Injective stability for odd-dimensional unitary K1

3.1 Hermitian form ring

Our concept closely follows [4]. Let R be an associative ring with identity 1. Recall that an anti-isomorphism ¯:R→R, a↦a¯ on R is a bijective map satisfying a+b¯=a¯+b¯, a⁢b¯=b¯⁢a¯ for all a,b∈R and 1¯=1. Let λ∈R with the property that a¯¯=λ⁢a⁢λ¯ for all a∈R. It is easy to see that λ¯=λ-1. The element λ is called a symmetry for ¯ and the triple (R,¯,λ) a ring with involution and symmetry. A Hermitian ring is a quadruple (R,¯,λ,μ), where (R,¯,λ) is a ring with involution and symmetry and μ∈R such that μ=μ¯⁢λ.

Let R∙ be the underlying set of the ring equipped with the multiplication of the ring, but not the addition of the ring. A (right) R∙-module is a not necessarily abelian group (G,∔) equipped with a map ∘:G×R∙→G, (x,a)↦x∘a, satisfying

1. x∘0=0 for all x∈G,

2. x∘1=x for all x∈G,

3. (x∘a)∘b=x∘(a⁢b) for all x∈G and a,b∈R,

4. (x∔y)∘a=(x∘a)∔(y∘a) for all x,y∈G and a∈R.

We treat ∘ as an operator with higher priority than ∔. An R-module is canonically an R∙-module, but not conversely. Let G and G′ be R∙-modules. By a homomorphism of R∙-modules, one means a group homomorphism f:G→G′ such that f⁢(x∘a)=f⁢(x)∘a for all x∈G and a∈R. By an R∙-submoduleN, we mean a subgroup N of G which is ∘-stable, that is, x∘a∈N for all x∈N and a∈R.

Let (R,¯,λ,μ) be a Hermitian ring. Consider the Heisenberg group𝔜, where 𝔜=R×R as a set, with a composition operation “∔”, satisfying

It is easy to see that the law ∔ is associative with identity (0,0), and the inverse is given by

Furthermore, supply 𝔜 with a right action of R∙ through (a,b)∘c=(a⁢c,c¯⁢b⁢c). 𝔜 becomes an R∙-module.

Let (R,+) have the R∙-module structure defined by a∘b=b¯⁢a⁢b. Define the trace maptr:𝔜→R by tr⁢((a,b))=a¯⁢μ⁢a+b+b¯⁢λ. It is easy to see that tr is a homomorphism of R∙-modules.

Set

An odd form parameter of (R,¯,λ,μ) is an R∙-submodule Δ of 𝔜 such that Δmin≤Δ≤Δmax. The pair ((R,¯,λ,μ),Δ) is called a Hermitian form ring and is abbreviated by (R,Δ).

3.2 Odd-dimensional unitary groups

Let (R,Δ) be a Hermitian form ring and n∈ℕ. Consider the right R-module M:=R2⁢n+1 with the fixed basis (e1,…,en,e0,e-n,…,e-1), where ei is the column vector whose i-th entry (1≤i≤-1) is 1 and whose other entries are 0. If u∈M, we call (u1,…,un,u-n,…,u-1)t∈R2⁢n the hyperbolic part of u and denote it by uhb. Define the maps

and

where u*=u¯t, uhb*=u¯hbt and p∈Mn⁢(R) is the matrix with 1 on the skew diagonal and 0 elsewhere. One checks easily the following lemma.

The odd-dimensional unitary groupU2⁢n+1⁢(R,Δ) is the group consisting of all elements σ∈GL2⁢n+1⁢(R) satisfying

for all u,v∈M. These groups are isomorphic to Petrov’s odd hyperbolic unitary groups such that V0=R and contain as special cases the groups GL2⁢n+1⁢(R), where R is any ring, the classical groups O2⁢n+1⁢(R) and Sp2⁢n+1⁢(R), where R is any commutative ring, and further all even-dimensional unitary groups U2⁢n⁢(R,Λ), where (R,Λ) is any form ring.

Set

Let ϵ⁢(i) be 1 for i=1,…,n and -1 for i=-n,…,-1.

Define the embedding Φ:GLn⁢(R)→U2⁢n+1⁢(R,Δ) by

3.3 The elementary subgroup

Let (R,¯,λ,μ) be a Hermitian ring. We defined an R∙-module structure on 𝔜. Let (R,¯,λ¯,μ¯) be the Hermitian ring defined in Remark 3.1 (1), and let 𝔜¯ be the Heisenberg group corresponding to (R,¯,λ¯,μ¯). Since the underlying set of both (R,¯,λ,μ) and (R,¯,λ¯,μ¯) is R×R, we get another R∙-module structure on R×R if we replace the Hermitian ring (R,¯,λ,μ) by (R,¯,λ¯,μ¯). We denote the group operation (resp. scalar multiplication) defined by (R,¯,λ,μ) and (R,¯,λ¯,μ¯) on R×R by ∔1 and ∔-1 (resp. ∘1 and ∘-1), respectively. Set

It is easy to see that ((R,¯,λ¯,μ¯),Δ-1) is a Hermitian form ring.

If i,j∈Θ, let ei⁢j denote the matrix in M2⁢n+1⁢(R) with 1 in the (i,j)-th position and 0 elsewhere. If i,j∈Θhb, i≠±j and a∈R, the element

is called an (elementary) short root matrix. If i∈Θhb and (a,b)∈Δ-ϵ⁢(i), the element

is called an (elementary) extra short root matrix. The extra short root matrices of the kind Ti⁢(0,b)=e+b⁢ei,-i are called (elementary) long root matrices. An element of U2⁢n+1⁢(R,Δ) is called elementary matrix if it is a short or extra short root matrix. The elementary subgroupE⁢U2⁢n+1⁢(R,Δ) is the subgroup of U2⁢n+1⁢(R,Δ) generated by all elementary matrices. The coset space

is called the odd-dimensional unitary K1-functor. Under the embedding

it is easy to see that ϕ2⁢n+12⁢n+3⁢(E⁢U2⁢n+1⁢(R,Δ))⊆E⁢U2⁢n+3⁢(R,Δ). Therefore, we have the stabilization map K⁢U1,n⁢(R,Δ)→K⁢U1,n+1⁢(R,Δ).

The elementary matrices satisfy the following properties:

Let i,j∈Θhb such that i≠±j. In the following, the usual ordering of the numbers -n,…,-1,0,1,…,n is used. Denote by Ti⁢j the subgroup consisting of all Ti⁢j⁢(a) with a∈R and by Ti the subgroup consisting of all Ti⁢(a,b) with (a,b)∈Δ-ϵ⁢(i). Let n≥3. We define the following subgroups of E⁢U2⁢n+1⁢(R,Δ) which will be used to establish a decomposition theorem in next section.

Denote by ⋊ the semidirect product. Set

3.4 Stable range condition

Let R be an associative ring with 1. A vector (a1,…,an)t with ai∈R is called (left) unimodular if there exist elements b1,…,bn∈R such that

The stable range conditionSn says that, for every unimodular vector

there exist elements b1,…,bn∈R such that

is unimodular. The stable rank of R, denoted by sr⁡(R), is the smallest number n such that Sn holds.