) = ⋂_{\begin{matrix}f:A→B \\ d(B)⩽i \end{matrix}} Ker(𝒢(A) → 𝒢(B)/ℰ(B))

for any i ⩾ 0. In general 𝒢(B)/ℰ(B) is just a coset space (not a group) and by definition, Ker(𝒢(A) → 𝒢(B)/ℰ(B)) = {a ∈ 𝒢(A) | image of a in 𝒢(B) lies in ℰ(B)}. It is clearly a subgroup of 𝒢(A). If there are no morphisms f : A → B such that d(B) ⩽ i then by definition 𝒢^{i}(A) = 𝒢(A). A trivial, but important consequence of the definition of 𝒢^{i} is that if i ⩾ dim(A) then 𝒢^{i}(A) = ℰ(A). Furthermore, if A and i are fixed and ℳ is a nonempty set of morphisms A → B such that for each B, dim(B) ⩽ i and 𝒢^{i}(A) = ⋂_{(A→B)∈ℳ} Ker(𝒢(A) → 𝒢(B)/ℰ(B)) then the induced

map 𝒢(A)/𝒢^{i}(A) → ∏_{(A→B)∈ℳ} 𝒢(B)/ℰ(B) is injective.

2.6. Definition. Let τ : 𝒮 → 𝒢 be as in (2.5). An arbitrary commutative square

\begin{matrix}A & ⟶ & B \\ ↓ & & ↓ \\ C & ⟶ & D \end{matrix}

is called τ-good, if the cube of groups

𝒮(A) ⟶ 𝒮(B)
↙  ↙
𝒮(C) ⟶ 𝒮(D)
↓ ↓
𝒢(A) − ; ⟶ 𝒢(B)
↓ ↙ ↓ ↙
𝒢(C) ⟶ 𝒢(D)

extends not necessarily functorially to a commutative cube of precrossed modules

(𝒮Ā → 𝒮(A)) ⟶(𝒮(B) ⟶^{1} 𝒮(B))
↙  ↙
(𝒮C̄ → 𝒮(C)) ⟶(𝒮(D) ⟶^{1} 𝒮(D))
↓  ↓
(𝒢Ā → 𝒢(A))− ⟶(𝒢(B) ⟶^{1} 𝒢(B))
↓  ↙  ↓  ↙
(𝒢C̄ → 𝒢(C)) ⟶(𝒢(D) ⟶^{1} 𝒢(D))

which satisfies all the conditions in (1.11) with J_{A} = 1 and Im(𝒮C̄ ⋊ 𝒮(C) → 𝒢C̄ ⋊ 𝒢(C)) is a normal subgroup of 𝒢C̄ ⋊ 𝒢(C). The natural transformation τ is called good, if 𝒮 and 𝒢 commute with arbitrary direct limits in 𝒞 and if each object A of 𝒞 such that 0 < d(A) < ∞ has a d-reduction

\begin{matrix}A & ⟶ & B_{i} \\ ↓ & & ↓ \\ C_{i} & ⟶ & D_{i} \end{matrix} (i ∈ I)

consisting of structure squares which are direct limits of τ-good squares.

2.7. Theorem. [Bk4] Let τ : 𝒮 → 𝒢 be a good natural transformation of functors 𝒮, 𝒢 : 𝒞 →((groups)) on a category with dimension (𝒞, 𝒮(𝒞), ℐ(𝒞), d) such that for each finite dimensional object A, ℰ(A) is normal in 𝒢(A). Then the dimension filtration 𝒢 ⩾ 𝒢^{0} ⩾ 𝒢^{1} ⩾ ⋯ ⩾ 𝒢^{n} ⩾ ⋯ ⩾ ℰ of τ on 𝒢 has the property that each 𝒢^{n} is normal in 𝒢, 𝒢^{n}(A) = ℰ(A) whenever n ⩾ d(A), and 𝒢^{0} ⩾ 𝒢^{1} ⩾ ⋯ is a descending central series i.e. for all objects A of 𝒞, the mixed commutator group [𝒢^{0}(A), 𝒢^{n}(A)] ⩽ 𝒢^{n+1}(A). Moreover, if 𝒢(A)/ℰ(A) is Abelian for all 0-dimensional objects A then 𝒢(A)/𝒢^{0}(A) is Abelian for all objects A.

Proof. The normality of each 𝒢^{n}(A) in 𝒢(A) follows from the definition of 𝒢^{n}(A) and from the normality of ℰ(B) in 𝒢(B) for each finite dimensional object B. Clearly ℰ(A) ⩽ 𝒢^{n}(A) and if d(A) ⩽ n then 𝒢^{n}(A) ⩽ Ker(𝒢(A) → 𝒢(A)/ℰ(A)) = ℰ(A). Thus d(A) ⩽ n implies 𝒢^{n}(A) = ℰ(A). To show that [𝒢^{0}(A), 𝒢^{n}(A)] ⩽ 𝒢^{n+1}(A)

(n ⩾ 0), it suffices, by the last assertion in (2.5), to consider the case d(A) ⩽ n + 1. If d(A) ⩽ n then 𝒢^{n}(A) = ℰ(A) = 𝒢^{n+1}(A) and the inclusion [𝒢^{0}(A), 𝒢^{n}(A)] ⩽ 𝒢^{n+1}(A) follows from the normality of ℰ(A). Suppose d(A) = n + 1. Let x ∈ 𝒢^{0}(A) and y ∈ 𝒢^{n}(A). We must show that [x, y] ∈ 𝒢^{n+1}(A). Let

\begin{matrix}A & ⟶ & B_{i} \\ ↓ & & ↓ \\ C_{i} & ⟶ & D_{i} \end{matrix} (i ∈ I)

be a d-reduction of A consisting of structure squares which are direct limits of τgood squares. There is canonical morphism A → \underrightarrow{lim}_{I} B, thanks to (2.2) (i). Since

d(\underrightarrow{lim}_{I} B) = 0 by (2.2) (ii), both x and y vanish in 𝒢(\underrightarrow{lim}_{I} B)/ℰ(\underrightarrow{lim}_{I} B). Since 𝒮 and

𝒢 commute with limits of infrastructure functors, so do ℰ and 𝒢/ℰ. Thus there is a j ∈ I such that x and y vanish in 𝒢(B_{j})/ℰ(B_{j}). Let ℰ̃(A) = {z ∈ 𝒢(A) | z vanishes in 𝒢(B_{j})/ℰ(B_{j}) and 𝒢(C_{j})/ℰ(C_{j})}. Since d(C_{j}) ⩽ n by (2.2) (iii), y vanishes in 𝒢(C_{j})/ℰ(C_{j}). Thus y ∈ ℰ̃(A). Since

A ⟶ B_{j}
↓  ↓
C_{j} ⟶ D_{j}

is a direct limit of τ-good squares and since 𝒮 and 𝒢 commute with arbitrary direct limits in 𝒞, we can assume that the square above is τ-good instead of a structure square and that x and y vanish in 𝒢(B_{j})/ℰ(B_{j}), and y ∈ ℰ̃(A). Let J(A) denote the subgroup of 𝒢(A) generated by x. Since the square above is τ-good, there is a cube of precrossed modules

(𝒮Ā → 𝒮(A)) ⟶(𝒮(B_{j}) ⟶^{1} 𝒮(B_{j}))
↙  ↙
(𝒮C̄_{j} → 𝒮(C_{j})) ⟶(𝒮(D_{j}) ⟶^{1} 𝒮(D_{j}))
↓  ↓
(𝒢Ā → 𝒢(A))− ⟶(𝒢(B_{j}) ⟶^{1} 𝒢(B_{j}))
↓  ↙  ↓  ↙
(𝒢C̄_{j} → 𝒢(C_{j})) ⟶(𝒢(D_{j}) ⟶^{1} 𝒢(D_{j}))

such that J(A) and this cube satisfy the hypotheses of (1.11). By the conclusion of (1.11), the action of J(A) on ℰ̃(A)/ℰ(A) by conjugation is trivial. Thus [x, y] ∈ ℰ(A) = 𝒢^{n+1}(A).

The last assertion of the theorem follows immediately from the last assertion of (2.5).

The theorem above says that the nilpotent class of 𝒢^{0}(A)/ℰ(A) is ⩽ d(A). Let N ∈ ℤ^{⩾0}. If 𝒢^{0}(A) = ℰ(A) whenever d(A) ⩽ N then it turns out that the nilpotent class of 𝒢^{0}(A)/ℰ(A) is ⩽ |d(A) − N|. One proves this in a trivial way, using dimension shifting which is defined as follows.

2.8. Definition. Let (𝒞, 𝒮(𝒞), 𝒮(𝒞), d) be a category with dimension. Let N ∈ ℤ^{⩾0}. For an object A of 𝒞, define

d[−N](A) = \{ \begin{matrix}0 & if d(A) ⩽ N \\ d(A) − N & if d(A) ⩾ N. \end{matrix}

Obviously (𝒞, 𝒮(𝒞), 𝒮(𝒞), d[−N]) is again a category with dimension.

2.9. Corollary. Let (𝒞, 𝒮(𝒞), ℐ(𝒞), d) be a category with dimension and τ : 𝒮 → 𝒢 a good natural transformation of functors 𝒮, 𝒢 : 𝒞 → ((groups)). Let N ∈ ℤ^{⩾0}. Let 𝒢 = 𝒢^{[−1]} ⩾ 𝒢^{[0]} ⩾ ⋯ ⩾ 𝒢^{[n]} ⩾ ⋯ ⩾ ℰ be the dimension filtration of τ on 𝒢 with respect to the dimension function d[−N]. Then the conclusions of (2.7) are valid for the filtration above. In particular 𝒢^{[n]}(A) = ℰ(A) whenever n ⩾ d[−N](A).

Proof. The corollary is an immediate consequence of (2.7).

3. Nets and net subgroups

3.1. Let J be a subset of the set ℕ of natural numbers, of order |J| = n (where n can be infinite) and let ν be an equivalence relation on J. Following [BV2], we denote by h(ν) the minimal order (possibly infinite) of the equivalence classes of ν. h(ν) is called the rank of ν. Recall that a square table σ = (σ_{ij})_{i,j∈J} is called a net over a ring R, with index set J if each σ_{ij} is an additive subgroup of R and σ_{ij}σ_{jk} ⊆ σ_{ik} for all i, j, k ∈ J. Let M(n, R) denote the ring, without unit element if n = ∞, of all n × n matrices with coefficients in R such that each row and each column contains only a finite number of nonzero elements. Enumerate the entries of a matrix in M(n, R) by elements in J × J. We shall usually identify a net σ with its net ring M(σ) which by definition is the subring of M(n, R) consisting of all matrices a such that a_{ij} ∈ σ_{ij} for all i, j ∈ J. Even if n is finite, M(σ) does not necessarily have a unit. A net σ is called a ν-net,if all σ_{ij} are ideals in R and

σ_{ij} = σ_{kl} whenever i ∼^{ν} k and j ∼^{ν} l.

3.2. The principal net subgroup G(σ) is the largest subgroup of GL(n, R) such that each matrix a ∈ G(σ) has the property that a_{ij} ≡ δ_{ij} mod σ_{ij} where δ_{ij} = 1 if i = j and δ_{ij} = 0 if i ≠ j.

Let e denote the identity matrix and e^{ij} the matrix with 1 in position (i, j) and 0 elsewhere. Let E(σ) denote the group generated by all elementary transvections t_{ij}(ξ) = e+ξe^{ij} where i ≠ j and ξ ∈ σ_{ij}. Denote by [ν, R] or simply [ν] the following net:

[ν]_{ij} = \{ \begin{matrix}R, if i ∼^{ν} j, \\ 0, otherwise. \end{matrix}

The net subgroups G(ν) = G(ν, R) := G([ν, R]) and E(ν) = E(ν, R) := E([ν, R]) corresponding to this net are called the principal and elementary block-diagonal groups, respectively. Thus the elementary block diagonal group E(ν) over a ring R is the subgroup of GL(n, R) generated by all elementary transvections t_{ij}(ξ) such that i ≠ j, i ∼^{ν} j, ξ ∈ R. The elementary net subgroup E(ν, σ) for a net σ corresponding to an equivalence ν is the normal closure E(σ)^{E(ν)} of the subgroup E(σ) by E(ν).

3.3. Clearly E(ν, σ) ⩽ G(σ) if and only if σ is a ν-net. Clearly G(σ) is normalized by E(ν) if and only if σ is a ν-net. For a ν-net σ, denote by σ + [ν] the net with ideals (σ + [ν])_{ij} = σ_{ij} if i ≁^{ν} j and (σ + [ν])_{ij} = R otherwise. If σ = σ + [ν] then σ is called a major ν-net. Clearly if σ is a major ν-net then E(σ) = E(ν, σ).

For the rest of this article, all nets are ν-nets.

Define C(ν, σ) to be the largest subgroup C of GL(n, R) such that

[C, E([ν] + σ)] ⩽ G(σ).

If σ is major then C(ν, σ) obviously coincides with the normalizer N(σ) of the principal net subgroup G(σ) in GL(n, R). On the other hand, if ν is the trivial equivalence (i.e. i ∼^{ν} j for all i, j ∈ J) then σ is a constant net (i.e. σ_{ij} = I for some ideal I and all i, j ∈ J) and E(ν, σ) = E(n, R, I), G(σ) = GL(n, R, I) and C(ν, σ) = C(n, R, I) are the corresponding relative subgroups of level I. In the above situation, σ is major means I = R.

The sandwich classification theorem shows that a subgroup of GL(n, R) is normalized by E(ν) if and only if it fits into a sandwich E(ν, σ) ⩽ … ⩽ C(ν, σ) for some ν-net σ. The subgroup G(σ) divides the sandwich E(ν, σ) ⩽ … ⩽ C(ν, σ) into two parts. We shall consider only the group G(σ)/E(ν, σ) in situations when E(ν, σ) is normal in G(σ). Denote this group by K_{1}(ν, σ). The kind of behavior we expect for net K_{1}- and K_{2}-groups is described in the next two propositions in the special case h(ν) = ∞.

3.4. Proposition. Let ν be an equivalence on ℕ with a finite number of equivalence classes each of infinite order.

(i) For a subgroup H ⩽ GL(R) normalized by E(ν), there exists a unique ν-net σ such that E(ν, σ) ⩽ H ⩽ G(σ).

(ii) [G(σ), G(ν)] = [G(σ), G([ν] + σ)] = E(ν, σ).

Proof. The second assertion of the proposition follows in the usual way from the block version of Whitehead lemma. We leave to the reader the formulation and proof of the block version. The first assertion follows easily from lemma 2 of [BV1] or lemma 3 of [S2].

The proposition shows that if h(ν) = ∞ then:

– C(ν, σ) coincides with G(σ).

– In terms of [S1], E(ν) is strongly polynormal in GL(R) and the subgroups E(ν, σ) as σ ranges over all ν-nets account for all E(ν)-perfect subgroups of GL(R).

– K_{1}(ν, σ) = G(σ)/E(ν, σ) lies in the center of G([ν] + σ)/E(ν, σ). In partic-

ular, the group K_{1}(ν, σ) is Abelian. 3.5. Let σ ⩾ [ν] be a major ν-net over a ring R, with index set J. Define the Steinberg group St(ν, σ) to be the group with generators x_{ij}(ξ), i ≠ j ∈ J, ξ ∈ σ_{ij} and the ordinary Steinberg relations: For any i, j, k, l ∈ J, i ≠ j, k ≠ l, i ≠ l, j ≠ k, one has:

(1) x_{ij}(ξ)x_{ij}(η) = x_{ij}(ξ + η).

(2) [x_{ij}(ξ), x_{kl}(η)] = 1.

(3) [x_{ij}(ξ), x_{jk}(η)] = x_{ik}(ξη). Let St(ν) = St(ν, R) := St([ν, R]). For an arbitrary ν-net σ, define St(σ) to be the subgroup of St(ν, σ +[ν]) generated by all generators x_{ij}(ξ) such that i ≠ j ∈ J and ξ ∈ σ_{ij}, and define St(ν, σ) to be the normal closure of St(σ) in St(ν, σ + [ν]), or equivalently the normal closure St(σ)^{St(ν,R)} of St(σ) by the subgroup Im(St(ν, R) → St(ν, σ + [ν])) of St(ν, σ + [ν]).

Let π denote the homomorphism π : St(ν, σ) → E(ν, σ), x_{ij}(ξ) ↦ t_{ij}(ξ), and define K_{2}(ν, σ) to be the kernel of π.

3.6. Proposition. Let R be a ring, ν be an equivalence on J and σ a major ν-net over R. If h(ν) ⩾ 5 then the group St(ν, σ) is centrally closed (i.e. any central extension of this group splits). If h(ν) = ∞ then St(ν, σ) is the universal perfect central extension of E(ν, σ).

Proof. The proof is absolutely the same as in [M, § 5]. (Note that to establish perfectness, we use σ ⩾ [ν].) The rest of the paper is devoted to investigating K_{1} and K_{2} of nets with finite index sets.

4. In the stable range.

Throughout this section, R denotes a ring and sr R its stable rank.

The standard description as in (3.4)(i) of E(ν)-normal subgroups of GL(n, R) is not known under a stable rank condition on rings (it was proved in [VvS] only under a weaker condition). Nevertheless the structure of the standard sandwiches is similar to that described in the previous section.

The normality of E(ν, σ) in N(σ) was proved in [VvS] for any net σ ⩾ [ν] whenever h(ν) ⩾ sup(sr R + 1, 3). Together with Corollary 5.2 this implies the commutator formula [C(ν, σ), E([ν] + σ)] = E(ν, σ). In this section, we shall prove a stronger commutator formula (4.3)(iii). The standard commutator formula will be obtained in the next section under a weaker condition. 4.1. In the rest of this section, ν will be an equivalence on ℕ and σ a ν-net over a ring R. Denote by ν^{(kh)} the equivalence on the set J = {k, …, h} got by restricting ν to J and by σ^{(kh)}, the ν^{(kh)}-net with index set J such that σ_{ij}^{(kh)} = σ_{ij} for all i, j ∈ J. If k = 1, we write ν^{(h)} and σ^{(h)} instead of ν^{(1h)} and σ^{(1h)}, respectively. For an equivalence χ, denote by |χ(m)| the order of the equivalence class of m.

4.2. Theorem. If |ν^{(n+1)}(n + 1)| ⩾ sr R + 1 then

G(σ^{(n+1)}) = E(ν^{(n+1)}, σ^{(n+1)}) G(σ^{(n)}).

If h(ν^{(n)}) ⩾ sr R and E(ν^{(n)}, σ^{(n)}) is normal in G(σ^{(n)}) then the canonical homomorphisms

K_{1}(ν, σ^{(n)}) ⟶ K_{1}(ν, σ^{(n+1)}) ⟶ ⋯ ⟶ K_{1}(ν, σ)

are surjective.

Proof. The proof is easy (see [Vv1]).

4.3. Theorem. (Injective stability for K_{1})

(i) If |ν^{(n+1)}(n + 1)| ⩾ sr R + 2 then

E(ν^{(n)}, σ^{(n)}) = G(σ^{(n)}) ∩ E(ν^{(n+1)}, σ^{(n+1)}).

(ii) If h(ν^{(n)}) ⩾ sr R + 1 and the set {1, …, n} has nonempty intersection with each equivalence class of ν then K_{1}(ν, σ^{(n)}) ≅ K_{1}(ν, σ).

(iii) If χ is an equivalence with h(χ) ⩾ sr R + 1 and τ is a χ-net over R then

[G(τ), G([χ] + τ)] = E(χ, τ).

Items (ii) and (iii) follow easily from (i). The proof of (i) will be given below and is based on ideas in [SuTu]. In the special case of the general linear group, it will appear in [Vv2]. The key step in [Vv2] and in our proof is the so-called Dennis–Vaserstein decomposition.

4.4. We introduce the following notation. Let a be a matrix and τ a net. Then – R^{n}(^{n}R) is the set of all rows (columns) of length n over R.

– a_{i∗} (a_{∗i}) is the i-th row (column) of a matrix a.

– a′_{i∗} (a′_{∗i}) is the i-th row (column) of a matrix a^{−1}.

– τ_{i∗} = {v ∈ R^{n}| v_{j} ∈ τ_{ij}}.

– τ_{∗i} = {u ∈ ^{n}R| u_{j} ∈ τ_{ji}}.

– t_{i∗}(v) = ∏_{j≠i} t_{ij}(v_{j}) where v ∈ R^{n}.

– t_{∗i}(u) = ∏_{j≠i} t_{ji}(u_{j}) where u ∈ ^{n}R.

4.5. Without lost of generality we may suppose that 1 ∼^{ν} n + 1. Consider the subgroups P = P (ν^{(n+1)}, σ^{(n+1)}) and Q = Q(ν^{(n+1)}, σ^{(n+1)}) of the group H = E(ν^{(n+1)}, σ^{(n+1)}) defined as follows:

\begin{aligned}P & = \{( \begin{matrix}1 & 0 \\ s & a \end{matrix} ) ∈ H| s ∈ ^{n}R, a ∈ E(ν^{(2,n+1)}, σ^{(2,n+1)})\} \\ Q & = \{( \begin{matrix}b & 0 \\ v & 1 \end{matrix} ) ∈ H| v ∈ R^{n}, b ∈ E(ν^{(n)}, σ^{(n)})\} . \end{aligned}

4.6. Theorem. (Dennis–Vaserstein decomposition) Let |ν^{(n+1)}(n + 1)| ⩾ sr R + 2. Then every element g ∈ E(ν^{(n+1)}, σ^{(n+1)}) can be written in the form g = yt_{1,n+1}(λ)z, where y ∈ P, λ ∈ σ_{11}, z ∈ Q.

We start with some lemmas.

4.7. Lemma. Suppose that |ν^{(n+1)}(n+1)| ⩾ sr R+2. Let u ∈ σ_{∗1}^{(n+1)}, w ∈ σ_{1∗}^{(n+1)}, w_{n+1} = 0, wu = 1 and λ ∈ σ_{11}. Then there exists x ∈ Q such that (xu)_{n+1} = 0 and t_{1,n+1}(−λ)xt_{1,n+1}(λ) ∈ P.

Proof. Since |ν^{(n)}(1)| ⩾ sr R + 1 and the column

(w_{1}u_{1}, u_{k}, w_{j}u_{j})_{k∼^{ν}1, j≁^{ν}1, 2⩽j,k⩽n}^{t}

is unimodular, there exists an α_{k} ∈ R such that the column (u_{k} + α_{k}w_{1}u_{1}, u_{j})^{t} is also unimodular. Hence for some β_{i} ∈ R (2 ⩽ i ⩽ n)

∑_{k∼^{ν}1, k≠1} u_{n+1}β_{k}(α_{k}w_{1}u_{1} + u_{k}) + ∑_{j≁^{ν}1} u_{n+1}β_{j}w_{j}u_{j} = u_{n+1}.

Clearly the elements β′_{k} = −u_{n+1}β_{k} are in σ_{n+1,n+1} = σ_{n+1,k} for k ∼^{ν} 1 and

β′_{j} = −u_{n+1}β_{j}w_{j} are in σ_{n+1,1}σ_{1j} ⊆ σ_{n+1,j} for j ≁^{ν} 1. Thus the matrices

x_{1} = ∏_{k∼^{ν}1, k≠1} t_{k1}(α_{k}w_{1}) and x_{2} = ∏_{i=2}^{n} t_{n+1,i}(β′_{i})t_{1i}(λβ′_{i})

are in Q. It is easy to see that (x_{1}x_{2}u)_{n+1} = 0 and t_{1,n+1}(−λ)x_{h}t_{1,n+1}(λ) ∈ P for h = 1, 2. Hence x = x_{1}x_{2} fits the assertion of the lemma.

Let X denote the subset in E(ν^{(n+1)}, σ^{(n+1)}) consisting of those elements g which can be expressed in the form g = yt_{1,n+1}(λ)z, where y ∈ P, λ ∈ σ_{11}, z ∈ Q.

4.8. Lemma. Assume that |ν^{(n+1)}(n + 1)| ⩾ sr R + 2. Fix j ∼^{ν} 1. Then every element g = yt_{1,n+1}(λ)z ∈ X can be expressed in the form g = ȳt_{1,n+1}(λ̄)z̄ as above where in addition z̄_{n+1,j} = 0.

Proof. Let u = z_{∗j} ∈ σ_{∗j} and w = z′_{j∗} ∈ σ_{j∗}. By the previous lemma, there exists a matrix x ∈ Q such that (xu)_{n+1} = 0 and t_{1,n+1}(−λ)xt_{1,n+1}(λ) ∈ P. Then

g = (yt_{1,n+1}(λ)x^{−1}t_{1,n+1}(−λ))t_{1,n+1}(λ)(xz)

is the desired factorization.

We are now ready to prove Dennis–Vaserstein decomposition.

Proof. We shall prove that X is normalized by E(ν^{(n+1)}). Since

Q^{E(ν^{(n+1)})} = E(ν^{(n+1)}, σ^{(n+1)}),

this will imply X = E(ν^{(n+1)}, σ^{(n+1)}).

Fix j ∼^{ν} 1, j ≠ 1, j ≠ n + 1. Clearly E(ν^{(n+1)}) is generated by all transvections

t_{ik}(ξ) with ξ ∈ R and either i ∼^{ν} k ≁^{ν} 1 or i = j ≠ k, k ∼^{ν} 1 or k = j ≠ i, i ∼^{ν} 1. Obviously if i ≠ 1 and k ≠ n + 1 then t_{ik}(ξ) normalizes X. Thus we only have to show that

c = (yt_{1,n+1}(λ)z)^{t_{j,n+1}(ξ)} ∈ X

for all y ∈ Q, z ∈ P, λ ∈ σ_{11} and ξ ∈ R (the case (i, k) = (1, j) can be proved in the same way).

By the last lemma, one may suppose that z_{n+1,j} = 0. Straightforward calculation shows that

c = [y ∏_{i=2}^{n} t_{i,n+1}((z_{ij} − δ_{ij})ξ)] t_{1,n+1}(λ + z_{1j}ξ) [( ∏_{i=1}^{n} t_{i,n+1}(−z_{ij}ξ))zt_{j,n+1}(ξ)] .

Since P is normalized by t_{j,n+1}(ξ) and z_{ij} − δ_{ij} ∈ σ_{ij} = σ_{i,n+1}, the factor in the first square brackets belongs to P. It is easy to verify that the factor in the last

square brackets is of the form ( \begin{matrix}a & 0 \\ v & 1 \end{matrix} ), where z = ( \begin{matrix}b & 0 \\ v & 1 \end{matrix} ) and a = b ∏_{i=1}^{n} t_{ji}(ξv_{i}).

Therefore it belongs to Q. Hence c ∈ X.

An analogue of this result holds also for the Steinberg group and allows one to prove surjective stability for net K_{2} (surjective stability for K_{2} is practically equivalent to injective stability for K_{1}, see [SuTu] for the case of GL(n, R)).

Proof of injective stability of K_{1}. It is obvious that E(ν^{(n)}, σ^{(n)}) ⊆ GL(σ^{(n)}) ∩ E(ν^{(n+1)}, σ^{(n+1)}). We want to prove the opposite inclusion. Indeed, take a matrix g ⊕ 1 ∈ GL(σ^{(n)}) ∩ E(ν^{(n+1)}, σ^{(n+1)}). Then by the decomposition theorem (4.6), it can be expressed as a product

( \begin{matrix}g & 0 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}1 & 0 & 0 \\ y^{21} & y^{22} & y^{23} \\ y^{31} & y^{32} & y^{33} \end{matrix} ) ( \begin{matrix}1 & 0 & λ \\ 0 & e & 0 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}z^{11} & z^{12} & 0 \\ z^{21} & z^{22} & 0 \\ z^{31} & z^{32} & 1 \end{matrix} )

where the matrices y, t_{1,n+1}(λ) and z are written in block form with respect to the partition (1, n − 1, 1). Multiplying the matrices on the right, we get λ = 0, y^{23} = 0 and y^{33} = 1. Hence, 1 ⊕ y^{22} ∈ E(ν^{(n)}, σ^{(n)}). Thus by definition of P and Q,

( \begin{matrix}y^{22} & y^{23} \\ 0 & 1 \end{matrix} ) ∈ E(ν^{(2,n+1)}, σ^{(2,n+1)}) and b = ( \begin{matrix}z^{11} & z^{12} \\ z^{21} & z^{22} \end{matrix} ) ∈ E(ν^{(n)}, σ^{(n)}).

Consequently g = ( \begin{matrix}1 & 0 \\ y^{21} & y^{22} \end{matrix} ) b = ( \begin{matrix}1 & 0 \\ y^{21} & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ 0 & y^{22} \end{matrix} ) b ∈ E(ν^{(n)}, σ^{(n)}) Q.E.D.

5. Over a quasi-finite ring

Following [Bk1], we call a ring quasi-finite, if it is a direct limit of module finite rings. In this section, we shall prove the standard classification of E(ν)-normal subgroups GL(n, A) over quasi-finite rings A, where ν is an equivalence on the set J = {1, ⋯, n} and h(ν) ⩾ 3. It will also be shown that K_{2}(ν, σ) is central in St(ν, σ) for any ν-net σ over a quasi-finite ring.

The standard classification theorem breaks into two parts, the sandwich theorem and the standard net commutator formula. The former follows by a direct limit argument from the sandwich theorem of I. Z. Golubchik [G] over module finite rings. So there is nothing to do here. The proof of the commutator formula over quasi-finite rings is reduced also by a direct limit argument to the case of a module finite ring. Module finite rings A over a commutative ring R have the property that if M is a maximal ideal in R and S = R \ M then S^{−1}A is semilocal (by definition, S^{−1}A/(Jacobson radical(S^{−1}A)) is semisimple) and thus its stable rank sr(S^{−1}A) = 1. We shall prove the commutator formula over R-algebras A such that given any maximal ideal M of R, there is a multiplicative set S ⊆ R \ M such that sup(sr(S^{−1}A) + 1, 3) ⩽ h(ν). The proof will use a localization method of Bak [Bk1]. Before beginning the proof, we want to reduce to the case where 1 ∈ σ_{ii} for any i ∈ J = {1, ⋯, n}.

5.1. Lemma. Suppose that subgroups D and H ⩽ C of a group G satisfy the following conditions:

(i) D is perfect, i.e. [D, D] = D.

(ii) H is D-perfect, i.e. [H, D] = H.

(iii) C is normalized by D.

(iv) D^{C} ⩽ HD.

(v) [C ∩ D, D] = D ∩ H.

Then [C, D] = [C, DH] = H.

Proof. First, note that [CD, HD] ⩽ HD by (ii) and (iv) and that the converse inclusion holds by (i) and (ii). In particular, HD is normal in CD. We are interested in the subgroup L = [C, D]. It is contained in C by (iii) and in [CD, HD] = HD. It follows that L ⩽ HD ∩ C = H(C ∩ D), and hence [L, D] ⩽ [H, D][C ∩ D, D] ⩽ H by (ii) and (v). On the the other hand, [L, D] ⩾ H by (ii). Lemma 1 from [S1] says that if D is perfect and [[C, D], D] is normal in C then [[C, D], D] = [C, D]. But

[[C, D], D] = [L, D] = [L, D^{DL}] = [L, D^{DH}] = [L, DH]

and the last group is normal in CD because L and DH are. Thus [C, D] = H. It follows that H is normalized by C and hence [C, DH] = H.

Recall that if σ is a net then by definition, N(σ) is the normalizer in GL(n, A) of G(σ).

5.2. Corollary. Let σ be a ν-net such that h(ν) ⩾ 3. Suppose that E(ν)^{N([ν]+σ)} ⩽ E([ν] + σ) and the standard commutator formula [C(|ν(i)|, R, σ_{ii}), E(|ν(i)|, R)] = E(|ν(i)|, R, σ_{ii}) holds for any i ∈ J. Then

[C(ν, σ), E(ν)] = [C(ν, σ), E([ν] + σ)] = E(ν, σ)

and E(ν, σ) is normal in G([ν] + σ).

Proof. Let D = E(ν), C = C(ν, σ) and H = E(ν, σ). Clearly CD ⩽ N([ν] + σ) and C ∩D is contained in the block diagonal group with diagonal blocks C(|ν(i)|, R, σ_{ii}). The first assertion follows now from the previous lemma. To prove the second one, note that G(σ) is normal in G = G([ν]+σ). Also E([ν]+σ) is normal in G, because it equals [G, D]. It follows that

[C(ν, σ)^{G}, E([ν] + σ)] ⩽ [C(ν, σ), E([ν] + σ)^{G}]^{G} ⩽ G(σ)^{G} ⩽ G(σ) .

By definition of C(ν, σ), the above implies that C(ν, σ)^{G} ⩽ C(ν, σ). Hence E(ν, σ) is normal in G.

5.3. Let R be a commutative ring, A an R-algebra, and σ a net over A. For a multiplicative subset S in R, we denote by S^{−1}R the localization of R in S, by S^{−1}A the S^{−1}R-algebra A ⊗_{R} S^{−1}R, and by S^{−1}σ the net over S^{−1}A of ideals (S^{−1}σ)_{ij} = σ_{ij} ⊗_{R} S^{−1}R. In the sequel, quotation marks will denote the image of an ideal or group under a homomorphism induced by the canonical homomorphism A → S^{−1}A, where S will be specified by context.

5.4. Lemma. Let A be an R-algebra, σ ⩾ [ν] a major net over A such that h(ν) ⩾ 3, and s ∈ R. Then for any u ∈ E(ν, ⟨s⟩^{−1}σ) there exists a positive integer m such that “E(s^{m}σ)”^{u} ⩽ “E(sσ)” .

Proof. The proof is the same as for Lemma 4.6 of [Bk1], because σ is major.

5.5. Lemma. Let A be an R-algebra such that the nilpotent radical of A is trivial and let s ∈ R. Then the ideal sA injects into ⟨s⟩^{−1}A and hence, GL(n, A, sA) injects into GL(n, ⟨s⟩^{−1}A).

Proof. Suppose that sξ ∈ sA goes to 0 in ⟨s⟩^{−1}A. Then there is m ∈ ℕ such that s^{m}sξ = 0. It follows that (sξA)^{m+1} = 0. Hence by the condition of the lemma, sξA = 0. Thus sξ = 0.

5.6. Lemma. (Vavilov, Stepanov [VvS]). Let A be a ring such that h(ν) ⩾ sup(sr A + 1, 3), σ ⩾ [ν] a major net over A, K a set of representatives of the equivalence classes of ν, and i ∈ J. Then each matrix a ∈ N(σ) can be decomposed as a product a = bu such that u ∈ E(ν, σ), b ∈ N(σ), and b_{ik} = 0 for k ∈ K.

5.7. Theorem. Let A be an R-algebra and ν be an equivalence on J = {1, …, n} (where n can be infinite). Suppose that for any maximal ideal M of R there is a multiplicative set S ⊆ R \ M such that h(ν) ⩾ sup(sr S^{−1}A + 1, 3). Then for any ν-net σ

[C(ν, σ), E(ν)] = [C(ν, σ), E([ν] + σ)] = E(ν, σ)

and E(ν, σ) is normal in G([ν] + σ). In particular if A is quasi-finite then the conclusions above hold for h(ν) ⩾ 3.

Proof. The last assertion of the theorem follows from the other assertions, because a quasi-finite ring is a direct limit \underrightarrow{lim} A_{i} of module finite rings A_{i}, C(ν,_{−}), E(ν,_{−}), and G(_{−}) commute with direct limits, and if A_{i} is module finite over R_{i} and S = R_{i}\M then sr(S^{−1}A_{i}) = 1. Thus sup(sr(S^{−1}A_{i}) + 1, 3) = 3.

We prove now the rest of the theorem. If ν is the trivial equivalence (cf. 3.3), the result has been proved by L.N.Vaserstein in [Vs1]. By Corollary 5.2, we have to prove only that E(ν)^{N([ν]+σ)} ⩽ E([ν] + σ). Thus we can assume that σ ⩾ [ν] is major.

Suppose first that the Jacobson radical Rad A is trivial. Let M be a maximal ideal in R, h ∼^{ν} i be distinct indexes from J, ξ ∈ A, and a ∈ N(σ). We want to show that there exists an α ∈ R \ M such that d = t_{hi}(ξαη)^{a} ∈ E(σ) for any η ∈ R. Choose a set K ⊂ J of representatives of the equivalence classes of ν and a multiplicative set S satisfying the conditions of the theorem. By Lemma 5.6, we can decompose the image of a in GL(n, S^{−1}A) into a product bu where u ∈ E(ν, S^{−1}σ), b ∈ N(S^{−1}σ) and b_{ik} = 0 for all k ∈ K. Since S^{−1}R = \underrightarrow{lim}⟨t⟩^{−1}R over all t ∈ S, there is an s ∈ S such that u ∈ E(s^{−1}σ) and b′_{pq} “σ_{qr}” b_{rj} ∈ s^{−1} “σ_{pj}” . By Lemma 5.5, GL(n, A, sA) injects into GL(n, ⟨s⟩^{−1}A). Thus by Lemma 5.4, there is an m ∈ ℕ such that “E(s^{m}σ)”^{u} ⩽ “E(sσ)” .

For an index l ∈ J consider the matrix

c^{(l)} = t_{hi}(ξs^{2m+2}ηb_{hl}b′_{lh})^{b} = e + b′_{∗h}ξs^{2m+2}ηb_{hl}b′_{lh}b_{i∗} =

= [t_{∗k}(b′_{∗h}ξs^{m+1}ηb_{hl}) , t_{k∗}(b′_{lh}s^{m+1}b_{i∗})] t_{k∗}(b′_{kh}ξs^{2m+2}ηb_{hl}b′_{lh}b_{i∗}) ∈ “E(s^{m}σ)”

where k ∈ K, k ∼^{ν} l (the formula above is just a variation of the Whitehead lemma). Set α = s^{2m+2}. Then d ∈ GL(n, A, sA) and the image of d in GL(n, ⟨s⟩^{−1}A) is equal to

t_{hi}(ξs^{2m+2}η)^{a} = (∏_{l∈J} c(l))^{u} ∈ “E(s^{m}σ)”^{u} ⩽ “E(s^{m′}σ)” .

Thus d ∈ E(sσ) ⩽ E(σ).

Let U be the set of all t ∈ R such that t_{hi}(ξtη)^{a} ∈ E(σ) for any η ∈ R. Obviously, U is an ideal in R. If U ≠ R then there exists a maximal ideal M ⩽ R containing U. But this is impossible because we have just proved that there exists an element α ∈ U which does not belong to M. The contradiction shows that U = R and hence t_{hi}(ξ)^{a} ∈ E(σ) for any h ≠ j ∈ J, ξ ∈ A and a ∈ N(σ).

Let now A be an arbitrary R-algebra, satisfying the conditions of the theorem. Clearly A/ Rad A fulfills the conditions of the theorem as well. Hence H = [E(ν), N(σ)] is contained in H′ = E(σ) GL(n, A, Rad R) ∩ G(σ) and is normal in N(σ). It is easy to see that H′ ⩽ E(σ) D(n, R) where D(n, R) denotes the subgroup of diagonal matrices and therefore [H, H] ⩽ [H′, H′] ⩽ E(σ). On the other hand [H, H] ⩾ E(σ), because [E(σ), E(ν)] = E(σ). Thus E(σ) is normal in N(σ), since it is the commutator subgroup [H, H] of the normal subgroup H of N(σ).

5.8. Standard Classification Theorem. Let A be a quasi-finite ring, ν an equivalence on J = {1, ⋯, n} (where n can be infinite) and h(ν) ⩾ 3. Then the E(ν)-normal subgroups of GL(n, A) are in one to one correspondence with the subgroups H of the sandwiches E(ν, σ) ⩽ H ⩽ C(ν, σ) where σ ranges over all ν-nets. Moreover each E(ν)-normal subgroup of GL(n, A) belongs to precisely one sandwich and the mixed commutator group [E(ν), C(ν, σ)] = [E([ν] + σ), C(ν, σ)] = E(ν, σ).

Proof. Since the functors E(ν,_{−}), G(_{−}), and GL(n,_{−}) commute with direct limits, we can reduce routinely the proof of the theorem to the case A is module finite over a commutative ring R. The commutator formula follows now from Theorem 5.7 and the fact that if M is a maximal ideal in R and S = R \ M then sr(S^{−1}A) = 1 because S^{−1}A is semilocal. From the commutator formula, it follows that if a subgroup H is contained in a sandwich E(ν, σ) ⩽ H ⩽ C(ν, σ) then [E(ν), H] ⩽ H. Thus H is E(ν)-normal. Furthermore a sandwich containing H must be unique, since E(ν, σ) = [E(ν), E(ν, σ)] ⩽ [E(ν), H] ⩽ [E(ν), C(ν, σ)] = E(ν, σ), i.e. E(ν, σ) = [E(ν), H], and obviously E(ν, σ) = E(ν, ρ) ⇔ σ = ρ. The fact that each E(ν)-normal subgroup of GL(n, A) is contained in a sandwich is a special case of Golubchik’s theorem [G].

Let K be an overgroup of E(ν) in G(ν). The K-normal subgroups of GL(n, A) are among the E(ν)-normal subgroups and the E(ν)-normal subgroups are classified in Theorem 5.8. If the conjugation action of K on C(ν, σ)/E(ν, σ) is trivial for each ν-net σ over A then the set of K-normal subgroups of GL(n, A) is the same as the set of E(ν)-normal subgroups. Thus the nilpotent class of the action of K on C(ν, σ)/E(ν, σ) is an obstruction to the sets above being equal and provides a rough measure of how much smaller the set of K-normal subgroups is. We shall investigate this nilpotent class in the next section.

5.9. Theorem. Let A be a quasi-finite ring, ν be an equivalence on J = {1, …, n} (where n can be infinite) and h(ν) ⩾ 4. Then for any major net σ ⩾ [ν], the group K_{2}(ν, σ) is contained in the center of St(ν, σ). Moreover there is a natural action of G(σ) on St(ν, σ) extending the action of G(σ) on E(ν, σ) via conjugation.

Proof. For module finite rings one can take the set

V = {(α, β)| α = a_{∗i} for some a ∈ E(ν, σ), i ∈ J, β ∈ σ_{i∗}, βα = 0}

and run it through the proof of the main theorem of [Tu], because σ is major. The quasi-finite case can be obtained by the standard direct limit procedure.

6. ν-nets as a category with dimension

In this section, we define the category of ν-nets over algebras and make it and its full subcategory of major ν-nets into categories with dimension. Then we show that the natural transformation π : St(ν,_{−}) → G(_{−}) is good on the category of major ν-nets. In the next section, we shall combine the goodness of π with Theorem 2.7 to conclude that for a major ν-net σ, the group G^{0}(σ)/E(ν, σ) has nilpotent class ⩽ dim(ν, σ) whenever dim(ν, σ) < ∞ and h(ν) ⩾ 4, and that for any ν-net σ, the group G^{0}(σ)/E(ν, σ) has nilpotent class ⩽ dim(ν, σ)+1, under the same conditions.

6.1. It will be important in this section to know exactly what is meant by an algebra and a morphism of algebras, because ground rings will frequently change. An algebra will mean a pair (R, A) where A is an associative ring with identity and R is a commutative ring with identity, together with a fixed ring homomorphism R → center(A) which preserves the identity. We let ((alg)) denote the category whose objects are all pairs (R, A) as above and whose morphisms f : (R, A) → (R′, A′) are all pairs of identity preserving ring homomorphisms R → R′ and A → A′ such that the diagram

\begin{matrix}A & ⟶ & A′ \\ ↑ & & ↑ \\ R & ⟶ & R′ \end{matrix}

commutes. An algebra (R, A) is called module finite (over R), if A is finitely generated as an R-module. Following [Bk1], we shall call an algebra (R, A) quasifinite (over R), if (R, A) is a direct limit \underrightarrow{lim}(R_{i}, A_{i}) in ((alg)) of module finite algebras (R_{i}, A_{i}). It is easy to check that this definition is equivalent to the one obtained by postulating that each R_{i} = R. The concept of quasi-finiteness for algebras is stronger than that for rings in §3, because it is not required in §3 that the ring homomorphisms A_{i} → A_{j}, where i ⩽ j, take the center(A_{i}) to the center(A_{j}). Let ν be a fixed equivalence on a subset J of ℕ. Define the category ((ν-nets)) as follows. An object is a triple (R, A, σ) where (R, A) is an algebra and σ is a ν-net in the sense of §3 over A. We shall frequently denote the triple (R, A, σ) by σ, if this does not lead to confusion. A morphism (R, A, σ) → (R′, A′, σ′) is an algebra homomorphism f : (R, A) → (R′, A′) such that f(σ_{ij}) ⩽ σ′_{ij} for all i, j ∈ J. We let ((major ν-nets)) denote the full subcategory of ((ν-nets)) consisting of all ν-nets (R, A, σ) such that σ is major in the sense of §3.

We want our structure squares in ((ν-nets)) to be pullback squares, since the functor G(_{−}) on ((ν-nets)) preserves pullback squares, and we want localizationcompletion squares to be included among our structure squares, since such squares behave well under Bass-Serre dimension. However localization-completion squares are not in general pullback squares, but only when R is Noetherian and A is module finite over R. To get around this problem, Bak [Bk4] has introduced the notion of finite completion and constructed localization-finite-completion squares which are always pullback squares and are well behaved under Bass-Serre dimension. We recall what is necessary for the current paper.

6.2. Let ((mod)) denote the category of modules M over commutative rings R. By definition, an object of ((mod)) is a pair (R, M). A morphism (R, M) → (R′, M′) is a pair (f, g) consisting of a ring homomorphism f : R → R′ and a homomorphism g : M → M′ of Abelian groups such that g(rm) = f(r)g(m) for all r ∈ R and m ∈ M. For an element s ∈ R, let M̂_{(s)} = \underleftarrow{lim}_{i⩾0} M/s^{i}M denote the completion of

M at s and let ⟨s⟩^{−1}M denote the module of ⟨s⟩-fractions of M at the multiplicative set ⟨s⟩ = {s^{i} | i ⩾ 0}. Let I(s, R, M) denote {(R_{i}, M_{i}) | R_{i} ⊆ R a finitely generated ℤ-subalgebra such that s ∈ R_{i}, M_{i} ⊆ M a finitely generated R_{i}-submodule}. Obviously I(s, R, M) is a directed, partially ordered set under inclusion. Define the finite completion M̃_{(s)} of M at s by M̃_{(s)} = \underrightarrow{lim}_{i∈I(s,R,M)}(M̂_{i})_{(s)}. Clearly M̃_{(s)} is

a module over R̃_{(s)}. Furthermore if R is finitely generated over ℤ and M is finitely generated over R then M̃_{(s)} = M̂_{(s)}. The basic facts concerning finite completion are contained in the next four lemmas.

For a fixed commutative ring R, let ((R-mod)) denote the subcategory of ((mod)) of all R-modules (R, M) and all morphisms (f, g) : (R, M) → (R, M′) such that f = identity.

6.3. Lemma. [Bk 4] Finite completion is an exact functor on ((R-mod)).

Proof. Any 3-term exact sequence in ((R-mod)) is a direct limit of 3-term exact sequences of finitely generated modules over finitely generated ℤ- subalgebras of R. Ordinary completion preserves such exact sequences and a direct limit of exact sequences is exact. The conclusion of the lemma follows.

6.4. If (R, M) ∈ ((mod)) and s ∈ R then

LF C(s, R, M) := \begin{matrix}(R, M) & ⟶ & (⟨s⟩^{−1}R, ⟨s⟩^{−1}M) \\ ↓ & & ↓ \\ (R̃_{(s)}, M̃_{(s)}) & ⟶ & (⟨s⟩^{−1}R̃_{(s)}, ⟨s⟩^{−1}M̃_{(s)}) \end{matrix}

is called the localization-finite-completion square of (R, M) at s.

6.5. Lemma. [Bk 4] Localization-finite-completion squares are fibre squares.

Proof. LF C(s, R, M) = \underrightarrow{lim}_{i∈I(s,R,M)} LF C(s, R_{i}, M_{i}) and it is well known classically

that each LF C(s, R_{i}, M_{i}) is a fibre square.

6.6. Definition-Lemma. [Bk 4] Let (R, A) be an R-algebra and s ∈ R. Define a multiplication on Ã_{(s)} as follows. Let x, y ∈ Ã_{(s)}. Choose α, β ∈ I(s, R, A) and elements x′ ∈ (Â_{α})_{(s)} and y′ ∈ (Â_{β})_{(s)} such that x′ and y′ represent x and y, respectively. Neither A_{α} nor A_{β} is necessarily closed under multiplication in A. Let

∏_{i⩾0} x_{i} ∈ ∏_{i⩾0} A_{α} represent x′ and ∏_{i⩾0} y_{i} ∈ ∏_{i⩾0} A_{β} represent y′. Choose γ ∈ I(s, R, A)

such that α ⩽ γ, β ⩽ γ, and A_{α}A_{β} ⊆ A_{γ}. Define x ∘ y to be the class in Ã_{(s)} of the element of (Â_{γ})_{(s)} defined by ∏_{i⩾0} x_{i}y_{i} ∈ ∏_{i⩾0} A_{γ}. Then the product x ∘ y is

independent of all choices made and makes Ã_{(s)} into an R̃_{(s)}-algebra.

Proof. Straightforward.

6.7. Corollary. [Bk4] If (R, A) is an R-algebra, s ∈ R, and 𝔄 ⊆ A is an ideal then 𝔄̃_{(s)} ⊆ Ã_{(s)} is an ideal.

Proof. This follows routinely from (6.3), (6.6), and the proof of (6.6).

6.8. Definition-Lemma. If (R, A, σ) ∈ ((ν − nets)) and s ∈ R then the square

LF C(s, R, A, σ) := \begin{matrix}(R, A, σ) & ⟶ & (⟨s⟩^{−1}R, ⟨s⟩^{−1}A, ⟨s⟩^{−1}σ) \\ ↓ & & ↓ \\ (R̃_{(s)}, Ã_{(s)}, σ̃_{(s)}) & ⟶ & (⟨s⟩^{−1}R̃_{(s)}, ⟨s⟩^{−1}Ã_{(s)}, ⟨s⟩^{−1}σ̃_{(s)}) \end{matrix}

is called the localization-finite-completion square of (R, A, σ) at s. Let LF C ((ν − nets)) (resp. LF C(( major ν − nets)) ) denote the class of all commutative squares in ((ν-nets)) (resp. ((major ν-nets))) which are isomorphic to a localization-finite-completion square. If S ⊆ R is a multiplicative set, give S the directed quasi-ordering (cf. §2) defined by s ⩽ r ⇔ ∃u ∈ S such that su = t. Let Frac((ν-nets)) (resp. Frac((major ν-nets))) denote the class of all functors F_{(S,R,A,σ)} : S → ((ν-nets)) (resp. ((major ν-nets))), s ↦ (⟨s⟩^{−1}R, ⟨s⟩^{−1}A, ⟨s⟩^{−1}σ), where (R, A, σ) ranges over all objects of ((ν-nets)) (resp. ((major ν-nets))) and S over all multiplicative subsets of R. Then (((ν-nets)), LF C((ν-nets)), Frac((νnets))) and ( ((major ν-nets)), LF C((major ν-nets)), Frac((major ν-nets))) are categories with structure in the sense of (2.1) whose structure squares are pullback squares. Let BS(R) denote the Bass-Serre dimension of R, cf. [Bk1]. Define

dim(R, A, σ) = \{ \begin{matrix}BS(R) & if A is quasi-finite over R \\ ∞ & otherwise. \end{matrix}

Then dim is a dimension function in the sense of (2.2) on the categories above. Moreover if dim(R, A, σ) = 0 then the stable rank sr(A) = 1.

Remark An important part of the definition of dim is that dim(R, A, σ) < ∞ implies (R, A) in quasi-finite.

Proof. Obviously we have categories with structure. Lemma 6.5 shows that all structure squares are pullback squares. It is clear that if (R, A) is quasi-finite then for any element s ∈ R and any multiplicative set S ⊆ R, every algebra in the localization-finite-completion square LF C(s, R, A) and the algebra (S^{−1}R, S^{−1}A) are quasi-finite. It follows therefore from the Induction Lemma 4.17 of [Bk1] that dim is a dimension function on both categories with structure. Suppose dim(R, A, σ) = 0. Since A is quasi-finite over R, A is a direct limit of module finite R-subalgebras A_{i}. Since BS(R) = 0, it follows that R is semilocal. Thus A_{i} is semilocal. Thus sr(A_{i}) = 1. But a direct limit of stable rank 1 rings has stable rank 1. Thus sr(A) = 1.

For the proof of the goodness lemma below we shall use smash products of rings and ν-nets. By definition, the smash product A ⋊ A of a ring A with itself has as elements those of the Cartesian product A × A. Addition is defined componentwise and multiplication by the rule (b, a)(b′, a′) = (bb′ + ab′ + ba′, aa′). If σ and ρ are ν-nets over A such that σρ ⊆ ρ and ρσ ⊆ ρ then one defines the smash product ρ ⋊ σ similarly and ρ ⋊ σ becomes a ν-net over A ⋊ A. If σ and ρ are major ν-nets, then so is ρ ⋊ σ. Let f : A ⋊ A → A, (b, a) ↦ a, and g : A ⋊ A → A, (b, a) ↦ b + a. Both f and g are ring homomorphisms and are commonly split by the homomorphism A → A ⋊ A, a ↦ (0, a). They induce morphisms f : ρ ⋊ σ → σ and g : ρ ⋊ σ → ρ + σ of ν-nets, and f is split by the ν-net map σ → ρ ⋊ σ, a ↦ (0, a). Moreover if ρ ⊆ σ then g is split by the same ν-net map.

Let ρ ⊆ σ be ν-nets over A such that σρ ⊆ ρ and ρσ ⊆ ρ. If L : ((ν-nets)) ⟶ ((groups)) is a functor, set L(σ, ρ) = Ker(L(f) : L(ρ⋊σ) → L(σ)). Give L(σ, ρ) the L(σ)-action defined by the conjugation action of L(ρ ⋊ σ) on L(σ, ρ) and the split

exact sequence L(σ, ρ) ↪ L(ρ ⋊ σ) ⟶^{L(f)} L(σ). Let h_{σ,ρ} denote the composition

of the homomorphisms L(σ, ρ) ↪ L(ρ ⋊ σ) ⟶^{L(g)} L(ρ + σ) = L(σ). Obviously h_{σ,ρ} : L(σ, ρ) → L(σ) is a precrossed module whose associated smash product group L(σ, ρ) ⋊ L(σ) (defined just prior to (1.11)) has a canonical identification L(σ, ρ) ⋊ L(σ) ≅ L(ρ ⋊ σ) with L(ρ ⋊ σ).

Let G(_{−}), E(ν,_{−}), and St(ν,_{−}) : ((ν-nets))→ ((groups)) be defined as in §3 and let π : St(ν,_{−}) → G(_{−}) denote the canonical natural transformation. Let ρ ⊆ σ be ν-nets on A such that σρ ⊆ ρ and ρσ ⊇ ρ. The proof of the goodness lemma below will use tacitly the following facts.

Since the square

ρ ⋊ σ ⟶^{f} σ
g↓  ↓
σ ⟶ σ/ρ

is a fibre square of ν-nets and the functor G(_{−}) preserves fibre squares, it follows that the map h_{σ,σ} : G(σ, σ) → G(σ) above is an isomorphism. Although the functor St(ν,_{−}) does not preserve fibre squares, using the relations (3.5) defining St(ν,_{−}), one can show (see for example the proof of [M] (6.1)) that h_{σ,σ} : St(ν, σ, σ) → St(ν, σ) is an isomorphism providing h(ν) ⩾ 4 and σ is major. It follows that h_{σ,σ} : E(ν, σ, σ) → E(ν, σ) is an isomorphism under the same conditions.

6.9. Goodness Lemma. The natural transformation π above is good in the sense of (2.6) on the category with structure (((major ν-nets)), LF C((major ν-nets)), Frac((major ν-nets))), providing h(ν) ⩾ 4.

Proof. It is straightforward to check that St and G commute with arbitrary direct limits. Any localization-finite-completion square LF C(s, R, A, σ) such that dim(R, A, σ) < ∞ is a direct limit of subsquares LF C(s, R_{i}, A_{i}, σ_{i}) where R_{i} is Noetherian and A_{i} is module finite over R_{i}. It suffices to show that each square LF C(s, R_{i}, A_{i}, σ_{i}) is π-good. Given i, one can show as in the proof of Lemma 4.10 in [Bk1] that there is an n ⩾ 0 such that s^{n}A_{i} is s-torsion free. It suffices therefore to prove the following:

(6.9.1) If (R, A) is quasi-finite, n ⩾ 0, and σ is a major ν-net such that s^{n}σ is s-torsion free then LF C(s, R, A, σ) is a π-good square.

By definition, it suffices to show that the commutative cube of precrossed modules

(St(ν, σ, s^{n}σ) ⟶^{h} St(ν, σ))⟶ (St(ν, ⟨s⟩^{−1}σ) ⟶^{1} St(ν, ⟨s⟩^{−1}σ))
↙  ↙
(St(ν, σ̃_{(s)}, s^{n}σ̃_{(s)}) ⟶^{h̃} St(ν, σ̃_{(s)})) ⟶(St(ν, ⟨s⟩^{−1}σ̃_{(s)}) ⟶^{1} St(ν, ⟨s⟩^{−1}σ̃_{(s)}))
↓  ↓
(G(s^{n}σ) ⟶^{h} G(σ)) − ⟶ (G(⟨s⟩^{−1}σ) ⟶^{1} G(⟨s⟩^{−1}σ))
↓ ↙ ↓ ↙
(G(s^{n}σ̃_{(s)}) ⟶^{h̃} G(σ̃_{(s)}))  ⟶ (G(⟨s⟩^{−1}σ̃_{(s)}) ⟶^{1} G(⟨s⟩^{−1}σ̃_{(s)}))

satisfies (1.11)(i), (ii) and (iii), and that the associated cube of smash product groups

St(ν, s^{n}σ ⋊ σ) ⟶St(ν, ⟨s⟩^{−1}σ ⋊ ⟨s⟩^{−1}σ)
↙  ↙
St(ν, s^{n}σ̃_{(s)} ⋊ σ̃_{(s)}) ⟶St(ν, ⟨s⟩^{−1}σ̃_{(s)} ⋊ ⟨s⟩^{−1}σ̃_{(s)})
↓ ↓
G(s^{n}σ ⋊ σ) − − → G(⟨s⟩^{−1}σ ⋊ ⟨s⟩^{−1}σ)
↓ ↙ ↓ ↙
G(s^{n}σ̃_{(s)} ⋊ σ̃_{(s)}) ⟶ G(⟨s⟩^{−1}σ̃_{(s)} ⋊ ⟨s⟩^{−1}σ̃_{(s)})

satisfies (1.10)(i) and (ii) and Im(St(ν, s^{n}σ̃_{(s)} ⋊ σ̃_{(s)}) → G(s^{n}σ̃_{(s)} ⋊ σ̃_{(s)})) is a normal subgroup of G(s^{n}σ̃_{(s)} ⋊ σ̃_{(s)}). Condition (1.10)(i) follows from Theorem 5.9. Condition (1.10) (ii) follows from the standard equations for elementary transvections, which are spelled out under (3.5)(1)–(3), and the fact that h(ν) ⩾ 3 and σ is major. A little extra care has to be taken in the case of E(ν, s^{n}σ̃_{(s)} ⋊ σ̃_{(s)}), because s^{n}σ̃_{(s)} ⋊ σ̃_{(s)} is not major, but only σ̃_{(s)}. The normality of E(ν, s^{n}σ̃_{(s)} ⋊ σ̃_{(s)}) in G(s^{n}σ̃_{(s)} ⋊ σ̃_{(s)}) follows from Theorem 5.7. Condition (1.11)(ii) follows from the fact that localization-finite-completion squares are fibre squares by Lemma 6.5 and G preserves fibre squares. Condition (1.11)(i) has 2 parts. We demonstrate first the injectivity of E(ν, σ̃_{(s)}, s^{n}σ̃_{(s)}) → E(ν, ⟨s⟩^{−1}σ̃_{(s)}). Since s^{n}σ̃_{(s)} is s-torsion free and finite-completion is exact by Lemma 6.3, it follows that s^{n}σ̃_{(s)} = \binom{}{s^{n}σ}_{(s)} is s-torsion free. Thus the canonical map s^{n}σ̃_{(s)} → ⟨s⟩^{−1}σ̃_{(s)} is injective. Thus the induced map G(s^{n}σ̃_{(s)}) → G(⟨s⟩^{−1}σ̃_{(s)}) is injective. Thus E(ν, σ̃_{(s)}, s^{n}σ̃_{(s)}) → E(ν, ⟨s⟩^{−1}σ̃_{(s)}) is injective. To demonstrate that the rest of condition (1.11)(i) and all of condition (1.11)(iii) are satisfied, we can reduce to the case that A is module finite over a Noetherian ring R, because any quasi-finite algebra is a direct limit of such algebras and the functors involved commute with direct limits. Under the assumption that A is module finite over a Noetherian ring R, conditions (1.11)(i) and (iii) are verified respectively in (6.12) and (6.13) below. This completes the proof Lemma 6.9.

6.10. Suppose that A is module finite over a Noetherian ring R. Let σ be a major ν-net over A. Let s ∈ R and let

σ ⟶^{φ} ⟨s⟩^{−1}σ
Ψ↓  ↓_{Ψ′}
σ̂_{(s)} ⟶_{φ′} ⟨s⟩^{−1}σ̂_{(s)}

denote the localization-completion square of (R, A, σ) at s. Our next goal is to show that for any nonnegative integer n, the canonical map

θ : St(ν, ⟨s⟩^{−1}σ)/ St(ν, σ, s^{n}σ) → St(ν, ⟨s⟩^{−1}σ̂_{(s)})/ St(ν, σ̂_{(s)}, s^{n}σ̂_{(s)})

of coset spaces is bijective.

For a major ν-net τ on A, let Y^{±1}(τ) denote the set of all formal products

∏_{k=1}^{l} y_{i_{k}j_{k}}(ξ_{k})^{η_{k}} where η_{k} = ±1 and ξ_{k} ∈ τ_{i_{k}j_{k}}. Consider the following formal prod-

ucts (i) y_{ij}(ξ)y_{ij}(ξ)^{−1}, y_{ij}(ξ)^{−1}y_{ij}(ξ) (ii) y_{ij}(ξ)y_{ij}(ζ)y_{ij}(ξ + ζ)^{−1}

y_{ij}(ξ)y_{pq}(ζ)y_{ij}(ξ)^{−1}y_{pq}(ζ)^{−1}, i ≠ q and j ≠ p y_{ij}(ξ)y_{jq}(ζ)y_{ij}(ξ)^{−1}y_{jq}(ζ)^{−1}y_{iq}(ξζ)^{−1}, i ≠ q.

Let Y^{±1}(τ) denote the union of Y^{±1}(τ) and the empty product. Y^{±1}(τ) has an

obvious rule of composition Y^{±1}(τ)×Y^{±1}(τ) → Y^{±1}(τ), ( ∏_{k=1}^{ℓ} y_{i_{k}j_{k}}(ξ_{k})^{η_{k}}, ∏_{k=1}^{ℓ′} y′_{i_{k}j_{k}}

(ξ′_{k})^{η′_{k}}) ↦ ∏_{k=1}^{ℓ} y_{i_{k}j_{k}}(ξ_{k})^{η_{k}} ∏_{k=1}^{ℓ′} y′_{i_{k}j_{k}}(ξ′_{k})^{η′_{k}}, such that if ∅ ∈ Y^{±1}(τ) denotes the

empty product and y ∈ Y^{±1}(τ) then y∅ = ∅y = y. If we form equivalence classes on Y^{±1}(τ) by deleting formal products under (i) from elements of Y^{±}(τ) or inserting formal products under (i) into elements of Y^{±1}(τ) then the rule of composition above is well defined on equivalence classes and the result is the free group on the symbols y_{ij}(ξ) such that ξ ∈ τ_{ij}. If we form equivalence classes by deleting formal products under (i) and (ii) from elements of Y^{±1}(τ) or inserting formal products under (i) and (ii) into elements of Y^{±1}(τ) then the composition on Y^{±1}(τ) is well defined on these equivalence classes and the result is the group St(ν, τ). If X is any set then it is evident that a map θ′ : Y^{±1}(τ) → X induces a map St(ν, τ) → X ⇔ it is constant on the equivalence classes above. (Note that if θ′ is constant on equivalence classes above then all of the formal products under (i) and (ii) and the empty product ∅ go to the same element of X. It is often helpful to think of this element as the base point of X.)

Let Y^{±1} = Y^{±1}(⟨s⟩^{−1}σ̂_{(s)}) and Y^{±1} = Y^{±1}(⟨s⟩^{−1}σ̂_{(s)}). Following Bak [Bk2]

(7.13), we call an element x = ∏_{k=1}^{ℓ} x_{i_{k}j_{k}}(ξ_{k}) ∈ St(ν, ⟨s⟩^{−1}σ) a good approxi-

mation for y = ∏_{k=1}^{ℓ} y_{i_{k}j_{k}}(ξ̂_{k})^{η_{k}} ∈ Y^{±1} if η_{k}ξ̂_{k} − Ψ′(ξ_{k}) ∈ φ′((s^{2^{k}+n}σ̂_{(s)})_{i_{k}j_{k}}). It

is routine to check that given k ⩾ 0 and ξ̂ ∈ (⟨s⟩^{−1}σ̂_{(s)})_{ij}, there is an element ξ ∈ (⟨s⟩^{−1}σ)_{ij} such that ξ̂ − Ψ′(ξ) ∈ φ′((s^{2^{k}+n}σ̂_{(s)})_{ij}). Thus any element of Y^{±1} has a good approximation. The good approximation of the empty formal product ∅ ∈ Y^{±1} is by definition the element 1 ∈ St(ν, ⟨s⟩^{−1}σ). The next lemma shows that the map

θ′ : Y^{±1} ⟶ St(ν, ⟨s⟩^{−1}σ)/ St(ν, σ, s^{n}σ)
y ⟼ φ(St(ν, s^{n}σ))(good approximation(y))

is well defined, providing h(ν) ⩾ 3, and that it induces a retract for θ.

6.11. Lemma. Suppose in the setting of (6.10) that h(ν) ⩾ 3. If x, x′ ∈ St(ν, ⟨s⟩^{−1}σ) are good approximations for the same element of Y^{±1} then x′x^{−1} ∈ φ(St(ν, σ, s^{n}σ)). It follows that the map θ′ is well defined. Furthermore θ′ is constant on equivalence classes of Y^{±1}(τ) and sends formal products representing naturally elements of φ′(St(ν, σ̂_{(s)}, s^{n}σ̂_{(s)})) to the trivial coset φ(St(ν, σ, s^{n}σ)) of St(ν, ⟨s⟩^{−1}σ)/ St(ν, σ, s^{n}σ). It follows that θ′ induces a map

St(ν, ⟨s⟩^{−1}σ̂_{(s)})/ St(ν, σ̂_{(s)}, s^{n}σ̂_{(s)}) → St(ν, ⟨s⟩^{−1}σ)/ St(ν, σ, s^{n}σ)

which is a retract of the canonical map θ in the other direction.

Proof. Suppose x = ∏_{k=1}^{ℓ} x_{i_{k}j_{k}}(ξ_{k}) and x′ = ∏_{k=1}^{ℓ} x_{i_{k}j_{k}}(ξ′_{k}). For 1 ⩽ h ⩽ ℓ, set

x_{h} = ∏_{k=h}^{l} x_{i_{k}j_{k}}(ξ_{k})  and  x′_{h} = ∏_{k=h}^{l} x_{i_{k}j_{k}}(ξ′_{k}),

so that x = x_{1} and x′ = x′_{1}. Since x and x′ are good approximations of the same element of Y^{±1}, we have the inclusion ψ′ ((ξ′_{k} − ξ_{k})s^{−2^{k}−n}) ∈ φ′ ((σ̂_{s})_{i_{k}j_{k}}) . Since the localization-completion square of ν-nets in (6.10) is fibred, it follows that (ξ′_{k} − ξ_{k})s^{−2^{k}−n} ∈ φ(σ_{i_{k}j_{k}}). Thus ξ′_{k} − ξ_{k} ∈ φ(s^{2^{k}+n}σ_{i_{k}j_{n}}). We shall prove by induction on h = l, l − 1, …, 1 that x′_{h}x_{h}^{−1} ∈ φ(St(ν, σ, s^{2^{h}+n}σ)). It has just been shown that the assertion is true if h = l. Consider the element

x′_{h}x_{h}^{−1} = x_{i_{h}j_{h}}(ξ′_{h} − ξ_{h})x_{i_{h}j_{h}}(ξ_{h})x′_{h+1}x_{h+1}^{−1}x_{i_{h}j_{h}}(−ξ_{h}).

Obviously the first factor belongs to φ(St(ν, σ, s^{2^{h}+n}σ)). By the induction assumption, x′_{h+1}x_{h+1}^{−1} ∈ φ(St(ν, σ, s^{2^{h+1}+n}σ)). Now, the same arguments as in Lemma 4.6 of [Bk1] or in Lemma 7.9 of [Bk2] show that x′_{h}x_{h}^{−1} ∈ φ(St(ν, σ, s^{2^{h}+n}σ)). This is where the assumption that h(ν) ⩾ 3 is required.

It follows now that θ′ induces a well defined map Y^{±1} → St(ν, ⟨s⟩^{−1}σ)/ St(ν, σ, s^{n}σ). If the formal product y′ is obtained from the formal product y by either deleting from y a formal product under (i) or (ii) or inserting into y a formal product under (i) or (ii) then it is clear that we can pick a good approximation x for y and x′ for y′ such that x = x′ in St(ν, ⟨s⟩^{−1}σ). For example, suppose y′ = ∅ and y = y_{ij}(ξ̂)y_{ij}(ζ̂)y_{ij}(ξ̂ + ζ̂)^{−1}. Then we could pick x = x_{ij}(ξ)x_{ij}(ζ)x_{ij}(−ξ − ζ) where ξ̂ − Ψ′(ξ), ζ̂ − Ψ′(ζ) ∈ φ′(s^{8+n}σ̂_{(s)}) and x′ = 1. Clearly x = 1 = x′. Thus θ′ induces a well defined map St(ν, ⟨s⟩^{−1}σ̂_{(s)}) → St(ν, ⟨s⟩^{−1}σ)/ St(ν, σ, s^{n}σ).

Suppose that y = ∏_{k=1}^{ℓ} y_{i_{k}j_{k}}(ξ̂_{k}) ∈ Y^{±1} such that ξ̂_{k} ∈ φ′(s^{n}σ̂_{(s)}) (1 ⩽ k ⩽ ℓ).

Then any good approximation x = ∏_{k=1}^{ℓ} x_{i_{k}j_{k}}(ξ_{k}) of y has the property that each

ξ_{k} ∈ φ(s^{n}σ). The proof has been given already in the first paragraph above. Thus x ∈ φ(St(ν, σ, s^{n}σ)). Finally suppose y ∈ Y^{±1} represents naturally an element of φ′(St(ν, σ̂_{(s)}, s^{n}σ̂_{(s)})) and x is a good approximation for y. Then one checks similarly that x ∈ φ(St(ν, σ, s^{n}σ)). Thus θ′ induces a well defined map

St(ν, ⟨s⟩^{−1}σ̂_{(s)})/ St(ν, σ̂_{(s)}, s^{n}σ̂_{(s)}) → St(ν, ⟨s⟩^{−1}σ)/ St(ν, σ, s^{n}σ).

Moreover if x = ∏_{k=1}^{ℓ} x_{i_{k}j_{n}}(ξ_{k}) ∈ St(ν, ⟨s⟩^{−1}σ) then x is a good approximation of the

formal product ∏_{k=1}^{ℓ} y_{i_{k}j_{k}}(Ψ′(ξ_{k})) which represents Ψ′(x). Thus θ′ induces a retract

of θ.

The next result is a generalization of the special case in [Bk2] (7.16) where ν has only one equivalence class.

6.12. Theorem. Suppose in the setting of (6.10) that h(ν) ⩾ 3. Then θ is a bijection, with inverse the map induced by θ′ in (6.11).

Proof. In view of (6.11), it suffices to show that θ is surjective. Let 1 ≠ ŷ ∈

St(ν, ⟨s⟩^{−1}σ̂_{(s)}) and let y = ∏_{k=1}^{ℓ} y_{i_{k}j_{k}}(ξ̂_{k}) ∈ Y^{±1} be a representative of ŷ. Let x

be a good approximation of y. The proof will be complete, if we can show that ŷΨ′(x)^{−1} ∈ φ′(St(ν, σ̂_{(s)}, s^{n}σ̂_{(s)})). But this is proved routinely using the methodology in the first paragraph of the proof of (6.11).

6.13. Lemma. Suppose in the setting of (6.10) that h(ν) ⩾ 3. Then the canonical map Ẽ(ν, σ, s^{n}σ)/E(ν, σ, s^{n}σ) → Ẽ(ν, σ)/E(ν, σ) is surjective.

Proof. Let z ≠ 1 ∈ Ẽ(ν, σ). By definition, Ψ(z) ∈ E(ν, σ̃_{(s)}). Reformulate the concept good approximation above by replacing St(ν, ⟨s⟩^{−1}σ̂_{(s)}) by St(ν, σ̂_{(s)}), St(ν, ⟨s⟩^{−1}σ) by St(ν, σ), and Y^{±1}(⟨s⟩^{−1}σ̂_{(s)}) by Y^{±1}(σ̂_{(s)}). Let y ∈ Y^{±1}(σ̂_{s}) represent Ψ(z). Let x′ ∈ St(ν, σ) be a good approximation for y. Let x denote the image of x′ in E(ν, σ). The methodology in the first paragraph of the proof of (6.11) shows that Ψ(z)Ψ(x)^{−1} ∈ E(ν, σ̂_{(s)}, s^{n}σ̂_{(s)}). Thus zx^{−1} ∈ Ẽ(ν, σ, s^{n}σ), by definition. Clearly zx^{−1} is a representative of the class of z in Ẽ(ν, σ)/E(ν, σ).

Let σ and ρ be ν-nets over A such that σρ ⊆ ρ and ρσ ⊆ ρ. Define

K_{1}(ν, σ, ρ) = Coker (St(ν, σ, ρ) → G(σ, ρ))
K_{2}(ν, σ, ρ) = Ker (St(ν, σ, ρ) → G(σ, ρ)) .

Note that h_{σ,ρ} : G(σ, ρ) → G(σ + ρ) maps G(σ, ρ) isomorphically onto G(ρ) ⩽ G(σ + ρ).

6.14. Theorem. Let A be a quasi-finite R-algebra and σ a major ν-net over A such that h(ν) ⩾ 4. Let n ⩾ 0 and s ∈ R be such that s^{n}σ is s-torsion free. Then corresponding to a localization-finite-completion square LFC(s, R, A, σ), there is a Mayer-Vietoris sequence of nonstable net K-groups

K_{2}(ν, ⟨s⟩^{−1}σ)  K_{1}(ν, ⟨s⟩^{−1}σ)
↘ ↗ ↘
K_{2}(ν, ⟨s⟩^{−1}σ̃_{(s)}) ⟶ K_{1}(ν, σ, s^{n}σ)  K_{1}(ν, ⟨s⟩^{−1}σ̃_{(s)}).
↗ ↘ ↗
K_{2}(ν, σ̃_{(s)}, s^{n}σ̃_{(s)})  K_{1}(ν, σ̃_{(s)}, s^{n}σ̃_{(s)})

Moreover the Mayer-Vietoris sequence is natural in LFC’s.

Proof. The conclusion of Theorem 1.8 provides the natural Mayer-Vietoris sequence we want. The hypotheses of Theorem 1.8 demand precisely that the commutative cube

St(ν, σ, s^{n}σ) ⟶St(ν, ⟨s⟩^{−1}σ)
↙  ↙
St(ν, σ̃_{(s)}, s^{n}σ̃_{(s)}) ⟶St(ν, ⟨s⟩^{−1}σ̃_{(s)})
↓ ↓
G(s^{n}σ) − ⟶ G(⟨s⟩^{−1}σ)
↓ ↙ ↓ ↙
G(s^{n}σ̃_{(s)}) ⟶ G(⟨s⟩^{−1}σ̃_{(s)})

satisfy conditions (1.4) (i) - (iv). By Lemma 1.5, it suffices to show that the canonical map St(ν, ⟨s⟩^{−1}σ)/ St(ν, σ, s^{n}σ) → St(ν, ⟨s⟩^{−1}σ̃_{(s)})/ St(ν, σ̃_{(s)}, s^{n}σ̃_{(s)}) is bijective, the map E(ν, σ̃_{(s)}, s^{n}σ̃_{(s)}) → E(ν, ⟨s⟩^{−1}σ̃_{(s)}) injective, and condition (1.4)(iv) satisfied. But this has been shown already in the proof of (6.9.1).

7. Nilpotent structure of net K_{1} and sandwich classification

Let St(ν,_{−}) and G(_{−}) : ((ν-nets)) →((groups)) denote the Steinberg net group functor and net group functor, respectively, defined in §3. Let π : St(ν,_{−}) → G(_{−}) denote the canonical natural transformation. Let G = G^{−1} ⩾ G^{0} ⩾ ⋯ ⩾ G^{i} ⩾ ⋯ ⩾ E(ν,_{−}) denote the dim-filtration of π on G defined in (2.5) for the dimension function dim defined in (6.8).

7.1. Lemma. If G = G^{⟨−1⟩} ⩾ G^{⟨0⟩} ⩾ G^{⟨1⟩} ⩾ ⋯ denotes the dim-filtration of π restricted to ((major ν-nets)) on G restricted to ((major ν-nets)) then G^{⟨i⟩} and G^{i} agree on ((major ν-nets)).

Proof. If σ → ρ is a morphism of ν-nets and σ is major then so is ρ.

Let J be a group which is operating automorphically on a group H. A chain H = H^{0} ⩾ H^{1} ⩾ ⋯ of subgroups of H is called a descending J-central series, if each H^{i} is invariant under the action of J and the subgroup D_{J}(H^{i}) := ⟨(^{x}h)h^{−1} | x ∈ J, h ∈ H⟩ ⩽ H^{i+1}.

The main result of the section is the following theorem which generalizes results in [Bk1] §5, particularly Theorem 6.5, concerning the special case that ν has only one equivalence class.

7.2. Theorem. Let σ denote any ν-net. If h(ν) ⩾ 2 then G^{0}(σ) is a normal subgroup of G([ν] + σ) and the action

G([ν] + σ) ↷ G(σ)/ G^{0}(σ)

of G([ν]+σ) on G(σ)/ G^{0}(σ) by conjugation is trivial. If h(ν) ⩾ 4 then the filtration

G(σ) ⩾ G(σ) ∩ G^{0}([ν] + σ) ⩾ G(σ) ∩ G^{1}([ν] + σ) ⩾ ⋯
⩾ G(σ) ∩ G^{i}([ν] + σ) ⩾ ⋯ ⩾ E(ν, σ)

is a descending G^{0}([ν] + σ)-central series of normal subgroups of G([ν] + σ) such that if i ⩾ dim(σ) then

G(σ) ∩ G^{i}([ν] + σ) = G(σ) ∩ E(ν, [ν] + σ), and
[G(σ) ∩ G^{i}([ν] + σ), C(ν, σ)] = E(ν, σ).

Furthermore under the notation of §4, the stabilization maps

G(σ^{(n)})/ G^{i}(σ^{(n)}) → G(σ^{(n+1)})/ G^{i}(σ^{(n+1)}), and
G(σ^{(n)})/ G(σ^{(n)}) ∩ G^{i}(([ν] + σ)^{(n)}) → G(σ^{(n+1)})/ G(σ^{(n+1)}) ∩ G^{i}(([ν] + σ)^{(n+1)})

are injective whenever |ν(n)| ⩾ sup(i + 2, 2). (Notice that this an injective stability result without any stability conditions on σ.) Finally if the underlying ring A of σ is commutative, σ is major, and SL_{n}(A) denotes the usual special linear group of rank n then

G(σ^{(n)}) ∩ G^{0}([ν] + σ) = G(σ^{(n)}) ∩ SL_{n}(A).

Proof. The proof of the last assertion is the same as that in [Bk1] (3.7) for the special case that ν has just one equivalence class.

The stabilization assertions follow directly from Theorem 4.3 and the definition of G^{i}.

To prove the first assertion, it suffices, by the last assertion in (2.5), to establish the special case that dim(σ) = 0. Let A denote the underlying ring of σ. By (6.8), sr(A) = 1. Thus the special case follows from (4.3)(iii).

It remains to prove the second assertion. To establish the descending G^{0}([ν]+σ)- central series, we can assume that σ = [ν] + σ is major. We would be finished, if we could apply Theorem 2.7. The hypotheses of (2.7) are precisely that the natural transformation π : St(ν,_{−}) → G(_{−}) is good on ((major ν-nets)) and that E(ν, σ) is normal in G(σ) whenever dim(σ) < ∞. But the former hypothesis is proved in Lemma 6.9 and the latter in Theorem 5.7. It follows also from Theorem 2.7 that G^{i}(σ) = E(ν, σ) whenever i ⩾ dim(σ). Thus for an arbitrary ν-net σ, G(σ) ∩ G^{i}([ν]+σ) = G(σ)∩E(ν, [ν]+σ) whenever i ⩾ dim(σ), because dim(σ) = dim([ν]+ σ). The inclusion [G(σ) ∩ E(ν, [ν] + σ), C(ν, σ)] ⩽ E([ν] + σ) follows immediately from Theorem 5.7.

Let N ⩾ 0. Define dim[−N]: ((ν-nets)) → ℤ^{⩾0} ∪ {∞} by

dim[−N](σ) = \{ \begin{matrix}0, & if dim(σ) ⩽ N \\ dim(σ) − N, & if dim(σ) ⩾ N . \end{matrix}

By (2.8) and (6.8), dim[−N] is a dimension function on the category with structure (((ν-nets)), LFC((ν-nets)), Frac((ν-nets))). Let G = G^{[−1]} ⩾ G^{[0]} ⩾ G^{[1]} ⩾ ⋯ denote the dim[−N]-filtration of the natural transformation π on G.

7.3. Corollary. Let N ⩾ 2. Let (R, A, σ) be a ν-net such that h(ν) ⩾ N ⩾ sr A+1. Then all of the conclusions of Theorem 7.2 hold after G^{i} is replaced by G^{[i]}.

Proof. The proof is identical to that of Theorem 7.2.

7.4 Theorem. Let σ denote any ν-net. If h(ν) ⩾ 3 then the action

G^{0}([ν] + σ) ↷ C(ν, σ)/ G^{0}(σ)

of G^{0}([ν] + σ) on C(ν, σ)/ G^{0}(σ) by conjugation is trivial. If h(ν) ⩾ 4 then the filtration C(ν, σ) ⩾ C(ν, σ) ∩ G^{0}([ν] + σ) ⩾ C(ν, σ) ∩ G^{1}([ν] + σ) ⩾ ⋯ ⩾ E(ν, σ) is a descending G^{0}([ν] + σ)-central series of normal subgroups of C(ν, σ) such that if i ⩾ dim(σ) then

C(ν, σ) ∩ G^{i}([ν] + σ) = C(ν, σ) ∩ E(ν, [ν] + σ), and
[C(ν, σ) ∩ G^{i}([ν] + σ), C(ν, σ)] = E(ν, σ).

Proof. To prove the first assertion, it suffices, by the last assertion of (2.5), to establish the special case that dim(σ) = 0. Under this assumption, G^{0}([ν] + σ) = E(ν, [ν] + σ) and by Theorem 5.7, [E(ν, [ν] + σ), C(ν, σ)] = E(ν, σ), i.e. the action of E(ν, [ν] + σ) on C(ν, σ)/ G^{0}(σ) by conjugation is trivial. The rest of the proof is the same as that of Theorem 7.2. 7.5 Corollary. Let N ⩾ 0. Let σ denote any ν-net. Then all of the conclusions of Theorem 7.4 hold after G^{i} is replaced by G^{[i]}. Proof. The proof is identical to that of Theorem 7.4.

The next result generalizes [Bk1] Theorem 6.25 concerning the special case ν has just one equivalence class. 7.6. Nilpotent Sandwich Classification Theorem. Let A be a quasi-finite ring and ν an equivalence on J = {1, ⋯, n} (where n can be infinite). Suppose h(ν) ⩾ 4. Then the E(ν)-normal subgroups of GL(n, A) are in one to one correspondence with the subgroups H of the disjoint sandwiches E(ν, σ) ⩽ H ⩽ C(ν, σ) where σ ranges over all ν-nets. Furthermore if F ⩽ G^{0}([ν] + σ) and F normalizes H then

H ⩾ H ∩ G^{0}([ν] + σ) ⩾ H ∩ G^{1}([ν] + σ) ⩾ ⋯ ⩾ E(ν, σ)

is a descending F -central series such that if i ⩾ dim(σ) then H ∩ G^{i}([ν] + σ) = H ∩ E(ν, [ν] + σ), and

[H ∩ G^{i}([ν] + σ), F ∩ C(ν, σ)] ⩽ E(ν, σ).

In particular, if F ⩽ C(ν, σ) ∩ G^{0}([ν] + σ) normalizes H and dim(σ) < ∞ then

H ⩾ H ∩ G^{0}([ν] + σ) ⩾ H ∩ G^{1}([ν] + σ) ⩾ ⋯ ⩾ H ∩ G^{dim(σ)}([ν] + σ) ⩾ E(ν, σ) is a descending F -central series. Proof. The theorem is an immediate consequence of Theorems 5.8 and 7.4. 7.7. Corollary. Let N ⩾ 0. If h(ν) ⩾ 4 then the conclusions of Theorem 7.6 remain valid after replacing dim by dim[−N] and G^{i} by G^{[i]}. Proof. The proof is exactly the same as that of Theorem 7.6.

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