Injective stability for odd unitary K₁

1 Introduction

The modern definition of odd unitary groups was given in [9] by Victor Petrov. His definition generalizes Anthony Bak’s unitary groups and split odd orthogonal groups, hence all classical Chevalley groups over arbitrary commutative rings. In [19], we introduced quadratic structures that may be used to construct these groups in a more geometric way.

The natural problem in this context is to prove stabilization for the K1-functor. For general linear groups, this was done in the groundbreaking paper [6] by Hyman Bass and also may be found in his book [7]. Injective stability for the linear K1-functor was reproved in [14] by Suslin and Tulenbaev using simple factorization of the elementary group. For the usual classical groups and Bak’s unitary groups, stability was proved in [2, 4, 15, 16] and in the unpublished paper [10], which can be found on Max-Albert Knus’s homepage. Finally, for Petrov’s unitary groups, the surjective part was already proved in [9], and the injective part in the main result of Yu Weibo’s paper [20]. All injectivity proofs for various unitary groups used the corresponding result for linear groups.

For unitary groups, the condition for injective stability is formulated in terms of the Λ-stable rank condition starting from [2]. This condition is weaker than all the previous ones; in particular, Λ-stable rank may be estimated by the absolute stable rank. For algebras R over a commutative ring K, the absolute stable rank may be bounded by the Bass–Serre dimension of K (more precisely, asr⁡(R)≤δ⁢(K)+1). On the other hand, in some cases, the injective stability was proved using only the stable rank of R. For Chevalley groups, it was done in Stein’s paper [13], and for even unitary groups, it is the main result of S. Sinchuk’s work [11].

In his paper [1], Bak proved the nilpotence of linear K1-groups using the localization-completion method. This result is much stronger than surjective stability and may be proved independently. The nilpotence of Bak’s unitary K1-groups was proved in the absolute case by Hazrat in [8] and in the relative case by Bak, Hazrat, and Vavilov in [3]. For Petrov’s unitary groups, it was done in a recent paper [22] of Weibo and Tang.

When the authors finished this work, it turned out that Weibo also proved injective stability under the stable rank condition; see [21]. In contrast to his paper, our approach is based on a more refined factorization than the usual Dennis–Vaserstein factorization of the elementary group (see [18] for details on various parabolic factorizations). Ultimately, this approach makes the proof self-contained without even using linear stability. Also, we use a more general definition of odd unitary groups. With this definition, the classical result for linear groups is a corollary of our main theorem, and in this case, our proof reduces to that of Suslin and Tulenbaev.

In the paper [19], we already implicitly used odd form rings in the definition of levels. A (unital special) odd form ring is a pair (R,Δ), where R is a ring with involution and Δ is Petrov’s odd form parameter on the right R-module R with the canonical hermitian form. As was shown in that paper, unitary groups of arbitrary regular quadratic bimodules may be considered as unitary groups of appropriate odd form rings. We show that the regularity condition is redundant.

In order to prove relative results for various groups, the common approach is Stein’s relativization technique from [12]. However, even for Bak’s unitary groups, it becomes complicated because the definition of Bak’s form rings is not truly algebraic (in the sense of universal algebra). This difficulty leads to the relativization with two parameters; see [5] for Bak’s unitary groups and [9] for Petrov’s unitary groups. Our odd form rings are defined using operations and axioms; hence the original Stein variant may be applied.

In Section 2, we review odd quadratic modules from [19], and in Sections 3–5, we develop the general theory of odd form rings. The main result of Section 6 is the following.

2 Quadratic and hermitian forms

Every ring in this paper is associative but not necessarily with 1. All commutative rings have identity elements and only trivial involutions; homomorphisms between them are unital. When we work with algebras over a fixed commutative ring K, we always consider only those bimodules that have the same K-module structure from the left and from the right. Recall that if R is a non-unital K-algebra, then R⋊K is a unital K-algebra with an ideal R; it is the K-module R⊕K with the multiplication (r,k)⁢(r′,k′)=(r⁢r′+r⁢k′+r′⁢k,k⁢k′).

If R is an arbitrary ring, then R∙ is the multiplicative semigroup of R (it is a monoid if R is unital), i.e. the set R with the multiplication operation. By C⁡(G) and C⁡(R), we denote the center of a group G or a ring R. The subgroups CG⁡(H) and NG⁡(H) are the centralizer and the normalizer of a subgroup H in a group G. We use the notation hg=g⁢h⁢g-1, hg=g-1⁢h⁢g and [g,h]=g⁢h⁢g-1⁢h-1 for elements g and h of an arbitrary group.

If R is a ring, then Rop={rop∣r∈R} is the opposite ring, rop⁢(r′)op=(r′⁢r)op. The same notation is used for the opposite modules (note that the opposite of a left module is right and vice versa). Clearly, (Rop)op≅R, and similarly for modules. An involution on a ring R is an additive map (-)¯:R→R such that r⁢r′¯=r′¯⁢r¯, and r¯¯=r (if R is unital it follows that 1¯=1). For example, if R=Sop×S for a ring S, then (aop,b)¯=(bop,a) is an involution.

Until the end of this section, all rings, ring homomorphisms and modules are unital.

Recall the definitions from [19]. Let R be an arbitrary ring and λ∈R*. A map (-)¯:R→R,r↦r¯ is called a λ-involution (or pseudo-involution) if it is additive,

For example, if R is commutative, then the identity map on R is a 1-involution (i.e. an involution) and a (-1)-involution simultaneously. Let MR be a right module. A map B:M×M→R is called a hermitian form if it is biadditive,

The hermitian form B is called regular (or non-degenerate) if MR is finitely generated projective and B induces an bijection M→HomR⁡(M,R),m↦B⁢(m,-). The module RR has the canonical regular hermitian form B1⁢(r,r′)=r¯⁢r′.

A quadratic structure on a ring R with a λ-involution is a right R∙-module A (i.e. an abelian group with a right action of the monoid R∙) with additive maps φ:R→A and tr:A→R such that

1. φ⁢(r¯⁢r′⁢r)=φ⁢(r′)⋅r, tr⁡(a⋅r)=r¯⁢tr⁡(a)⁢r,

2. tr⁡(φ⁢(r))=r+r¯⁢λ,

3. tr⁡(a)=tr⁡(a)¯⁢λ, φ⁢(r)=φ⁢(r¯⁢λ),

4. a⋅(r+r′)=a⋅r+φ⁢(r¯′⁢tr⁡(a)⁢r)+a⋅r′.

For example, if Λ≤R is a form parameter (i.e. Λ is an additive subgroup, {r-r¯⁢λ}≤Λ≤{r∣r+r¯⁢λ=0} and r¯⁢Λ⁢r≤Λ for all r), then A=R/Λ is a quadratic structure with φ⁢(r)=r+Λ and tr⁡(r+Λ)=r+r¯⁢λ.

Let (MR,B) be a hermitian module and A a quadratic structure on R. A quadratic form on M with values in A is a map q:M→A such that

1. q⁢(m⁢r)=q⁢(m)⋅r,

2. tr⁡(q⁢(m))=B⁢(m,m),

3. q⁢(m+m′)=q⁢(m)+φ⁢(B⁢(m′,m))+q⁢(m′).

Such a triple (M,B,q) is called a quadratic module. The unitary group of a quadratic module (M,B,q) is

In the case of Bak’s quadratic forms, we have A=R/Λ for a form parameter Λ,

for a sesquilinear map Q, and q⁢(m)=φ⁢(Q⁢(m,m)). Quadratic modules with regular hermitian forms are also called quadratic spaces.

As was shown in [19], Petrov’s quadratic forms are almost the same as our quadratic forms if A is generated by the images of φ and q. Indeed, recall that the Heisenberg group of (M,B) is Heis⁡(M,B)=M×R with the operation

(the identity element 0˙=(0,0) and the inverses -˙⁡(m,r)=(-m,-B⁢(m,m)-r)). The monoid R∙ acts on the group Heis⁡(M,B) from the right via

and there are natural maps

A Petrov’s odd form parameter is an R∙-subgroup ℒ≤Heis⁡(M,B) such that

Then φ, tr and q are well-defined on A=Heis⁡(M,B)/ℒ. Conversely, if a quadratic structure A with a fixed quadratic form q is generated by the images of φ and q, then A is isomorphic to Heis⁡(M,B)/ℒ for a unique ℒ.

Recall also that if (P,B,q) and (P′,B′,q′) are quadratic modules over a ring R with a λ-involution and a quadratic structure A, then

is a quadratic module, where

Conversely, if a quadratic module (P,B,q) splits as a direct sum of orthogonal submodules P=P1⊕P2 (i.e. B⁢(P1,P2)=0), then P=P1⊥P2.

Let P be a finitely generated projective module over a ring R with a λ-involution and a quadratic structure A. Then Q=HomR(P,R)op is also an R-module and the map 〈-,=〉:Q×P→R,(xop,p)↦x⁢(p) is the canonical pairing. Now the module H⁢(P)=Q⊕P has a hermitian form

and a quadratic form

The module H⁢(P) is called a hyperbolic space (clearly, the hermitian form of H⁢(P) is regular), P and Q are the lagrangians of H⁢(P). Conversely, if M is a quadratic space and there is a direct sum decomposition M=P⊕Q such that B|Q×Q=0=B|P×P and q|Q=0=q|P, then M≅H⁢(P).

Note that if (N,B|N×N,q|N)≤(M,B,q) is a regular submodule of a quadratic module, then M canonically splits as an orthogonal sum of N and its orthogonal complement N⊥={m∈M∣B⁢(m,N)=0}. We usually consider the situation when a given quadratic module (M,B,q) has several pairwise orthogonal hyperbolic submodules Mi=H⁢(Pi), 1≤i≤n, i.e.

The orthogonal complement to all Mi is denoted by M0. If Pi=R, 0<|i|≤n, then M is exactly an odd hyperbolic space of rank n from [9].

3 Odd form rings

Let (M,B,q) be a quadratic module over a unital ring R with a λ-involution and a quadratic structure A. Consider the ring

clearly, T possess an involution (xop,y)¯=(yop,x). If B is regular, then

Also, there is the odd form parameter

It is easy to see that U⁢(M,B,q)≅U⁢(T,B1,qΞ), where

Conversely, let T be a unital involution ring, let Ξ≤Heis⁡(T,B1) be an odd form parameter, and let U⁢(T,B1,qΞ) be the corresponding unitary group. We call the pair (T,Ξ) a special unital odd form ring. Clearly, there are natural maps

For any g∈U⁢(T,B1) (i.e. if g∈T* and g-1=g¯), let

then U⁢(T,B1,qΞ)={g∈T*∣g-1=g¯,γ⁢(g)∈Ξ}. Note that π⁢(γ⁢(g))=g-1 and ρ⁢(γ⁢(g))=g¯-1. In other words, we may express the unitary group using only Ξ instead of the full Heisenberg group.

Now we define odd form rings using axioms. A pair (R,Δ) is called an odd form ring if R is a ring with involution (non-unital in general), Δ is a group with a right R∙-action, and there are maps ϕ:R→Δ, π:Δ→R, ρ:Δ→R such that

1. π is a group homomorphism, π⁢(u⋅x)=π⁢(u)⁢x,

2. ϕ is a group homomorphism, ϕ⁢(x¯⁢y⁢x)=ϕ⁢(y)⋅x, ϕ⁢(z)=0 if z=z¯,

3. u∔ϕ⁢(x)=ϕ⁢(x)∔u, u∔v=v∔u∔ϕ⁢(-π⁢(u)¯⁢π⁢(v)),

4. ρ⁢(u∔v)=ρ⁢(u)-π⁢(u)¯⁢π⁢(v)+ρ⁢(v), ρ⁢(u)¯=ρ⁢(-˙⁡u), ρ⁢(u⋅x)=x¯⁢ρ⁢(u)⁢x,

5. π⁢(ϕ⁢(x))=0, ρ⁢(ϕ⁢(x))=x-x¯,

6. u⋅(x+y)=u⋅x∔ϕ⁢(y¯⁢ρ⁢(u)⁢x)∔u⋅y.

Note that, in every odd form ring, we have

An odd form ring (R,Δ) is called unital if R is unital and u⋅1=u for all u∈Δ; for such odd form rings, the identity u⋅(-1)∔u=ϕ⁢(ρ⁢(u)) holds.

Clearly, any special unital odd form ring is an odd form ring. Conversely, if (R,Δ) is an odd form ring and (π,ρ):Δ→R×R is injective, then (R,Δ) is called special (and if R is unital, then (R,Δ) is a special unital odd form ring as in the definition above). Our odd form rings are preferable in comparison with the special ones since they behave better under Stein’s relativization and other algebraic constructions like tensor products.

An odd form ideal of an odd form ring (R,Δ) is a pair (I,Γ) such that I⁢⊴⁢R is a two-sided ideal, Γ⁢⊴⁢Δ is a normal subgroup,

where Γmin=Δ⋅I∔ϕ⁢({r∣r-r¯∈I}) and Γmax={u∈Δ∣π⁢(u),ρ⁢(u)∈I} (clearly, (I,Γmin) and (I,Γmax) are odd form ideals). If (I,Γ)⁢⊴⁢(R,Δ) is an odd form ideal, then (R,Δ)/(I,Γ)=(R/I,Δ/Γ) is an odd form ring. This does not always hold for special odd form rings since the factor may not be special.

We say that an odd form ring (R,Δ) is an odd form algebra over a commutative ring K if R is an involution K-algebra and (R⋊K,Δ) is a unital odd form ring (i.e. the action of K∙ on Δ is defined and satisfies appropriate identities). Any odd form algebra (R,Δ) naturally becomes an ideal in a unital odd form algebra (R⋊K,Δ). Any odd form ring is an odd form ℤ-algebra.

Now let us define the unitary group of an odd form ring (R,Δ). Since the ring may be non-unital, we should use quasi-regular elements of R instead of invertible ones. Also, we need to explicitly use the odd form parameter; hence we define the unitary group as a certain subset of R×Δ. The components of g∈R×Δ are denoted as β⁢(g)∈R and γ⁢(g)∈Δ. To simplify further computations, it is useful to set α⁢(g)=β⁢(g)+1∈R⋊ℤ (we may consider α⁢(g) as an element of R⋊K if (R,Δ) is an odd form K-algebra, or even as an element of R itself if R is unital). Finally,

The first equation in the definition of U⁢(R,Δ) may be written as

The group operation is given by α⁢(g⁢h)=α⁢(g)⁢α⁢(h), γ⁢(g⁢h)=γ⁢(g)⋅α⁢(h)∔γ⁢(h). Note that if (R,Δ) is special unital, then we may identify an element g∈U⁢(R,Δ) with α⁢(g)∈R.

For example, if (R,Δ) is obtained from a quadratic module (M,B,q), then U⁢(R,Δ)≅U⁢(M,B,q). Also, if R is a C*-algebra, then

is an odd form parameter, (R,Δ) is a special odd form ℝ-algebra (it is not a ℂ-algebra by our definition since the involution is anti-linear) and U⁢(R,Δ) is isomorphic to the unitary group of R.

If (R,Δ) is an arbitrary odd form ring, M is a right R-module, then (R,Δ⊕M) is also an odd form ring with π⁢(u⊕m)=π⁢(u) and ρ⁢(u⊕m)=ρ⁢(u). It is easy to see that U⁢(R,Δ⊕M)≅M⋊U⁢(R,Δ). In particular, various affine groups like Kn⋊GL⁢(n,K) are unitary groups of certain non-special odd form rings.

Now we formulate Stein’s relativization in our context. Let (I,Γ)⁢⊴⁢(R,Δ) be an odd form ideal. The double (R,Δ)×(R/I,Δ/Γ)(R,Δ) is the fiber product of (R,Δ) with itself over (R/I,Δ/Γ). In other words,

The projections from the double to (R,Δ) are denoted by p1 and p2, the factor-map (R,Δ)→(R/I,Δ/Γ) is denoted by q, and the diagonal map from (R,Δ) into the double is denoted by d (so pi∘d=id). The double is canonically isomorphic to (I⋊R,Γ⋊Δ), where I⋊R=I⊕R as an abelian group, Γ⋊Δ=Γ×Δ as a set (actually, it is the semi-direct product of groups), and the operations are given by

1. (x,y)⁢(x′,y′)=(x⁢x′+x⁢y′+y⁢x′,y⁢y′),

2. (u,v)∔(u′,v′)=(u∔u′∔ϕ⁢(-π⁢(v)¯⁢π⁢(u′)),v∔v′),

3. (u,v)⋅(x,y)=(u⋅x∔u⋅y∔v⋅x∔ϕ⁢(y¯⁢(ρ⁢(u)+ρ⁢(v))⁢x),v⋅y),

4. π⁢(u,v)=(π⁢(u),π⁢(v)), ϕ⁢(x,y)=(ϕ⁢(x),ϕ⁢(y)), ρ⁢(u,v)=(ρ⁢(u)-π⁢(u)¯⁢π⁢(v),ρ⁢(v)).

The isomorphism (I⋊R,Γ⋊Δ)→(R×R/IR,Δ×Δ/ΓΔ) is given by

and in terms of (I⋊R,Γ⋊Δ), we have p1⁢(x,y)=x+y, p1⁢(u,v)=u∔v, p2⁢(x,y)=y, p2⁢(u,v)=v, d⁢(x)=(0,x) and d⁢(u)=(0˙,u).

Now if (I,Γ)⁢⊴⁢(R,Δ) is an odd form ideal, then there is a left exact sequence

Obviously, U⁢(I,Γ)⁢⊴⁢U⁢(R,Δ). It is easy to see that the sequence

is split exact with the section d, i.e. is a semi-direct product (where p1-1 takes values in Ker⁡(p2)). It follows from the fact that the functor (R,Δ)↦U⁢(R,Δ) commutes with fibered products.

4 Idempotents

We start with a construction of the free odd form algebra.

Let (I,Γ)⁢⊴⁢(R,Δ) be an odd form ideal in a unital odd form ring, and let e∈R be a hermitian idempotent (i.e. e=e¯ and e2=e). It is easy to see that (Ie,Γee)⊆(I,Γ) is an odd form subalgebra, where

Usually, we apply this to an arbitrary non-unital odd form K-algebra

If (R,Δ) is obtained from a quadratic module (M,B,q), then there is a natural bijection between hermitian idempotents of R and orthogonal summands N≤M; under this bijection, (Re,Δee) are obtained from (N,B|N×N,q|N).

Now let us see what happens when a quadratic module (M,B,q) has a family of orthogonal hyperbolic summands, M=M0⊥H⁢(P1)⊥⋯⊥H⁢(Pn) for some n≥0. Let Ei∈End⁡(M) be the canonical projections on Pi for 0<|i|≤n and on M0 for i=0; then Ei form a complete system of orthogonal idempotents. Let ei=(E-iop,Ei) for -n≤i≤n; then

Moreover, these ei also form a complete system of orthogonal idempotents, and ei¯=e-i. Also, (ei,0)∈Ξ for all i≠0, where Ξ is the odd form parameter considered above, since q|Pi=0.

We say that η=(e-,e+,q-,q+) is a hyperbolic pair in an odd form ring (R,Δ) if e- and e+ are orthogonal idempotents in R, e+¯=e-, q- and q+ lie in Δ,