J. Pure & Appl. Alg. 150(2000) 107-121

Stability for Hermitian K_{1}

Anthony Bak

Dept. of Math., University of Bielefeld, 33615 Bielefeld, Germany E-mail: bak@mathematik.uni-bielefeld.de

Tang, Guoping

Dept. of Math., Shaanxi Normal University 710062, Xian, Shaanxi, P.R.China

Abstract

The general Hermitian group GH_{2n} and its elementary subgroup EH_{2n} are the analogs in the theory of Hermitian forms of the general linear group GL_{n} and its elementary subgroup E_{n}. This article proves that the canonical map GH_{2n}/EH_{2n} ⟶ GH_{2(n+1)}/EH_{2(n+1)} is an isomorphism whenever n is large with respect to a suitable stable range condition for rings with involution.

1 Introduction

An open question since the 1960’s is whether stability theorems for K_{0} and K_{1} of projective modules and quadratic forms have analogs for Hermitian forms. This paper establishes an analog for K_{1} and a companion article [BT] an analog for K_{0}.

The long time required for demonstrating a K_{1}-analog is explained by the lack of a notion of elementary subgroup in the general Hermitian group, which is necessary in formulating a K_{1}-stability result. This subgroup was discovered recently by the second author [T].

The general Hermitian group GH_{2n}(R, a_{1}, …, a_{r}) is the analog in the theory of Hermitian forms of the general linear group GL_{n}(R) in the theory of projective modules. It is by definition the group of isomorphisms of an orthogonal sum 𝕄(a_{1}) ⊥ … ⊥ 𝕄(a_{n}) of metabolic planes 𝕄(a_{i}) where a_{i} = 0 for all i > r. There is an obvious stabilization homomorphism GH_{2n}(R, a_{1}, …, a_{r}) ⟶ GH_{2(n+1)}(R, a_{1}, …, a_{r}). The elementary Hermitian group EH_{2n}(R, a, …, a_{r}) is a subgroup of GH_{2n}(R, a, …, a_{r}), which is generated by certain functorially defined matrices called elementary Hermitian matrices. Some generators are very complex and require many nonzero off diagonal coefficients. The stabilization homomorphism takes EH_{2n}(R, a_{1}, …, a_{r}) to EH_{2(n+1)}(R, a_{1}, …, a_{r}). We define KH_{1,n}(R, a_{1}, …, a_{r}) = GH_{2n}(R, a_{1}, …, a_{r})/EH_{2n}(R, a_{1}, …, a_{r}). A priori KH_{1,n}(R, a_{1}, …, a_{r}) is just a coset space. The stabilization homomorphism above induces a stabilization map KH_{1,n}(R, a_{1}, …, a_{r}) ⟶ KH_{1,n+1}(R, a_{1}, …, a_{r}).

The stability theorem is proved under a stable range condition which is weaker than its predecessors and easier to apply. We describe this condition. Let R be an associative ring with identity 1 and involution a ↦ ā. Let λ ∈ center(R) such that λλ̄ = 1. Let max^{λ}(R) = {a ∈ R|a = −λā}. The ring R is said to satisfy the max^{λ}(R)-stable range condition max^{λ}(R)S_{m} of degree m if R satisfies the usual stable range condition SR_{m} of H. Bass and if given a (right) unimodular vector (a_{1}, ⋯, a_{(m+1)}, b_{1}, ⋯, b_{(m+1)}) of length 2(m + 1), there is an (m + 1) × (m + 1) −λ̄-Hermitian matrix γ such that (a_{1}, ⋯, a_{m+1}) + (b_{1}, ⋯, b_{m+1})γ is a unimodular of length m + 1.

The main result is as follows.

Theorem 1.1 Let R and max^{λ}(R) be as above. Suppose that R satisfies the stable range condition max^{λ}(R)S_{m}. Then for all n > m + r,

KH_{1,n}(R, a, …, a_{r})

is a group, the canonical map

KH_{1,n−1}(R, a_{1}, …, a_{r}) ⟶ KH_{1,n}(R, a_{1}, …, a_{r})

is surjective, and the canonical homomorphism

KH_{1,n}(R, a_{1}, …, a_{r}) ⟶ KH_{1,n+1}(R, a_{1}, …, a_{r})

is an isomorphism.

The rest of the article is organized as follows. In §2, we recall in detail the definitions of GH_{2n} and EH_{2n}, and of important subgroups of EH_{2n} which are used in establishing a decomposition of EH_{2n} when stable range conditions are imposed on R. In §3, we define a generalization of the stable range condition above, which uses form parameters, and show that it is weaker than its predecessors, namely the unitary stable range condition and that developed by W. van der Kallen, B. Magurn and L. Vaserstein. In §4, we prove our main result Theorem 1.1. An important tool in the proof is the decomposition theorem for EH_{2n}, which is also proved in the section.

2 Preliminaries on GH and EH

The basic references for the general Hermitian group GH and its elementary subgroup EH are [B] and [T]. The groups GH and EH are the analogs for Hermitian forms of the general quadratic group GQ and its elementary subgroup EQ in the theory of quadratic forms. Whereas the groups GQ and EQ have been known for a long time and their quotient KQ_{1} = GQ/EQ intensively studied, the group EH has been only recently discovered. Investigation of the quotient group KH_{1} = GH/EH and of the higher Hermitian K-groups KH_{i} defined using the Volodin construction is only beginning now. The topic K_{1}-stability for quadratic forms was treated already in the late 1960’s by A. Bak, H. Bass, and A. Roy, and in the early 1970’s by M. Kolster and L. Vaserstein. The fact that KQ-groups defined with respect to the maximal form parameter agree with KH-groups defined for r = 0, by [B, Theorem (1.1) and (1.3)], leads one to conjecture that stability results for KQ_{1}-groups have analogs for KH_{1}-groups.

We recall now the definitions of the groups GH and EH and of subgroups of EH which will be used in obtaining in §4 a decomposition of EH under the max^{λ}(R)-stable range condition.

We fix the following notation. Let R be an associative ring with identity 1 and involution a ↦ ā; thus \overline{ab} = b̄ā and ¯ ā = a for all a, b ∈ R. If α = (a_{ij}) denotes an m × n matrix with coefficients a_{ij} ∈ R, let ᾱ denote its conjugate transpose; thus ᾱ = (a′_{kl}) is the n × m matrix such that a′_{kl} = ā_{lk}.

Let r and n be natural numbers such that n ≥ r. Let λ ∈ center(R) such that λλ̄ = 1. Let a_{1}, …, a_{n} be a sequence of elements in R such that a_{i} = λā_{i} for all 1 ≤ i ≤ n and a_{r+1} = a_{r+2} = ⋯ = a_{n} = 0. In the context we are working, it makes sense letting r = 0 mean that a_{1} = ⋯ = a_{n} = 0. So we shall do this. Let

A_{1} = r × r diagonal matrix ( \begin{matrix}a_{1} \\ & . & . & . \\ & & & & a_{r} \end{matrix} ) ,

A = n × n diagonal matrix ( \begin{matrix}a_{1} \\ & . & . & . \\ & & & & a_{r} \\ & & & & & 0 \\ & & & & & & . & . & . \\ & & & & & & & & & 0 \end{matrix} ) ,

I = an identity matrix .

Define the n-th general Hermitian group of the elements a_{1}, …, a_{r} by

GH_{2n}(R, a_{1}, …, a_{r}) = \{σ ∈ GL_{2n}(R)| σ̄ ( \begin{matrix}A & λI \\ I & 0 \end{matrix} ) σ = ( \begin{matrix}A & λI \\ I & 0 \end{matrix} )\} .

A typical element of this group is denoted by a 2n × 2n matrix

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

where α, β, γ and δ are n × n block matrices. There is an obvious embedding

GH_{2n}(R, a_{1}, …, a_{r}) ⟶ GH_{2(n+1)}(R, a_{1}, …, a_{r})

( \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} ) .

and one defines

GH(R, a_{1}, …, a_{r}) = lim_{n≥r^{⟶}} GH_{2n}(R, a_{1}, …, a_{r}).

Let min^{−λ}(R) = {a + λā|a ∈ R}. For any a_{1}, …, a_{r} as above, let

C = { ^{t}(x_{1}, …, x_{r}) ∈ ^{t}(R^{r}) | ∑_{i=1}^{r} x̄_{i}a_{i}x_{i} ∈ min^{−λ}(R)}.

In order to deal effectively with technical difficulties caused by the elements a_{1}, …, a_{r}, we shall finely partition a typical matrix

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

of GH_{2n}(R, a_{1}, …, a_{r}) into the form

(2.1)  ( \begin{matrix}α_{11} & α_{12} & β_{11} & β_{12} \\ α_{21} & α_{22} & β_{21} & β_{22} \\ γ_{11} & γ_{12} & δ_{11} & δ_{12} \\ γ_{21} & γ_{22} & δ_{21} & δ_{22} \end{matrix} )

where α_{11}, β_{11}, γ_{11}, δ_{11} are r × r matrices, α_{12}, β_{12}, γ_{12}, δ_{12} are r × (n − r) matrices, α_{21}, β_{21}, γ_{21}, δ_{21} are (n − r) × r matrices, and α_{22}, β_{22}, γ_{22}, δ_{22} are (n − r) × (n − r) matrices. By [T, 3.4],

(2.2) the columns of α_{11} − I, α_{12}, β_{11}, β_{12}, β̄_{11}, β̄_{21}, δ̄_{11} − I and δ̄_{21} belong to C.

Letting GQ_{2n}(R, max^{λ}(R)) denote the general quadratic group [B, §3] over R for maximal form parameter max^{λ}(R), one checks straightforward that the subgroup of GH_{2n}(R, a_{1}, …, a_{r}) consisting of

(2.3) \{( \begin{matrix}I & 0 & 0 & 0 \\ 0 & α_{22} & 0 & β_{22} \\ 0 & 0 & I & 0 \\ 0 & γ_{22} & 0 & δ_{22} \end{matrix} ) ∈ GH_{2n}(R, a_{1}, …, a_{r})\} ≅ GQ_{2(n−r)}(R, max^{λ}(R)).

We identify now functorically defined elements of GH_{2n}, which will be used to generate EH_{2n}. The first 3 kinds of generators are taken for the most part from GQ_{2(n−r)}(R, max^{λ}(R)) which is embedded as in (2.3) as a subgroup of GH_{2n} and the last 2 kinds are motivated by the result (2.2) concerning the columns of a matrix in GH_{2n}.

Let

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

denote the 2n × 2n matrix with 1 along the diagonal, a in the (i, n + j)’th position, −λā in the (j, n + i)’th position, and 0 elsewhere. If i = j, this forces of course that a = −λā. 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, this forces of course that a = −λ̄ā.

For ζ = ^{t}(x_{1}, ⋯, x_{r}) ∈ C, let

ζ_{f} ∈ R such that ζ_{f} + λζ̄_{f} = ∑_{i=1}^{r} x̄_{i}a_{i}x_{i}.

The element ζ_{f} is not in general unique. Define

Hm_{i}(ζ) = ( \begin{matrix}I & α_{12} & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & −Ā_{1}α_{12} & I & 0 \\ 0 & γ_{22} & −ᾱ_{12} & I \end{matrix} ) (ζ ∈ C and r + 1 ≤ i ≤ n)

to be the 2n × 2n matrix such that α_{12} is the r × (n − r) matrix with ζ as its (i − r)’th column and all other column’s zero, and γ_{22} is the (n − r) × (n − r) matrix with ζ̄_{f} in its (i − r, i − r)’th position and 0 elsewhere. Define

r_{i}(ζ) = ( \begin{matrix}I & 0 & 0 & β_{12} \\ 0 & I & −λβ̄_{12} & β_{22} \\ 0 & 0 & I & −Ā_{1}β_{12} \\ 0 & 0 & 0 & I \end{matrix} ) (ζ ∈ C and r + 1 ≤ i ≤ n)

to be the 2n × 2n matrix such that β_{12} is the r × (n − r) matrix with ζ as its (i − r)’th column and all other columns 0, and β_{22} is the (n − r) × (n − r) matrix with λζ̄_{f} in its (i − r, i − r)’th position and 0 elsewhere.

Each of the matrices above is called an elementary Hermitian matrix for the elements a_{1}, …, a_{r}.

One can show by direct computation as in [T, §4] that each elementary Hermitian matrix is in GH_{2n}(R, a_{1}, …, a_{r}).

Define the n’th elementary Hermitian group

EH_{2n}(R, a_{1}, …, a_{r})

of the elements a_{1}, …, a_{r} to be the subgroup of GH_{2n}(R, a_{1}, …, a_{r}) generated by all elementary Hermitian matrices. It is obvious that the embedding GH_{2n}(R, a_{1}, …, a_{r}) ⟶ GH_{2(n+1)}(R, a_{1}, …, a_{r}) takes EH_{2n}(R, a_{1}, …, a_{r}) to EH_{2(n+1)}(R, a_{1}, …, a_{r}) and one defines

EH(R, a_{1}, …, a_{r}) = lim_{n≥r^{⟶}} EH_{2n}(R, a_{1}, …, a_{r}).

It is customary to identify GH_{2(n−1)}(R, a_{1}, …, a_{r}) and EH_{2(n−1)}(R, a_{1}, …, a_{r}), respectively, with their images in GH_{2n}(R, a_{1}, …, a_{r}) and EH_{2n}(R, a_{1}, …, a_{r}).

The following subgroups of EH_{2n}(R, a_{1}, …, a_{r}) will be used to establish a decomposition of EH_{2n}(R, a_{1}, …, a_{r}) under stable range conditions. Let

C_{n} = < Hϵ_{in}(a), r + 1 ≤ i < n; l_{in}(a), 1 ≤ i ≤ n and Hm_{n}(ζ), a ∈ R, ζ ∈ C >
R_{n} = < Hϵ_{nj}(a), 1 ≤ j < n; r_{nj}(a), r + 1 ≤ j ≤ n and r_{n}(ζ), a ∈ R, ζ ∈ C >
P_{n} = {σσ_{1} | σ ∈ EH_{2(n−1)}(R, a_{1}, …, a_{r}) and σ_{1} ∈ C_{n}}
Q_{n} = < Hm_{j}(ζ), Hϵ_{ij}(a), r + 1 ≤ i, j ≤ n, i ≠ j; and l_{ij}(a), 1 ≤ i, j ≤ n, a ∈ R, ζ ∈ C > .

Lemma 2.4 Suppose n ≥ 2. Suppose σ ∈ GH_{2n}(R, a_{1}, …, a_{r}) 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. In particular, if n > r then σ ∈ GH_{2(n−1)}(R, a_{1}, …, a_{r}).

Proof Let

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

By [T, (3.1)]

( \begin{matrix}α & β \\ γ & δ \end{matrix} )^{−1} = ( \begin{matrix}δ̄ + β̄A & λβ̄ \\ ᾱĀ − Āβ̄A + λ̄γ̄ − Āδ̄ & ᾱ − Aβ̄ \end{matrix} ) .

Using the equation

( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ( \begin{matrix}δ̄ + β̄A & λβ̄ \\ ᾱĀ − Āβ̄A + λ̄γ̄ − Āδ̄ & ᾱ − Aβ̄ \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

and the fact that A is a diagonal matrix, one deduces routinely the conclusion of the lemma. □

Lemma 2.5 GH_{2(n−1)}(R, a_{1}, …, a_{r}) normalizes EH_{2n}(R, a_{1}, …, a_{r}).

Proof By [T, (8.3)], R_{n} and C_{n} generates EH_{2n}(R, a_{1}, …, a_{r}). But it is obvious that R_{n} and C_{n} are normalized by GH_{2(n−1)}(R, a_{1}, …, a_{r}). □

The following corollary obvious.

Corollary 2.6 EH(R, a_{1}, …, a_{r}) is a normal subgroup of GH(R, a_{1}, …, a_{r}).

Furthermore according to the Hermitian Whitehead Lemma [T, §5], EH(R, a_{1}, …, a_{r}) is the commutator subgroup of GH(R, a_{1}, …, a_{r}). One defines

KH_{1,n}(R, a_{1}, …, a_{r}) = GH_{2n}(R, a_{1}, …, a_{r})/EH_{2n}(R, a_{1}, …, a_{r})

and

KH_{1}(R, a_{1}, …, a_{r}) = GH(R, a_{1}, …, a_{r})/EH(R, a_{1}, …, a_{r}).

Whereas KH_{1}(R, a_{1}, …, a_{r}) is an abelian group, KH_{1,n}(R, a_{1}, …, a_{r}) is in general just a coset space.

3 Λ-stable range condition

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 A. 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.

Suppose that R has an involution a ↦ ā. Let λ ∈ center(R) such that λλ̄ = 1. Let min^{λ}(R) = {a − λā|a ∈ R} and max^{λ}(R) = {a ∈ R|a = −λā}. A form parameter Λ is an additive subgroup of R such that

1) aΛā ⊆ Λ for all a ∈ R,

2) min^{λ}(R) ⊆ Λ ⊆ max^{λ}(R). Clearly the extremes in (2) satisfy (1) so that they are form parameters. Let

𝕄_{m}(Λ) ( resp. 𝕄_{m}(Λ̄))

denote the set of all m × m matrices γ such that γ = −λγ̄ and the diagonal coefficients of γ lie in Λ (resp. γ = −λ̄γ̄ and the diagonal coefficients of γ lie in Λ̄).

Definition 3.1 Let Λ be a form parameter on R. R is said to satisfy the Λ-stable range condition ΛS_{m} if it satisfies SR_{m} and if 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.

Lemma 3.2 The following conditions are equivalent for a ring R with involution and form parameter Λ ⊆ R.

(3.2.1) R satisfies ΛS_{m},

(3.2.2) R satisfies SR_{m} and given any unimodular vector (a_{1}, …, a_{m+1}, b_{1}, …, b_{m+1}) there is a 2(m + 1) × 2(m + 1) matrix

σ = ( \begin{matrix}I & 0 \\ γ & I \end{matrix} )

where I is the (m + 1) × (m + 1) identity matrix and γ ∈ 𝕄_{m+1}(Λ̄) such that vσ = (a′_{1}, …, a′_{m+1}, b′_{1}, …, b′_{m+1}) and (a′_{1}, …, a′_{m+1}) is unimodular.

(3.2.3) R satisfies SR_{m} and given any unimodular vector (a_{1}, …, a_{m+1}, b_{1}, …, b_{m+1}) there is a 2(m + 1) × 2(m + 1) matrix

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

where ϵ an invertible (m + 1) × (m + 1) matrix and γϵ^{−1} ∈ 𝕄_{m+1}(Λ̄) such that vσ = (a′_{1}, …, a′_{m+1}, b′_{1}, …, b′_{m+1}) and (a′_{1}, …, a′_{m+1}) is unimodular.

Proof It is clear that (3.2.1) ⟺ (3.2.2) ⇒ (3.2.3). Suppose that (3.2.3) holds. We show that (3.2.2) holds. Let

ρ = ( \begin{matrix}I & 0 \\ γϵ^{−1} & I \end{matrix} )

and vρ = (a′′_{1}, …, a′′_{m+1}, b′′_{1}, …, b′′_{m+1}). Since

σ = ρ ( \begin{matrix}ϵ & 0 \\ 0 & ϵ̄^{−1} \end{matrix} ) ,

it is clear that (a′_{1}, …, a′_{m+1}) = (a′′_{1}, …, a′′_{m+1})ϵ. Thus (a′_{1}, …, a′_{m+1})ϵ^{−1} = (a′′_{1}, …, a′′_{m+1}). Since(a′_{1}, …, a′_{m+1}) is unimodular and ϵ is invertible, it follows that (a′′_{1}, …, a′′_{m+1}) is unimodular. □

Lemma 3.3 ΛS_{m} ⇒ ΛS_{n} for all n ≥ m.

Proof We shall use the matrix notation introduced in §2 with r = 0. Let n > m. Clearly SR_{n} holds. By induction on n, we can assume that R satisfies ΛS_{n−1}. We shall show that R satisfies the formulation of ΛS_{n} given in (3.2.3).Let v = (a_{1}, ⋯, a_{n+1}, b_{1}, ⋯, b_{n+1}) be a unimodular vector. By SR_{n} (cf. [K, Chap VI, Remark 1.5.1]), there are elements x_{1}, …, x_{n} ∈ R such that if σ = Hϵ_{1n+1}(−x̄_{1}) ⋯ Hϵ_{nn+1}(−x̄_{n}) then vσ = (a′_{1}, ⋯, a′_{n+1}, b′_{1}, ⋯, b′_{n+1}) and (a′_{1}, ⋯, a′_{n+1}, b′_{1}, ⋯, b′_{n}) is unimodular. Again by SR_{n}, there exists elements y_{1}, …, y_{n} ∈ R such that if ρ = Hϵ_{n+1,1}(y_{1}) ⋯ Hϵ_{n+1,n}(y_{n}) then vσρ = (a′′_{1}, ⋯, a′′_{n+1}, b′′_{1}, ⋯, b′′_{n+1}) and (a′′_{1}, ⋯, a′′_{n}, b′′_{1}, ⋯, b′′_{n}) is unimodular. By ΛS_{n−1}, there is a 2n × 2n matrix

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

where γ′ ∈ 𝕄_{n}(Λ̄) such that (a′′_{1}, ⋯, a′′_{n}, b′′_{1}, ⋯, b′′_{n})τ′ = (c_{1}, ⋯, c_{n}, d_{1}, ⋯, d_{n}) and (c_{1}, ⋯, c_{n}) is unimodular. Let τ denote the 2(n + 1) × 2(n + 1) stabilization of τ′. Thus

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

where

γ = ( \begin{matrix}γ′ & 0 \\ 0 & 0 \end{matrix} ) ∈ 𝕄_{n+1}(Λ̄).

Clearly vσρτ = (c_{1}, ⋯, c_{n+1}, d_{1}, ⋯, d_{n+1}) and (c_{1}, ⋯, c_{n+1}) is unimodular because (c_{1}, ⋯, c_{n}) is. □

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}}.

Lemma 3.4 Let R be a ring with involution. Then AS_{m} ⇒ ΛS_{m} for any form parameter Λ on R.

Proof We show first that SR_{m} holds. Let (a_{1}, ⋯, a_{m+1}) be a unimodular vector. By AS_{m}, there are elements x_{1}, ⋯, x_{m} such that 𝔍{a_{1}, ⋯, a_{m+1}} = 𝔍{a_{1} +a_{m+1}x_{1}, ⋯, a_{m} +a_{m+1}x_{m}}. Since R = 𝔍{a_{1}, ⋯, a_{m+1}}, it follows that (a_{1} + a_{m+1}x_{1}, ⋯, a_{m} + a_{m+1}x_{m}) is unimodular. Thus SR_{m} holds.

We shall use now the equivalent formulation of ΛS_{m} given in (3.2.3). Let v = (a_{1}, ⋯, a_{2(m+1)}) be a unimodular vector. Let p = m + 3 and q = 2(m+1). By AS_{m}, there exist elements x_{1}, ⋯, x_{m} ∈ R such that if σ_{1} = Hϵ_{1,m+1}(−x̄_{1}) ⋯ Hϵ_{m,m+1}(−x̄_{m}) then vσ_{1} = (a_{1}^{(1)}, ⋯, a_{q}^{(1)}) and 𝔍{a_{p−1}^{(1)}, ⋯, a_{q−1}^{(1)}} = 𝔍\{a_{p−1}^{(1)}, ⋯, a_{q}^{(1)}\} = 𝔍\{a_{p−1}, ⋯, a_{q}\}. Let 2 ≤ n ≤ m + 1 (= \frac{q}{2}) and suppose that for each 1 ≤ i < n, we have found a q × q matrix

σ_{i} = ( \begin{matrix}ϵ_{i} & 0 \\ γ_{i} & ϵ̄_{i}^{−1} \end{matrix} )

as in (3.2.3) such that if vσ_{1} ⋯ σ_{i} = (a_{1}^{(i)}, ⋯, a_{q}^{(i)}) then 𝔍{a_{p−i}^{(i)}, ⋯, a_{q−i}^{(i)}} = 𝔍{a_{p−i}^{(i)}, ⋯, a_{q}^{(i)}}. We construct now a q × q matrix σ_{n} with the same properties. By AS_{m}, there exist elements y_{1}, ⋯, y_{m} ∈ R such that if

σ_{n} = Hϵ_{1, \frac{q}{2} −n+1}(−ȳ_{1}) ⋯ Hϵ_{\frac{q}{2} −n, \frac{q}{2} −n+1}(−ȳ_{\frac{q}{2} −n})l_{\frac{q}{2} −n+1, \frac{q}{2} −n+2}(y_{\frac{q}{2} −n+1}) ⋯ l_{\frac{q}{2} −n+1, \frac{q}{2}} (y_{m}), i.e.

σ_{n} = ( \begin{matrix}1 & & & −ȳ_{1} \\ & ... & & ⋮ \\ & & 1 & −ȳ_{\frac{q}{2} −n} \\ & & & 1 \\ & & & & 1 \\ . & . & . \\ & & & & & & 1 \\ & & & & & & & 1 \\ . & . & . \\ & & & & & & & & & 1 \\ & & & 0 & y_{\frac{q}{2} −n+1} & ⋯ & y_{m} & y_{1} & ⋯ & y_{\frac{q}{2} −n} & 1 \\ & & & −λ̄ȳ_{\frac{q}{2} −n+1} & & & & & & & & 1 \\ & & & ⋮ & & & & & & & & & ... \\ & & & −λ̄ȳ_{m} & & & & & & & & & & 1 \end{matrix} )

then vσ_{1} ⋯ σ_{n} = (a_{1}^{(n)}, ⋯, a_{q}^{(n)}) and 𝔍{a_{p−n}^{(n)}, ⋯, a_{q−n}^{(n)}} = 𝔍{a_{p−n}^{(n)}, ⋯, a_{q−n+1}^{(n)}} = 𝔍{a_{p−n}^{(n−1)}, ⋯, a_{q−n+1}^{(n−1)}}. By the induction the hypothesis, 𝔍{a_{p−n+1}^{(n−1)}, ⋯, a_{q−n+1}^{(n−1)}} = 𝔍{a_{p−n+1}^{(n−1)}, ⋯, a_{q}^{(n−1)}} and clearly, 𝔍{a_{p−n+1}^{(n−1)}, ⋯, a_{q}^{(n−1)}} = 𝔍{a_{p−n+1}^{(n)}, ⋯, a_{q}^{(n)}}. Thus 𝔍{a_{p−n}^{(n)}, ⋯, a_{q−n}^{(n)}} = 𝔍{a_{p−n}^{(n)}, ⋯, a_{q}^{(n)}}. By induction, we can construct a sequence σ_{1}, ⋯ σ_{m+1} of matrices σ_{i} as above such that 𝔍{a_{2}^{(m+1)}, ⋯, a_{m+1}^{(m+1)}} = 𝔍{a_{2}^{(m+1)}, ⋯, a_{2(m+1)}^{(m+1)}}. Thus 𝔍{a_{1}^{(m+1)}, ⋯, a_{m+1}^{(m+1)}} = 𝔍{a_{1}^{(m+1)}, ⋯, a_{2(m+1)}^{(m+1)}} = R. □

Let Λ be a form parameter on R. The Λ-unitary stable range condition ΛUS_{m} (cf. [HO, p. 526]) says that SR_{m} holds and that if (a_{1}, …, a_{m}, b_{1}, ⋯, b_{m}) is a unimodular vector then there is 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 any n ≥ m.

Lemma 3.5 ΛUS_{m} ⇒ ΛS_{m}.

Proof We use the formulation of ΛS_{m} given in (3.2.3). Let v = (a_{1}, ⋯, a_{m+1}, b_{1}, ⋯, b_{m+1}) be a unimodular vector. As in the proof of (3.3), we can find a product of hyperbolic elementary matrices Hϵ_{ij}(x) such that after multiplying v on the right by this product, we can assume (a_{1}, ⋯, a_{m}, b_{1}, ⋯, b_{m}) is unimodular. By ΛUS_{m}, there is 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. Let σ = Hϵ_{1,m+1}(x′_{1}) ⋯ Hϵ_{m,m+1}(x′_{m})l_{1,m+1}(y′_{1}) ⋯ l_{m+1,m+1}(y′_{m+1}) where x′_{i} = x_{i}(1 − a_{m+1}), y′_{i} = y_{i}(1 − a_{m+1}) (1 ≤ i ≤ m) and y′_{m+1} = − ∑_{i=1}^{m} \overline{(1 − a_{m+1})}x̄_{i}y_{i}(1 − a_{m+1}). Then vσ = (a′_{1}, ⋯, a′_{m+1}, b′_{1}, ⋯, b′_{m+1}) has the property that a′_{m+1} = 1. Thus (a′_{1}, ⋯, a′_{m+1}) is unimodular. By (3.2.3), we are finished. □

Lemma 3.6 Suppose that R is module finite over a subring k ⊆ center(R). Let Max(k) denote the maximal ideal spectrum of k in the Zariski topology. Define d_{k}(R) = dimension(Max(k)), cf. [Bs, pp. 92-102]. Suppose that Max(k) is Noetherian and d_{k}(R) is finite. Then R satisfies ΛS_{d_{k}(R)+1} for any form parameter Λ in R.

Proof By [MKV, Theorem (3.1)], R satisfies AS_{d_{k}(R)+1}. Thus by (3.4), we are finished. □

4 Proof of Theorem 1.1

Throughout this section, R denotes an associative ring with identity and involution a ↦ ā. λ denotes an element in the center(R) such that λλ̄ = 1 and max^{λ}(R) = {a ∈ R|a = −λā}. It will be assumed throughout that

R satisfies the stable range condition max^{λ}(R)S_{m}.

Lemma 4.1 Let n ≥ r + m + 1. Then for any σ ∈ GH_{2n}(R, a_{1}, …, a_{r}), there is an element τ ∈ Q_{n} such that στ has 1 in its (n, n)’th position.

Proof Let

σ = ( \begin{matrix}α_{11} & α_{12} & β_{11} & β_{12} \\ α_{21} & α_{22} & β_{21} & β_{22} \\ γ_{11} & γ_{12} & δ_{11} & δ_{12} \\ γ_{21} & γ_{22} & δ_{21} & δ_{22} \end{matrix} )

be the 4 × 4 block matrix description of σ given in (2.1) and (2.2). Since σ^{−1} ∈ GH_{2n}(R, a_{1}, …, a_{r}) and therefore also has such a description, there are r × (n − r) matrices x_{1}, y_{1} and (n − r) × (n − r) matrices x_{2}, y_{2} such that α_{21}x_{1} + α_{22}x_{2} + β_{21}y_{1} + β_{22}y_{2} = I and the columns of x_{1} lie in C. Thus (α_{21}x_{1}, α_{22}, β_{21}y_{1}, β_{22}) is a unimodular vector in (M_{(n−r)}(R))^{4}. Let v_{i} denote the bottom row of α_{2i} (i = 1, 2) and w_{i} denote the bottom row of β_{2i} (i = 1, 2). Then (v_{1}x_{1}, v_{2}, w_{1}, w_{2}) is the bottom row of (α_{21}x_{1}, α_{22}, β_{21}, β_{22}) and hence is unimodular in R^{3(n−r)+r}. Since the stable range condition SR_{m} holds and n − r ≥ m + 1, there exists (cf. [K, Chap.VI, Remark 1.5.1]) an (n − r) × (n − r) matrix z_{1} such that (v_{2} + v_{1}x_{1}z_{1}, w_{1}, w_{2}) is unimodular in R^{2(n−r)+r}. Since the columns of x_{1} belong to C, it follows straightforward that the columns of x_{1}z_{1} belong to C. Let ζ_{i} denote the i’th column of x_{1}z_{1} and let z_{2} be the (n − r) × (n − r) matrix defined by

( \begin{matrix}I & x_{1}z_{1} & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & −Ā_{1}x_{1}z_{1} & I & 0 \\ 0 & z_{2} & −z̄_{1}x̄_{1} & I \end{matrix} ) = ∏_{i=1}^{n−r} Hm_{r+i}(ζ_{i}) ∈ Q_{n}.

Set

τ_{1} = ( \begin{matrix}I & x_{1}z_{1} & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & −Ā_{1}x_{1}z_{1} & I & 0 \\ 0 & z_{2} & −z̄_{1}x̄_{1} & I \end{matrix} ) ( \begin{matrix}I & 0 & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & Ā_{1}x_{1}z_{1} & I & 0 \\ −λ̄z̄_{1}x̄_{1}A_{1} & 0 & 0 & I \end{matrix} ) ∈ Q_{n}.

Then the n’th row of στ_{1} is

(v_{1} − λ̄w_{2}z̄_{1}x̄_{1}A_{1}, v_{2} + v_{1}x_{1}z_{1} + w_{2}(z_{2} − z̄_{1}x̄_{1}Ā_{1}x_{1}z_{1}), w_{1} − w_{2}z̄_{1}x̄_{1}, w_{2}) .

Let (v′_{1}, v′_{2}, w′_{1}, w′_{2}) denote this row. Then (v′_{2}, w′_{1}, w′_{2}) is unimodular in R^{2(n−r)+r}, because (v_{2} + v_{1}x_{1}z_{1}, w_{1}, w_{2}) is. Since R satisfies SR_{m} and n − r ≥ m + 1, there exists an r × (n − r) matrix z_{3} such that (v′_{2} + w′_{1}z_{3}, w′_{2}) is unimodular in R^{2(n−r)}. Set

τ_{2} = ( \begin{matrix}I & 0 & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & z_{3} & I & 0 \\ −λ̄z̄_{3} & 0 & 0 & I \end{matrix} ) ∈ Q_{n}.

Then the n’th row of στ_{1}τ_{2} is (v′_{1} − w′_{2}(λ̄z̄_{3}), v′_{2} + w′_{1}z_{3}, w′_{1}, w′_{2}) and (v′_{2} + w′_{1}z_{3}, w′_{2}) is unimodular. Let (v′′_{1}, v′′_{2}, w′′_{1}, w′′_{2}) denote this row. Thus (v′′_{2}, w′′_{2}) is unimodular in R^{2(n−r)}. Since R satisfies max^{λ}(R)S_{m} and n − r ≥ m + 1, there exists a matrix γ ∈ 𝕄_{n−r}(\overline{max^{λ}(R)}) such that v′′_{2} + w′′_{2}γ is unimodular in R^{n−r}. Set

τ_{3} = ( \begin{matrix}I & 0 & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & 0 & I & 0 \\ 0 & γ & 0 & I \end{matrix} ) = ∏_{r+1≤i≤j≤n} l_{ij}(a_{ij}) ∈ Q_{n}

where a_{ij} is the (i − r, j − r)’th coefficient of γ. Since R satisfies SR_{m} and (n − r) ≥ m + 1, there is by [Bs, Theorem 5.3.3] a product ϵ of elementary (n − r) × (n − r) matrices such that (v′′_{2} + w′′_{2}γ)ϵ = (0, ⋯, 0, 1). Set

τ_{4} = ( \begin{matrix}I & 0 & 0 & 0 \\ 0 & ϵ & 0 & 0 \\ 0 & 0 & I & 0 \\ 0 & 0 & 0 & ϵ̄^{−1} \end{matrix} ) ∈ Q_{n}.

Then στ_{1}τ_{2}τ_{3}τ_{4} has n’th row (v′′_{1}, (0, ⋯, 0, 1), w′′_{1}, w′′′_{2}). □

Recall the subgroups P_{n}, Q_{n}, and R_{n} of EH_{2n}(R, a_{1}, …, a_{r}), which are defined in §2.

Definition 4.2 Let ϕ ∈ EH_{2n}(R, a_{1}, …, a_{r}). 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 ≥ r + m + 2. Then every element of EH_{2n}(R, a_{1}, …, a_{r}) has a P RQ-decomposition, i.e. EH_{2n}(R, a_{1}, …, a_{r}) = P_{n}R_{n}Q_{n}.

Proof Let ϕ ∈ EH_{2n}(R, a_{1}, …, a_{r}). 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 generators θ of EH_{2n}(R, a_{1}, …, a_{r}) and show using reduced P RQ-decompositions that θP_{n}R_{n}Q_{n} ⊆ P_{n}R_{n}Q_{n}. It follows trivially that EH_{2n}(R, a_{1}, …, a_{r}) = 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} ∈ EH_{2(n−1)}(R, a_{1}, …, a_{r}). Since n ≥ r + 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 EH_{2n}(R, a_{1}, …, a_{r}) (under the stabilization map EH_{2(n−1)}(R, a_{1}, …, a_{r}) ⟶ EH_{2n}(R, a_{1}, …, a_{r})) 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} ∈ EH_{2(n−1)}(R, a_{1}, …, a_{r}) ⊆ 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 r + 1 ≤ i ≤ n − 1), r_{nj}(a) = [Hϵ_{nj}(a), r_{ji}(1)] (a ∈ R and r + 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 ∈ max^{λ}(R)), r_{n}(ζ) = ⋯ [r_{j}(ζ), Hϵ_{nj}(−1)]r_{jn}(ζ_{f}), (ζ ∈ C, r + 1 ≤ j ≤ n − 1),

show that P_{n} and the matrices Hϵ_{n,n−1}(a) (a ∈ R) generate EH_{2n}(R, a_{1}, …, a_{r}). 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} ∈ EH_{2(n−1)}(R, a_{1}, …, a_{r}). A straightforward computation shows that Hϵ_{n,n−1}(a)σ_{3}Hϵ_{n,n−1}(−a) ∈ P_{n} and it is clear that EH_{2(n−1)}(R, a_{1}, …, a_{r}) 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 ∈ GH_{2n}(R, a_{1}, …, a_{r}). By (4.1), there is a τ_{1} ∈ Q_{n} ⊆ EH_{2n}(R, a_{1}, …, a_{r}) such that the (n, n)’th coefficient of στ_{1} is 1.

Clearly there is a matrix τ_{2} = ∏_{i=1}^{n−1} Hϵ_{ni}(x_{i}) such that στ_{1}τ_{2} has 0 in the first

(n − 1) entries of its n’th row and 1 in the n’th entry of this row. From (2.2),

it follows that there is a matrix τ_{3} = ∏_{i=1}^{n} ℓ_{in}(y_{i}) ∏_{i=r+1}^{n−1} ϵ_{in}(y′_{i})Hm_{n}(ζ) such that

τ_{3}στ_{1}τ_{2} has the same n’th row as στ_{1}τ_{2} and the same n’th column as the 2n × 2n identity matrix. For any matrix

( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ GH_{2n}(A, a_{1}, ⋯, a_{r}),

it follows from the identity

( \overline{\begin{matrix}α & β \\ γ & δ \end{matrix}} ) ( \begin{matrix}A & λI \\ I & 0 \end{matrix} ) ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) = ( \begin{matrix}A & λI \\ I & 0 \end{matrix} )

that

(i) ᾱAβ + γ̄β + λᾱδ = λI, and

(ii) β̄Aβ + δ̄β + λβ̄δ = 0.

From (i), we obtain that the (2n, 2n)’th coefficient of τ_{3}στ_{1}τ_{2} is 1. From (2.2),

it follows now that there is a matrix τ_{4} = ∏_{i=1}^{n−1} Hϵ_{ni}(z_{i}) ∏_{i=r+1}^{n} r_{in}(z′_{i})r_{n}(ξ) such that τ_{4}τ_{3}στ_{1}τ_{2} has the same n’th row and n’th column as τ_{3}στ_{1}τ_{2} and the same 2n’th column as the 2n × 2n identity matrix. It follows now from (ii) that τ_{4}τ_{3}στ_{1}τ_{2} has the same n’th row as the 2n × 2n identity matrix. Thus τ_{4}τ_{3}στ_{1}τ_{2} ∈ GH_{2(n−1)}(R, a_{1}, …, a_{r}), by (2.4). Let ρ = τ_{4}τ_{3}στ_{1}τ_{2}. By (2.5), ρ normalizes EH_{2n}(R, a_{1}, …, a_{r}). Since σ = τ_{3}^{−1}τ_{4}^{−1}ρτ_{2}^{−1}τ_{1}^{−1}, it follows that σ normalizes EH_{2n}(R, a_{1}, …, a_{r}). Thus KH_{1,n}(R, a_{1}, …, a_{r}) is a group and the map KH_{1,n−1}(R, a_{1}, …, a_{r}) ⟶ KH_{1,n}(R, a_{1}, …, a_{r}) is surjective. By induction on n − m − r, we obtain that the map KH_{1,m+r}(R, a_{1}, …, a_{r}) ⟶ KH_{1,n}(R, a_{1}, …, a_{r}) is surjective.

Let ϕ ∈ GH_{2n}(R, a_{1}, …, a_{r}) ∩ EH_{2(n+1)}(R, a_{1}, …, a_{r}). 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} ∈ EH_{2n}(R, a_{1}, …, a_{r}) and τ_{1} ∈ C_{(n+1)} ⊆ Q_{(n+1)}. Obviously ϕ = σ_{1}(τ_{1}ατ) and τ_{1}ατ ∈ Q_{(n+1)} ∩ GH_{2n}(R, a_{1}, …, a_{r}). It suffices to show that τ_{1}ατ ∈ EH_{2n}(R, a_{1}, …, a_{r}). 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} ) .

From the definition of Q_{n+1}, ϵ_{1} is an (n+1)×(n+1) matrix in the elementary group E_{n+1}(R), of the form

ϵ_{1} = ( \begin{matrix}I & α_{2} \\ 0 & ϵ′_{1} \end{matrix} )

where α_{2} is an r × (n + 1 − r) matrix whose columns lie in C. Furthermore since τ_{1}ατ lies in GH_{2n}(R, a_{1}, …, a_{r}), the (n+1−r)’th column of α_{2} is trivial and the last row and column of ϵ′_{1} are the same as those of the (n + 1 − r) identity matrix. Let ξ_{i} denote the i’th column of α_{2} (1 ≤ i ≤ n − r) and set τ_{2} = ∏_{i=1}^{n−r} Hm_{r+i}(−ξ_{i}). Then τ_{2} ∈ Q_{n} and

τ_{2}τ_{1}ατ = ( \begin{matrix}ϵ_{2} & 0 \\ γ_{2} & ϵ̄_{2}^{−1} \end{matrix} )

where

ϵ_{2} = ( \begin{matrix}I & 0 \\ 0 & ϵ′_{2} \end{matrix} ) ∈ E_{n+1}(R)

and ϵ′_{2} ∈ GL_{n+1−r}(R) whose last row and column are the same as those of the (n + 1 − r) identity matrix. Thus ϵ′_{2} ∈ E_{n+1}(R) ∩ GL_{n−r}(R). Since n − r ≥ m + 1 and A satisfies SR_{m}, we obtain by stability for K_{1} of the general linear group [Bs, Theorem 5.4.2 ] that ϵ′_{2} ∈ E_{n−r}(R). Set

τ_{3} = ( \begin{matrix}ϵ_{2}^{−1} & 0 \\ 0 & ϵ̄_{2} \end{matrix} ) .

Then τ_{3} ∈ Q_{n} and

τ_{3}τ_{2}τ_{1}ατ = ( \begin{matrix}I & 0 \\ γ_{3} & I \end{matrix} ) .

Since the matrix on the right hand side of the equality lies in GH_{2n}(R, a_{1}, …, a_{r}), it must lie in Q_{n}. Thus τ_{1}ατ lies in Q_{n}. □

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