K-Theory 29 (2003), 1 - 11

Stability for Quadratic K_{1}

Anthony Bak^{∗}

Dept. of Math., University of Bielefeld, 33501 Bielefeld, Germany

E-mail: bak@mathematik.uni-bielefeld.de

Viktor Petrov*

Dept. of Math and Mech., St. Petersburg State University 198904, Peterhof, Russia E-mail: victor@vp11701.spb.edu

Tang, Guoping^{†}

Dept. of Math., Northwestern Polytechnical University 710072, Xian, P.R.China E-mail: tanggp@nwpu.edu.cn

dedicated to Hyman Bass on his 70’th birthday

21.02.2003

Abstract

The general quadratic group GQ_{2n} and its elementary subgroup EQ_{2n} are analogs in the theory of quadratic forms of the general linear group GL_{n} and its elementary subgroup E_{n}. This article proves that the stabilization map GQ_{2n}/EQ_{2n} ⟶ GQ_{2(n+1)}/EQ_{2(n+1)} is an isomorphism whenever n ≥ ΛS + 1 and ΛS denotes the Λ-stable rank of rings with antiinvolution. As a corollary, a result is obtained which has been anticipated since the late 1960’s: Over rings of finite Bass-Serre dimension d, the stabilization map is an isomorphism whenever n ≥ d + 2.

∗The first two authors were partially support by DAAD, INTAS 00-566 and the partnership program Bielefeld University - St. Petersburg State University.

†The third author is supported by the National Natural Science Foundation of China (No. 10271094)

1 Introduction

Let R denote an associative ring with identity and antiinvolution. Let Λ denote a form parameter on R defined with respect to the antiinvolution. Let GQ_{2n}(R, Λ) denote the general quadratic group of rank n over the form ring (R, Λ) and let EQ_{2n}(R, Λ) denote its elementary subgroup. The role of the group GQ_{2n} and its elementary subgroup EQ_{2n} in the theory of nonsingular quadratic modules is analogous to that of the general linear group GL_{n} and its elementary subgroup E_{n} in the theory of finitely generated projective modules.

Let KQ_{1,n}(R, Λ) = GQ_{2n}(R, Λ)/EQ_{2n}(R, Λ). There is a natural embedding GQ_{2n}(R, Λ) → GQ_{2(n+1)}(R, Λ) of groups which induces a map KQ_{1,n} (R, Λ) → KQ_{1,n+1}(R, Λ) of coset spaces called the stabilization map. Our main result is the following.

Theorem 1.1 Suppose R satisfies the Λ-stable range condition ΛS_{m} where m ≥ 2. If n > m then

KQ_{1,n}(R, Λ)

is a group, the stabilization map

KQ_{1,n−1}(R, Λ) → KQ_{1,n}(R, Λ)

is surjective, and the stabliziation map

KQ_{1,n}(RΛ) → KQ_{1,n+1}(R, Λ)

is an isomorphism of groups.

Stability results of the kind above have been available in the literature for a long time, beginning with the papers [B1] and [B2] for surjective stability and the papers [V1] and [V2] for surjective and injective stability. These papers and later publications employed notions of stable range and dimension which are stronger than Λ-stable range. Their precise relation to Λ-stable range is given in the theorem below.

Several of the notions of stable range and dimension are defined for any associative ring with identity. They are Bass-Serre dimension BS, Krull dimension K dim of Rentschler and Gabriel, max Krull dimension K max of Stafford, dim(max spec) = length of longest chain of Jacobson prime ideals, and absolute stable range AS_{m} of level m. For an associative ring R with identity, antiinvolution, and form parameter Λ, let ΛUS_{m} denote the unitary stable range condition of level m on (R, Λ).

For any associative ring R with identity, let 𝔍(0) denote its Jacobson radical.

Theorem 1.2 Suppose R is a ring with antiinvolution and form parameter Λ. Then R satisfies ΛS_{m}, whenever one of the following conditions is satisfied.

(1.2.1) R satisfies AS_{m}. (1.2.2) R satisfies ΛUS_{m}. (1.2.3) m = 1 + BS(R). (1.2.4) R is right Noetherian and m = 1 + K dim(R/𝔍(0)).

(1.2.5) R is a P I ring, either R is finitely generated as an algebra over a

Noetherian subring in center (R) or R is module finite over a J-Noetherian subring in center (R), and m = 1 + dim(max spec(R)).

(1.2.6) R is strongly right J-Noetherian and m = 1 + K max(R).

In view of the theorems above, one might expect that existing surjective (resp. injective) stability results in the literature begin at m (resp. m + 1). However, this is true only for surjective stability. Injective stability results in the literature begin at m + 2. It follows that Theorem 1.1 provides not just a qualitative, but also a quantitative advance for stability results. Its corollary that the stabilization map is an isomorphism whenever n ≥ 2 + BS(R) has been sought since the late 1960’s and is a direct analog for GQ of Bass’ original stability theorem for GL. Moreover Theorem 1.1 as well as its analog for Hermitian forms in [BT] shows that Λ-stable range is the right analog for rings with antiinvolution of Bass’ notion of stable range for rings.

The literature records several times dissatisfaction with the proof of injective stability, cf. [HO, p. 521] and [K, p. vi]. Moreover accepted proofs such as M. Saliani’s in [K, VI (4.7.1)] are still long and complicated. In the current paper, we provide a one and a half page proof of injective stability, which is self contained modulo a basic elementary knowledge of quadratic modules over form rings, such as presented in the first 3 sections of chapter V of the standard text [HO] of Hahn and O’Meara or in sections 1-5 of chapter I and section 4 of chapter VI of the textbook [K] of Knus. (Either of these texts can be supplemented by sections 1-6 in chapter I and sections 1-5 in chapter II of [Bs 2], sections 1 and 2 in [BV], and sections 1B, 2 and 3 in [B3].) Although the presentations in these sources assume that the symmetry λ ∈ center (R), it is a minor detail dropping this assumption, or the reader can simply assume in the current paper that λ ∈ center (R).

The rest of the paper is organized as follows. In §2, we recall the notions of antiinvolution, form parameter, general quadratic group, and elementary quadratic group, and prove a couple elementary facts concerning the last 2 notions. In §3, we recall the notions of stable rank and dimension used above and prove Theorem 1.2. In §4, we provide an elementary proof of Theorem 1.1, using just the material in §2.

2 Preliminaries on GQ and EQ

In this section, we recall the definition of the general quadratic group GQ_{2n} and its elementary subgroup EQ_{2n} and prove a couple easy lemmas.

Let R denote an associative ring with identity. Recall that an antihomomorphism h : R → R is an additive map such that h(ab) = h(b)h(a) for all a, b ∈ R. An antiinvolution − : R → R is an antiisomorphism for which there is an element λ ∈ R with the property that λaλ̄ = ¯ ā for all a ∈ R. Setting a = 1, we obtain λλ̄ = 1. We claim that λ̄λ = 1, from which it will follow that λ is invertible in R and λ^{−1} = λ̄. Clearly ¯ λ̄ = λλλ̄ = λ(λλ̄) = λ. Thus 1 = (λλ̄)(λλ̄) = λ(λ̄λ)λ̄ = \overline{\overline{λ̄λ}} = ¯ ¯ λ̄¯λ̄ = λ̄λ. Note that this argument doesn’t work, if in the definition above the condition λaλ̄ = ¯ ā is replaced by the condition a = λ̄¯āλ. In the latter case, we would have to insist additionally that λ is invertible in R. The element λ is called a symmetry of the antiinvolution - and is obviously unique up to an element c ∈ center (R) such that cc̄ = 1. An antiinvolution with symmetry λ will be called a λinvolution.

Let − : R → R denote a λ-involution on R. Let max^{λ}(R) = {a ∈ R | a = −āλ} and min^{λ}(R) = {a − āλ | a ∈ R}. On checks straightforward that max^{λ}(R) and min^{λ}(R) are closed under the operation a ↦ x̄ax for any x ∈ R. A λ- form parameter on R is an additive subgroup Λ of R such that

min^{λ}(R) ⊆ Λ ⊆ max^{λ}(R),

x̄Λx ⊆ Λ for all x ∈ R.

Let M denote a right R-module. A sesquilinear form on M is a biadditive map B : M × M → R such that B(va, wb) = āB(v, w)b for all v, w ∈ M and a, b ∈ R. To any sesquilinear form B on M, we associate a λ-Hermitian form

< , >_{B}: M × M → R

defined by

< v, w >_{B}= B(v, w) + \overline{B(w, v)}λ

and a Λ- quadratic form

q_{B} : M → R/Λ

defined by

− q_{B}(v) = [B(v, v)].

The triple (M, < , >_{B}, q_{B}) is called a Λ- quadratic module. It is nonsingular if M is finitely generated and projective over R and the map M → Hom_{R}(M, R), m ↦ ⟨m, ⟩_{B}, is bijective. A morphism of Λ-quadratic modules is an R-linear map of right R-modules which preserves the associated λ-Hermitian and Λ-quadratic forms.

Let M denote a free right R-module with ordered basis e_{1}, ⋯, e_{n}, e_{−1}, ⋯, e_{−n}. Let φ denote the unique sesquilinear form on M such that the 2n × 2nmatrix

(φ(e_{i}, e_{j})) = ( \begin{matrix}0 & 0 \\ I & 0 \end{matrix} ) .

One checks straightforward that the Λ-quadratic module (M, < , >_{φ}, q_{φ}) is nonsingular. Using the basis of M, one can identify Aut (M, < , >_{φ}, q_{φ}) with a subgroup of the general linear group GL_{2n}(R). This subgroup is denoted by

GQ_{2n}(R, Λ)

and is called the general quadratic group of rank n. Its role in the theory of nonsingular quadratic forms is analogous to that of the general linear group in the theory of finitely generated projective modules. The following lemma provides a matrix characterization of the elements of GQ_{2n}(R, Λ) and can be used as a definition of GQ_{2n}(R, Λ).

If α = (a_{ij}) denotes an m × m matrix with coefficients a_{ij} ∈ R, let ᾱ = (a′_{ij}) where a′_{ij} = ā_{ji}. It is easy to see that the rule α ↦ ᾱ defines a λ-involution on the ring 𝕄_{m}(R) of all m × m matrices with coefficients in R.

Lemma 2.1 [B3, (3.1) and (3.4)] An invertible 2n × 2n matrix ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈

GL_{2n}(R) is an element of GQ_{2n}(R, Λ) ⇔ the following conditions hold.

(2.1.1) \overline{( \begin{matrix}α & β \\ γ & δ \end{matrix} )} ( \begin{matrix}& λI \\ I \end{matrix} ) ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) = ( \begin{matrix}& λI \\ I \end{matrix} ) .

(2.1.2) The diagonal coefficients of γ̄α and δ̄β lie in Λ. Moreover condition (2.1.1) is equivalent to the following condition.

(2.1.1)′  ( \begin{matrix}α & β \\ γ & δ \end{matrix} )^{−1} = ( \begin{matrix}δ̄ & β̄λ \\ λ̄γ̄ & λ̄ᾱλ \end{matrix} ) .

There is an obvious embedding

\begin{matrix}GQ_{2n}(R, Λ) & ⟶ & GQ_{2(n+1)}(R, Λ) \\ ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) & ⟼ & ( \begin{matrix}α & 0 & β & 0 \\ 0 & 1 & 0 & 0 \\ γ & 0 & δ & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) \end{matrix}

and using this map, we shall frequently want to consider GQ_{2n}(R, Λ) as a subgroup of GQ_{2(n+1)}(R, Λ).

We recall next the elementary quadratic matrices Hϵ_{ij}(a), r_{ij}(a) and l_{ij}(a) in GQ_{2n}(R, Λ). Let

Hϵ_{ij}(a) (a ∈ R and 1 ≤ i, j ≤ n, i ≠ j)

denote the 2n×2n matrix with 1 along the diagonal, a in the (i, j)’th position, −ā in the (n + j, n + i)’th position, and 0 elsewhere. Let

r_{ij}(a) (a ∈ R and 1 ≤ i, j ≤ n)

denote the 2n × 2n matrix with 1 along the diagonal, a in the (i, n + j)’th position, −āλ in the (j, n + i)’th position, and 0 elsewhere. If i = j, it is required additionally that a ∈ Λ. Let

l_{ij}(a) (a ∈ R and 1 ≤ i, j ≤ n)

denote the 2n × 2n matrix with 1 along the diagonal, a in the (n + i, j)’th position, −λ̄ā in the (n + j, i)’th position, and 0 elsewhere. If i = j, it is required additionally that a ∈ Λ̄. Using Lemma 2.2, one checks straightforward that each matrix above lies in GQ_{2n}(R, Λ). The subgroup generated by these matrices is denoted by

EQ_{2n}(R, Λ)

and is called the elementary quadratic group. The stabilization map GQ_{2n}(R, Λ) → GQ_{2(n+1)}(R, Λ) above induces a stabilization map EQ_{2n}(R, Λ) → EQ_{2(n+1)}(R, Λ) which will be used frequently to identify EQ_{2n}(R, Λ) with its image in EQ_{2(n+1)}(R, Λ).

The following subgroups of EQ_{2n}(R, Λ) will be used to establish a decomposition of EQ_{2n}(R, Λ) under stable range conditions. Let

C_{n} = < Hϵ_{in}(a), 1 ≤ i < n; l_{in}(a), 1 ≤ i ≤ n, a ∈ R >
R_{n} = < Hϵ_{nj}(a), 1 ≤ j < n; r_{nj}(a), 1 ≤ j ≤ n, a ∈ R >
P_{n} = {σσ_{1} | σ ∈ EQ_{2(n−1)}(R, Λ) and σ_{1} ∈ C_{n}}
Q_{n} = < Hϵ_{ij}(a), 1 ≤ i, j ≤ n, i ≠ j; l_{ij}(a), 1 ≤ i, j ≤ n, a ∈ R > .

Lemma 2.2 Let n ≥ 2. Suppose σ ∈ GQ_{2n}(R, Λ) such that the n’th row and n’th column of σ are identical with the n’th row and n’th column of the 2n × 2n identity matrix, respectively. Then the 2n’th row and 2n’th column of σ are identical with the 2n’th row and 2n’th column of the 2n×2n identity matrix, respectively. Consequently σ ∈ GQ_{2(n−1)}(R, Λ).

Proof Let

σ = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) .

By (2.1.1)′ ,

( \begin{matrix}α & β \\ γ & δ \end{matrix} )^{−1} = ( \begin{matrix}δ̄ & β̄λ \\ λ̄γ̄ & λ̄ᾱλ \end{matrix} ) .

Using the equation

( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ( \begin{matrix}δ̄ & β̄λ \\ λ̄γ̄ & λ̄ᾱλ \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) ,

one deduces straightforward the conclusion of the lemma. □

Lemma 2.3 GQ_{2(n−1)}(R, Λ) normalizes EQ_{2n}(R, Λ).

Proof By [B3, (3.16)], R_{n} and C_{n} generate EQ_{2n}(R, Λ). But it is obvious that R_{n} and C_{n} are normalized by GQ_{2(n−1)}(R, Λ). □

3 Λ-stable range condition

In this section, we recall the Λ-stable range condition and other stable range conditions and notions of dimension used in the article and prove Theorem 1.2.

Let R be an associative ring with identity. A vector (a_{1}, …, a_{n}) with coefficients a_{i} ∈ R is called right unimodular if there are elements b_{1}, …, b_{n} ∈ R such that a_{1}b_{1} + ⋯ + a_{n}b_{n} = 1. The stable range condition SR_{m} of H. Bass in the formulation of L. Vaserstein says that if (a_{1}, …, a_{m+1}) is a unimodular vector then there exist elements b_{1}, …, b_{m} ∈ R such that (a_{1} + a_{m+1}b_{1}, …, a_{m} + a_{m+1}b_{m}) is unimodular. It follows easily that SR_{m} ⇒ SR_{n} for any n ≥ m. The stable rank SR(R) of R is the smallest number m such that SR_{m} holds.

Let R be an associative ring with identity and λ-involution a ↦ ā. Let Λ be a λ-form parameter on R. Let 𝕄_{n}(Λ̄) denote the set of all n×n matrices γ such that γ = −λ̄γ̄ and the diagonal coefficients of γ lie in Λ̄. The Λ-stable range condition ΛS_{m} of A. Bak and G. Tang says that R satisfies SR_{m} and that given any unimodular vector (a_{1}, ⋯, a_{m+1}, b_{1}, ⋯, b_{m+1}) ∈ R^{2m+2} there exists a matrix γ ∈ 𝕄_{m+1}(Λ̄) such that (a_{1}, ⋯, a_{m+1}) + (b_{1}, ⋯, b_{m+1})γ is unimodular. By [BT, (3.3)], ΛS_{m} ⇒ ΛS_{n} for all n ≥ m. The Λ-stable rank ΛS(R) of R is the smallest number m such that ΛS_{m} holds.

We recall next two other stable range conditions which have been used to prove stability results for quadratic forms and record a lemma stating that each is stronger than ΛS_{m}. The lemma has the consequence that stability results based on ΛS_{m} are stronger that those based on the other stable range conditions.

If S is a set of elements of R, let

𝔍(S) = ( ⋂_{𝔐⫆S} 𝔐) ⋂ R

where 𝔐 runs through all maximal right ideals of R. Note that a vector (a_{1}, …, a_{n}) is unimodular ⟺ 𝔍{a_{1}, …, a_{n}} = R. The absolute stable range condition AS_{m} of M. Stein, W. van der Kallen, B. Magurn, and L. Vaserstein says if (a_{1}, ⋯, a_{m+1}) is a vector then there are elements x_{1}, ⋯, x_{m} ∈ R such that a_{m+1} ∈ 𝔍{a_{1} + a_{m+1}x_{1}, ⋯, a_{m} + a_{m+1}x_{m}}, i.e. 𝔍{a_{1}, ⋯, a_{m+1}} = 𝔍{a_{1} + a_{m+1}x_{1}, ⋯, a_{m} + a_{m+1}x_{m}}. It is not difficult to check that AS_{m} ⇒ AS_{n} for all n ≥ m. The absolute stable rank AS(R) of R is the smallest number m such that AS_{m} holds.

Let Λ be a form parameter on R. The Λ-unitary stable range condition ΛUS_{m} (cf. [HO, p. 526]) says R satisfies SR_{m} and given any unimodular vector (a_{1}, …, a_{m}, b_{1}, ⋯, b_{m}) there exists a vector (x_{1}, …, x_{m}, y_{1}, ⋯, y_{m}) such that x_{1}ȳ_{1}+…+x_{m}ȳ_{m} ∈ Λ and a_{1}x_{1}+…+a_{m}x_{m}+b_{1}y_{1}+⋯+b_{m}y_{m} = 1. It is not difficult to show that ΛUS_{m} ⇒ ΛUS_{n} for all n ≥ m. The Λ-unitary stable rank ΛUS(R) is the smallest number m such that ΛUS_{m} holds.

Lemma 3.1 [BT, (3.4) - (3.5)] The following holds for any ring R with form parameter Λ. (3.1.1) AS_{m} ⇒ ΛS_{m} (3.1.2) ΛUS_{m} ⇒ ΛS_{m}.

If P denotes a partially ordered set, let dev(P) denote its deviation, cf. [MR, 6.1.2], and let ℓ(P) denote the length of the longest chain in P. Let R denote an associative ring with identity. A right ideal 𝔄 in an associative ring R with identity is called a J-ideal (J stands for Jacobson), if 𝔄 = 𝔍(𝔄). Let ℒ(R) (resp. ℒ_{J}(R)) denote the lattice of all right ideals (resp. J-ideals) of R. Define the Krull dimension (in the sense of Rentschler and Gabriel) K dim(R) = devℒ(R), cf. [MR, 2.2], and the max Krull dimension (in the sense of Stafford) K max(R) = devℒ_{J}(R), cf. [S, §1]. Let Prime_{J}(R) denote the partially ordered set of all prime J-ideals of R and define dim(max spec (R)) = ℓ(Prime_{J}(R)), cf. [S, §0]. Let k denote a commutative ring. Let max spec(k) denote the set of all maximal ideals of k under the Zariski topology. If Y is a subspace of max spec(k), let Irr(Y) denote the partially ordered set of all irreducible closed subsets of Y. Define bs(k) to be the smallest natural number m such that max spec(k) is a finite union of Noetherian subspaces Y such that ℓ(Y) ≤ m. If no such natural number exists then set bs(k) = ∞. For an associative ring R with identity, define the Bass-Serre dimension BS(R) to be the smallest number m such that R is module finite over a commutative subring k ⊆ center (R) and bs(k) ≤ m.

Proof of Theorem 1.2 (1.2.1) and (1.2.2) are the assertions of Lemma 3.1.

(1.2.3) By [KMV, Theorem 3.7], R satisfies AS_{m} and so we are finished

thanks to (1.2.1).

(1.2.4) (resp. (1.2.5), (1.2.6)) By [S, Theorem A (i)] (resp. [S, Theorem A

(ii)], [S, Theorem B]), R satisfies AS_{m} and so we are finished again by (1.2.1). □

4 Proof of Theorem 1.1

Throughout this section, R is an associative ring with identity and λ-involution a ↦ ā and Λ is a λ-form parameter on R. It will be assumed throughout that

R satisfies the Λ-stable range condition ΛS_{m}.

This implies by definition that R satisfies Bass’ stable range condition SR_{m}. Recall the subgroup Q_{n} of EQ_{2n}(R, Λ), which was defined in §2.

Lemma 4.1 Let n ≥ m + 1. Then for any σ ∈ GQ_{2n}(R, Λ), there is an element τ ∈ Q_{n} such that στ has 0 in the first (n − 1) entries of its n’th row and 1 in the n’th entry of this row.

Proof Let

σ = ( \begin{matrix}α & β \\ γ & δ \end{matrix} )

be the 2 × 2 block matrix description of σ provided in Lemma 2.2. Clearly (α, β) is a unimodular vector in (𝕄_{n}(R))^{2}. Let v denote the bottom row of α and w the bottom row of β. Then (v, w) is the bottom row of (α, β) and hence is unimodular in R^{2n}.

Since R satisfies ΛS_{m} and n ≥ m + 1, there exists a matrix γ ∈ 𝕄_{n}(\overline{Λ}) such that v + wγ is unimodular in R^{n}. Set

τ_{1} = ( \begin{matrix}I & 0 \\ γ & I \end{matrix} ) = ∏_{1≤i≤j≤n} l_{ij}(a_{ij}) ∈ Q_{n}

where a_{ij} is the (i, j)’th coefficient of γ. Since R satisfies SR_{m} and n ≥ m+1, there is by [Bs1, Theorem 5.3.3] a product ϵ of elementary n × n matrices such that (v + wγ)ϵ = (0, ⋯, 0, 1). Set

τ_{2} = ( \begin{matrix}ϵ & 0 \\ 0 & ϵ̄^{−1} \end{matrix} ) ∈ Q_{n}.

Then στ_{1}τ_{2} has n’th row ((0, ⋯, 0, 1), w). □

Recall the subgroups P_{n}, Q_{n}, and R_{n} of EQ_{2n}(R, Λ), which are defined in §2.

Definition 4.2 Let ϕ ∈ EQ_{2n}(R, Λ). A P RQ-decomposition of ϕ is a product decomposition ϕ = σατ where σ ∈ P_{n}, α ∈ R_{n}, and τ ∈ Q_{n}.

Decomposition Theorem 4.3 Let n ≥ m + 2. Then every element of EQ_{2n}(R, Λ) has a P RQ-decomposition, i.e. EQ_{2n}(R, Λ) = P_{n}R_{n}Q_{n}.

Proof Let ϕ ∈ EQ_{2n}(R, Λ). A P RQ-decomposition σατ of ϕ will be called reduced if the (n − 1, n)’th coefficient of σ is 0. The strategy of the proof is as follows. First we show that if ϕ has a P RQ-decomposition then it has a reduced one. Then we identify a set of generators θ of EQ_{2n}(R, Λ) and show using reduced P RQ-decompositions that θP_{n}R_{n}Q_{n} ⊆ P_{n}R_{n}Q_{n}. It follows trivially that EQ_{2n}(R, Λ) = P_{n}R_{n}Q_{n}.

Let σατ be a P RQ-decomposition of ϕ. Write

σ = ( \begin{matrix}σ_{11} & σ_{12} & σ_{13} & 0 \\ 0 & 1 & 0 & 0 \\ σ_{31} & σ_{32} & σ_{33} & 0 \\ σ_{41} & σ_{42} & σ_{43} & 1 \end{matrix} )

and set

σ_{1} = ( \begin{matrix}σ_{11} & σ_{13} \\ σ_{31} & σ_{33} \end{matrix} ) .

By definition, σ_{1} ∈ EQ_{2(n−1)}(R, Λ). Since n ≥ m + 2, it follows from (4.1) that there is a τ_{1} ∈ Q_{n−1} such that the (n − 1, n − 1)’th coefficient of σ_{1}τ_{1} is 1. It is obvious that if τ_{1} is identified with its image in EQ_{2n}(R, Λ) (under the stabilization map EQ_{2(n−1)}(R, Λ) ⟶ EQ_{2n}(R, Λ)) then τ_{1} ∈ Q_{n} and the (n − 1, n − 1)’th coefficient of στ_{1} is 1. Furthermore τ_{1} ∈ P_{n} ∩ Q_{n} and τ_{1} normalizes R_{n}. Thus (στ_{1})(τ_{1}^{−1}ατ_{1})(τ_{1}^{−1}τ) is a P RQ-decomposition of ϕ such that the (n − 1, n − 1)’th coefficient of στ_{1} is 1. Choose x ∈ R such that the (n − 1, n)’th coefficient of στ_{1}Hϵ_{n−1,n}(x) is 0. Choose y ∈ R such that the (n, n − 1)’th coefficient of τ_{1}^{−1}ατ_{1}Hϵ_{n,n−1}(y) is 0. Let τ_{2} = Hϵ_{n−1,n}(x) and τ_{3} = Hϵ_{n,n−1}(y). Then τ_{2}^{−1}(τ_{1}^{−1}ατ_{1}τ_{3})τ_{2} = σ_{2}α_{1} for some σ_{2} ∈ EQ_{2(n−1)}(R, Λ) ⊆ P_{n} and some α_{1} ∈ R_{n}. Thus ϕ = σατ = (στ_{1}τ_{2})(τ_{2}^{−1}(τ_{1}^{−1}ατ_{1}τ_{3})τ_{2})(τ_{2}^{−1}τ_{3}^{−1}τ_{1}^{−1}τ) = (στ_{1}τ_{2}σ_{2})α_{1}(τ_{2}^{−1}τ_{3}^{−1}τ_{1}^{−1}τ) which is a reduced P RQ-decomposition of ϕ.

The relations

Hϵ_{ni}(a) = [Hϵ_{n,n−1}(a), Hϵ_{n−1,i}(1)] (a ∈ R and 1 ≤ i ≤ n − 1), r_{nj}(a) = [Hϵ_{nj}(a), r_{ji}(1)] (a ∈ R and 1 ≤ i ≠ j ≤ n − 1), r_{nn}(a)r_{n−1,n}(−a) = [r_{n−1,n−1}(a), Hϵ_{n,n−1}(1)], (a ∈ Λ),

show that P_{n} and the matrices Hϵ_{n,n−1}(a) (a ∈ R) generate EQ_{2n}(R, Λ). Obviously P_{n}(P_{n}R_{n}Q_{n}) ⊆ P_{n}R_{n}Q_{n}. Let σατ be a reduced P RQ-decomposition. Since the (n − 1, n)’th coefficient of σ is 0, σ can be expressed as a product σ = σ_{3}σ_{4} where σ_{3} ∈ C_{n} such that the (n − 1, n)’th coefficient of σ_{3} is 0 and σ_{4} ∈ EQ_{2(n−1)}(R, Λ). A straightforward computation shows that Hϵ_{n,n−1}(a)σ_{3}Hϵ_{n,n−1}(−a) ∈ P_{n} and it is clear that EQ_{2(n−1)}(R, Λ) normalizes R_{n}. Thus Hϵ_{n,n−1}(a)σατ = (Hϵ_{n,n−1}(a)σ_{3}Hϵ_{n,n−1}(−a)σ_{4}) (σ_{4}^{−1}Hϵ_{n,n−1}(a)σ_{4}α)τ which is a P RQ-decomposition. □

Proof of Theorem (1.1) Let σ ∈ GQ_{2n}(R, Λ). By (4.1), there is a τ_{1} ∈ Q_{n} ⊆ EQ_{2n}(R, Λ) such that στ_{1} has 0 in the first (n − 1) entries of its n’th row and 1 in the n’th entry of this row. It follows that there is a matrix τ_{3} =

∏_{i=1}^{n} ℓ_{in}(y_{i}) ∏_{i=1}^{n−1} Hϵ_{in}(y′_{i}) such that τ_{3}στ_{1} has the same n’th row as στ_{1}τ_{2} and the

same n’th column as the 2n × 2n identity matrix. Now we can find a matrix

τ_{2} = ∏_{i=1}^{n} r_{in}(z_{i}) such that τ_{3}στ_{1}τ_{2} has the same n’th row and n’th column as

2n × 2n identity matrix. Thus the element ρ = τ_{3}στ_{1}τ_{2} ∈ GQ_{2(n−1)}(R, Λ), by (2.2). So ρ normalizes EQ_{2n}(R, Λ), by (2.3). Since σ = τ_{3}^{−1}τ_{4}^{−1}ρτ_{2}^{−1}τ_{1}^{−1}, it follows that σ normalizes EQ_{2n}(R, Λ). Thus KQ_{1,n}(R, Λ) is a group and the map KQ_{1,n−1}(R, Λ) ⟶ KQ_{1,n}(R, Λ) is surjective.

Let ϕ ∈ GQ_{2n}(R, Λ) ∩ EQ_{2(n+1)}(R, Λ). Let σατ be a P_{(n+1)}R_{(n+1)}Q_{(n+1)}- decomposition of ϕ. Since the (n + 1)’th row of σ coincides with that of the 2(n + 1) × 2(n + 1) identity matrix, it follows that the (n + 1)’th row of σατ coincides with the (n + 1)’th row of ατ. Thus the (n + 1)’th row of ατ coincides with that of the 2(n + 1) × 2(n + 1) identity matrix. Write

τ = ( \begin{matrix}ϵ & 0 \\ γ & ϵ̄^{−1} \end{matrix} ) .

If (v, w) denotes the (n + 1)’th row of α then the (n + 1)’th row of ατ is

(v, w) ( \begin{matrix}ϵ & 0 \\ γ & ϵ̄^{−1} \end{matrix} ) = (vϵ + wγ, wϵ̄^{−1}).

Thus wϵ̄^{−1} = 0. Since ϵ̄^{−1} is invertible, w = 0. Thus α ∈ Q_{n+1}. Write σ = σ_{1}τ_{1} where σ_{1} ∈ EQ_{2n}(R, Λ) and τ_{1} ∈ C_{(n+1)} ⊆ Q_{(n+1)}. Obviously ϕ = σ_{1}(τ_{1}ατ) and τ_{1}ατ ∈ Q_{(n+1)} ∩ GQ_{2n}(R, Λ). It suffices to show that τ_{1}ατ ∈ EQ_{2n}(R, Λ). In fact, we shall show that τ_{1}ατ ∈ Q_{n}.

Write

τ_{1}ατ = ( \begin{matrix}ϵ_{1} & 0 \\ γ_{1} & ϵ̄_{1}^{−1} \end{matrix} ) .

Since τ_{1}ατ ∈ GQ_{2n}(R, Λ), it follows that the last row and column of γ_{1} are zero and ϵ_{1} ∈ GL_{n}(R). From the definition of Q_{n+1}, ϵ_{1} is an (n + 1) × (n + 1) matrix in the elementary group E_{n+1}(R). Thus ϵ_{1} ∈ GL_{n}(R) ∩ E_{n+1}(R). To complete the proof, it suffices to show that ϵ_{1} ∈ E_{n}(R). Since R satisfies SR_{m} (because it satisfies ΛS_{m}) and n ≥ m + 1, it follows by stability for K_{1} of the general linear group [Bs1, Theorem 5.4.2 ] that GL_{n}(R) ∩ E_{n+1}(R) = E_{n}(R). □

References

[B1] A. Bak, The stable structure of quadratic modules, Thesis, Columbia University, 1969

[B2] A. Bak, On modules with quadratic forms, in: Algebraic K-Theory and its Applications, Lecture Notes in Mathematics 108 (1969), 55 - 66

[B3] A. Bak, K-theory of Forms, Ann. Math. Studies 98 (1981), Princeton Univ. Press, Princeton, N.J.

[BT] A. Bak and Guoping Tang, Stability for Hermitian K_{1}, J. Pure & Appl. Alg. 150 (2000), 107 - 121

[BV] A. Bak and N. Vavilov, Structure of Hyperbolic Unitary Groups I : Elementary Subgroups, Alg. Coll. 7:2 (2000), 159 - 196

[Bs1] H. Bass, Algebraic K-theory, Benjamin, New York, 1968

[Bs2] H. Bass, Unitary Algebraic K-theory, in: Algebraic K-Theory, III: Hermitian K-Theory and Geometric Applications, Lecture Notes in Mathematics 343 (1973), 57 - 265

[HO] A. Hahn O.T.O’Meara, The Classical Groups and K-theory, Grundlehren der Mathematischen Wissenschaften 291. Berlin, Heidelberg, New York. Springer 1989

[K] Max-Albert Knus, Quadratic and Hermitian Forms over Rings, Grundlehren der Mathematischen Wissenschaften 294. Berlin, Heidelberg, New York. Springer 1991

[KMV] W. van der Kallen, B. Magurn, and L. N. Vaserstein, Absolute stable rank and Witt cancellation for non-commutative rings, Invent. Math. 91 (1988), 525-542

[MR] J. C. McConnel and J. C. Robson, Noncommutative Noetherian Rings, Pure & Appl. Math., John Wiley, New York, 1987

[S] J. T. Stafford, Absolute Stable Rank and Quadratic Forms over Noncommutative Rings, K-Theory 4 (1990), 121 - 130 [V1] L. N. Vaserstein, Stabilization of unitary and orthogonal groups over

a ring, Math. USSR Sbornik 10 (1970), 307 - 326

[V2] L. N. Vaserstein, Stabilization for classical groups over rings, Math. USSR Sbornik 22 (1975), 271 - 303