Normality for Elementary Subgroup Functors

ANTHONY BAK Department of Mathematics, University of Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany

NIKOLAI VAVILOV Department of Mathematics and Mechanics, University of Sanct Petersburg, Petrodvorets, 198904, Russia, and Department of Mathematics, University of Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany

Abstract We define a notion of group functor G, on categories of graded modules, which unifies all previous concepts of a group functor G possessing a notion of elementary subfunctor E. We show under a general condition, which is easily checked in practice, that the elementary subgroup E(M) of G(M) is normal for all weak and quasi-weak Noetherian objects M in the source category of G. This result includes all previous ones on Chevalley and classical groups G of rank ≧ 2 over a commutative or module finite ring M and settles positively the still unanswered cases of normality for these group functors.

1. Introduction.

Beginning with H. Bass’ pioneering paper [Bs1], much attention has been given to determining whether or not the elementary subgroup E(M) of a Chevalley or classical group G(M) over a module finite ring M is normal in G(M). Bass’ result [Bs1] was valid for G = GL_{n} (general linear group of rank n) under the condition that n was large with respect to the dimension of the ring M. It was immediately asked whether or not similar looking subgroups of other classical groups were also normal. This question was settled by H. Bass [Bs2] in the symplectic case and in general by A. Bak [B1], [B2], again under the assumption that the rank of the functor was large with respect to the dimension of the ring. The approach taken in [B1], [B2] introduced a generalization GQ_{2n} (referred to by the notation U_{2n} in [B1], [B2], [Bs2] and by GQ_{2n} in [B3]) of the notion of classical group, which depended on the concept of a form parameter and allowed treating in a uniform way all the classical and classical-like groups other than those in the GL_{n}-family. The groups corresponding to the minimum form parameter are treated also in L. Vaserstein [V]. After a gap of roughly ten years, A. Suslin showed in [S], [Tu] that the elementary subgroup E_{n}(M) of GL_{n}(M) is normal whenever n ≥ 3 and M is commutative or module finite. Following his results, several papers Kopeiko [K], Kopeiko-Suslin [KS], Taddei [T1], [T2], Golubchik-Mikhalev [GM] and Vavilov [Vv1], [Vv2] were devoted to analogous questions for Chevalley groups and general quadratic groups GQ_{2n}. A detailed survey of developments to-date is given in [Vv3, §9]. However, the articles above left open the case of twisted Chevalley groups and the general case (in classical language, nonsplit case) of the general quadratic group GQ_{2n}(M, Λ) (n ≥ 2) for an arbitrary module finite ring M with involution and form parameter Λ. They are settled in (3.12) and (3.10) respectively, as consequences of our Main Theorem (2.9). The result (3.10) is included in those announced in [B4, §1].

The approach we take to prove our results is inspired by the technique used in [B4], particularly in [B4, §4], for establishing results concerning GL_{n}. This technique hinges for the most part only on a simple minded, flexible, and general concept of group functor, which is divorced from the confines of linear algebra and at the same time contains as special cases the group functors mentioned in the previous paragraph. Specifically, we consider group functors G on a category Λ Λ Λ of graded modules M (for example, a category of rings or form rings) such that G(M) contains homogeneous components of M in a functorial way. The elementary subgroup E(M) of G(M) is defined as the subgroup of G(M) generated by the embedded homogeneous components of M. This construction obviously defines a subfunctor E ↣ G, which we christen the elementary subfunctor of G. In the case of Chevalley or general quadratic groups G, it is clear that the elementary matrices define embeddings into G of the ring and form parameter over which G is defined and that the elementary subgroup of G is indeed the subgroup generated by these embedded entities. On the other hand, any Chevalley or general quadratic group is itself embedded set theoretically, but still functorially, into a direct sum of copies of the ring over which it is defined. So, we assume in the general setting that our groups G(M) embed set theoretically and functorically into a direct sum ⊕M of copies of M. This extra structure coupled with the notion of weak Noetherian module (2.6) is enough to proof our main result (2.9) concerning the normality of E(M) in G(M), in just a few words.

The rest of the article is organized as follows. In §2, we define in detail the concepts we use and prove the Main Theorem (2.9) concerning the normality of E(M). In §3, we explain how Chevalley and general quadratic groups fit into our setting and then apply our main theorem to these groups.

2. Main theorem.

We fix the following notation for the remainder of the article. We let G G G denote the category of all groups and group homomorphisms. The letter ℐ will denote an index set and we let M M M(ℐ) be equal to the category whose objects are all pairs (k_{M}, M) where k_{M} is a commutative, associative ring with identity and M = ⊕_{i∈ℐ}M_{i} is an ℐ-graded, right k_{M}-module. A morphism

(k_{M}, M) ⟶ (k_{M′}, M′) in M M M(ℐ) is any pair g : k_{M} ⟶ k_{M′}, f : M ⟶ M′ where g is a ring homomorphism preserving the identity, f a graded homomorphism of abelian groups of degree zero, and f(ma) = f(m)g(a) for all m ∈ M and a ∈ k_{M}. We let M M M = M M M(ℐ_{0}) where ℐ_{0} is a set containing exactly one element. There is a canonical inclusion M M M ⊂ G G G and for each i ∈ ℐ, there is a canonical coordinate functor λ_{i} : M M M(ℐ) ⟶ M M M, M ↦ M_{i}, and ((g, f) : (k_{M}, M) ⟶ (k_{M′}, M′)) ↦ ((g, f|_{M_{i}}) : (k_{m}, M_{i}) ⟶ (k_{M′}, M′_{i})).

The letter Λ Λ Λ will always denote a subcategory of M M M(ℐ), for some ℐ, such that if (k_{M}, M) ∈ Obj(Λ Λ Λ) and S ⊂ k_{M} is a multiplicative set then (k_{S^{−1}M}, S^{−1}M) ∈ Obj(Λ Λ Λ) where k_{S^{−1}M} := S^{−1}k_{M} and the canonical morphism (k_{M}, M) ⟶ (k_{S^{−1}M}, S^{−1}M) is in Mor (Λ Λ Λ).

Definition 2.1 A Λ Λ Λ-group functor (on Λ Λ Λ) consists of a functor G : Λ Λ Λ ⟶ G G G and for each i ∈ ℐ, a natural inclusion ϵ_{i} : λ_{i} ⟶ G of functors. If M ∈ Λ Λ Λ and U ⊆ M is a homogeneous subgroup of M, let E(M, U) denote the subgroup of G(M) generated by all subgroups ϵ_{i}^{U}(U_{i}) ⊂ G(M)(i ∈ ℐ) where U_{i} := i’th homogeneous component of U. Let E(M) = E(M, M). Clearly, the construction M ↦ E(M) defines a subfunctor E ↣ G. If M ∈ Λ Λ Λ and s ∈ k_{M}, define A(M, s) = {σ ∈ G(M)| given p ∈ ℤ^{≥0}, ∃q ∈ ℤ^{≧0} such that σE(M, Ms^{q})σ^{−1} ⊆ E(M, Ms^{p}) and σ^{−1}E(M, Ms^{q})σ^{−1} ⊆ E(M, Ms^{p})}. Clearly, A(M, s) is a subgroup of G(M).

Lemma 2.2 Let G be a Λ Λ Λ-group functor and let M ∈ Λ Λ Λ. Suppose that for each element σ ∈ G(M) and each maximal ideal 𝔭 ⊆ k_{M}, there is an element s_{σ,𝔭} ∈ k_{M} \ 𝔭 such that σ ∈ A(M, s_{σ,𝔭}). Then E(M) ◁ G(M).

Proof Let i ∈ ℐ, m ∈ M_{i}, and σ ∈ G(M). It suffies to show that σϵ_{j}(m)σ^{−1} ∈ E(M). For each maximal ideal 𝔭 of k_{M}, choose s_{𝔭}(= s_{σ,𝔭}) as above. By hypothesis, there is a q_{𝔭}≥ 0 such that σϵ_{i}(Ms_{𝔭}^{q_{𝔭}})σ^{−1} ∈ E(M). Since {s_{𝔭}^{q_{𝔭}}|𝔭 maximal ideal of k_{M}} is not contained in any maximal of k_{M}, there is a finite set 𝔭_{1}, ..., 𝔭_{ℓ} of maximal ideals of k_{M} and of elements a_{1}, ..., a_{ℓ} of k_{M} such

that ∑_{j=1}^{ℓ} a_{j}s_{j}^{q_{𝔭_{j}}} = 1. But, then σϵ_{i}(m)σ^{−1} = ∏_{j=1}^{ℓ} σϵ_{i}(ma_{j}s_{j}^{q_{𝔭_{j}}})σ^{−1} ∈ E(M).

Q.E.D.

Definition 2.3 Let G be a Λ Λ Λ-group functor. If M ∈ Λ Λ Λ and s ∈ k_{M}, let < s > denote the multiplicative set generated by s and let “Ms” denote the image of Ms in < s >^{−1} M under the canonical homomorphism M ⟶< s >^{−1} M. Define B(M, s) = {σ ∈ G(< s >^{−1} M)| for each p ∈ ℤ^{≥0}, there is a q ∈ ℤ^{≥0} such that σE(< s >^{−1} M, “Ms^{q}”)σ^{−1} ⊆ E(< s >^{−1} M, “Ms^{p}”) and σ^{−1}E(< s >^{−1} M, “Ms^{q}”)σ ⊆ E(< s >^{−1} M, “Ms^{p}”). Clearly, B(M, s) is a subgroup of G(< s >^{−1} M). Define \overline{E}(M, Ms) = normal closure of E(M, Ms) in G(M).

Lemma 2.4 Let G be a Λ Λ Λ-group functor. Let M ∈ Λ Λ Λ and s ∈ k_{M}. Then the canonical homomorphism Φ : G(M) ⟶ G(< s >^{−1} M) takes A(M, s) into B(M, s). Moreover, if for some p ∈ ℤ^{≧0}, Φ |_{\overline{E}(M,Ms^{p})} is injective then Φ^{−1}(B(M, s)) = A(M, s).

Proof Straightforward.

Lemma 2.5 Let G be a Λ Λ Λ-functor. Let M ∈ Λ Λ Λ. Suppose that the following conditions hold: (2.5.1) For each element σ ∈ G(M) and each maximal ideal 𝔭 of k_{M}, there is an element s_{σ,𝔭} ∈ k_{M} \ 𝔭 such that the image of σ in G(< s_{σ,𝔭} >^{−1} M) lies in B(M, s).

(2.5.2) For each element s ∈ k_{M} \ ⋂_{\begin{matrix}𝔭 \\ maximal \end{matrix}} 𝔭, there is a p ∈ ℤ^{≧0} such that

\overline{E}(M, Ms^{p}) injects into G(< s >^{−1} M) under the canonical homomorphism G(M) ⟶ G(< s >^{−1} M). Then E(M) ◁ G(M).

Proof This follows immedeately from (2.2) and (2.4).

Definition 2.6 An object (k_{M,}M) ∈ M M M(ℐ) will be called weak Noetherian if given s ∈ k_{M} \ ⋂_{\begin{matrix}𝔭 \\ maximal \end{matrix}} 𝔭, there is an element p ∈ ℤ^{≧0} such that Ann_{M}(s^{p}) =

Ann_{M}(s^{p+1}) (where Ann_{M}(s^{p}) = {m ∈ M | ms^{p} = 0}). Note that the index set ℐ need not be finite; for example, if each M_{i} is Noetherian over k_{M} and the number of isomorphism classes [M_{i}] of modules M_{i} is finite then (k_{M}, M) is weak Noetherian. An object (k_{M}, M) ∈ Λ Λ Λ will called quasi-weak Noetherian (in Λ Λ Λ) if it is a direct limit of weak Noetherian objects in Λ Λ Λ. An object (k_{M}, M) ∈ M M M(ℐ) will be called Noetherian if M is Noetherian over k_{M}. An object (k_{M}, M) ∈ Λ Λ Λ will be called quasi-Noetherian (in Λ Λ Λ) (or quasi-finite as in [B4, §3]) if it is a direct limit of Noetherian objects in Λ Λ Λ. Clearly, the property of being Noetherian (resp. quasi-Noetherian) implies the property of being weak Noetherian (resp. quasi-weak Noetherian).

The next lemma generlizes [B 4, (4.10)].

Key Lemma 2.7 Let G be a Λ Λ Λ-group functor. Let M ∈ Λ Λ Λ. Suppose the following:

(2.7.1) For each element s ∈ k_{M} \ ⋂_{\begin{matrix}𝔭 \\ maximal \end{matrix}} 𝔭 and p ∈ ℤ^{≧0}, there is a commu-

tative diagram of sets

\overline{E}(M, Ms^{p}) ↣ G(M) ⟶^{Φ} G(< s >^{−1} M)

↓  ↓  ↓

(⊕M)s^{p} ↣ ⊕M ⟶_{Ψ}< s >^{−1} (⊕M)

where \overline{E}(M, Ms^{p}) is as in (2.3), ⊕M is a finite direct sum of M’s, the horizontal maps are the canonical ones, and the left vertical map is injective. Then if M is weak Noetherian over k_{M}, the group \overline{E}(M, Ms^{p}) maps injectively for some p into G(< s >^{−1} M), under Φ.

Proof It suffices to show that for some p ∈ ℤ^{≥0}, Ψ maps Ms^{p} injectively into < s >^{−1} M. But, this is an easy exercise which is done, for example, in the proof of [B4, (4.10)]. Q.E.D.

Theorem 2.8 Let G be a Λ Λ Λ-group functor. Let M ∈ M M M and suppose that (2.6.1) and the following are satisfied: (2.8.1) For each element σ ∈ G(M) and each maximal ideal 𝔭 of k_{M}, there is an element s_{σ,𝔭} ∈ k_{M} \ 𝔭 such that the image of σ in G(< s_{σ,𝔭} >^{−1} M), under the canonical homomorphism, lies in B(M, s_{σ,𝔭}). Then if M is weak Noetherian, E(M) ◁ G(M).

Proof This follows immediately from (2.5) and (2.7).

Main Theorem 2.9. Let G be a Λ Λ Λ-group functor. Suppose that conditions (2.7.1) and (2.8.1) hold for each object M′ ∈ Λ Λ Λ which is weak Noetherian.

If G commutes with direct limits then E(M) ◁ G(M) for every quasi-weak Noetherian object M of Λ Λ Λ.

Proof This follows immediately from (2.8) and the definition of direct limit.

3. Applications to Chevalley and general quadratic groups.

In this section, we show that Chevalley, twisted Chevalley, and general quadratic groups are Λ Λ Λ-groups, for suitable Λ Λ Λ, satisfying (2.7.1) and that they satisfy (2.8.1) under the usual conditions excluding small ranks. It will follow then from the Main Theorem (2.9) that their elementary subgroups are normal.

The demonstration that (2.7.1) is fulfilled will be essentially trivial. The proof that (2.8.1) is satisfied will be executed by showing that the following condition fulfilled.

Condition 3.1 Let ℐ,M M M(ℐ), and Λ Λ Λ ⊂ M M M(ℐ) be as in section 2. Let G be a Λ Λ Λ-group functor. Let M ∈ Λ Λ Λ and if 𝔭 is a maximal ideal of k_{M}, let S_{𝔭} denote the multiplicative set k_{M} \𝔭. Suppose the following holds for G, M, and each maximal ideal 𝔭 of k_{M}. (3.1.1) For each j ∈ ℐ, there is a subset ℐ(j) of ℐ such that the following holds: Given s ∈ S_{𝔭} and p ∈ ℤ^{≥0}, there is a q ∈ ℤ^{≥0} such that for each m_{j} ∈ M_{j} (= λ_{j}(M)) and i ∈ ℐ(j), ϵ_{j}(\frac{m}{s})ϵ_{i}(M_{i}s^{q})ϵ_{j}(\frac{m}{s})^{−1} ⊆ E(< s >^{−1} M, “Ms^{p} ”). (3.1.2) Given p ∈ ℤ^{≥0}, there is a q ∈ ℤ^{≥0} such that for all i, j ∈ ℐ, ϵ_{i}(M_{i}s^{q}) ⊆< ϵ_{i′}(M_{i′}s^{p})|i′ ∈ ℐ(j) > where ℐ(j) is as in (3.1.1). (3.1.3) If σ ∈ G(M), there is an element s ∈ S_{𝔭} such that the image “σ” of σ in G(< s >^{−1} M) has a product decomposition “σ ”= δϵ where ϵ ∈ E(< s >^{−1} M) and δ has the property that given i ∈ ℐ and p ∈ ℤ^{≧0}, there is a q ∈ ℤ^{≧0} such that δϵ_{i}(“M_{i}s^{q} ”)δ^{−1} ⊆ E(< s >^{−1} M, “Ms^{p} ”) and δ^{−1}ϵ_{i}(“M_{i}s^{q} ”)δ ⊆ E(< s >^{−1} M, “Ms^{p}”).

Lemma 3.2 Let G a Λ Λ Λ-group and M ∈ Λ Λ Λ such that (2.7.1) and (3.1) hold. Then if M is weak Noetherian over k_{M}, E(M) ◁ G(M). Furthermore, if for all weak Noetherian objects in Λ Λ Λ, (2.7.1) and (3.1) hold and G commutes with direct limits then for all quasi-weak Noetherian objects M ∈ Λ Λ Λ, E(M) ◁ G(M).

Proof Conditions (3.1.1) and (3.1.2) show that for each maximal ideal 𝔭 of k_{M} and element s ∈ k_{M} \𝔭, E(< s >^{−1} M) ⊆ B(M, s). Since the element δ in (3.1.3) clearly lies in B(M, s), it follows from (3.1.1) - (3.1.3) that condition (2.8.1) is satisfied. Thus, for M weak Noetherian over k_{M}, E(M) ◁ G(M) by (2.8). The second assertion of the lemma follows from (2.9). Q.E.D.

Let alg(ℐ) denote the subcategory of M M M(ℐ) of all objects (k_{A}, A) such that A_{i} (= λ_{i}(A)) is an associative k_{A}-algebra with identity , A_{i} = A_{j} for all i, j ∈ ℐ, and each morphism f : (k_{A}, A) ⟶ (k_{A′}, A′) is a direct sum f = ⊕_{i∈I}f_{i} of algebra homomorphism f_{i} preserving the identity such that

f_{i} = f_{j} for all i, j ∈ ℐ. Let alg = alg (ℐ_{0}) where ℐ_{0} is some index set with exactly one element. There are mutually inverse canonical functors Ψ(ℐ) :

alg ⟶ alg (ℐ), (k_{A}, A) ⟼ (k_{A}, ⊕_{i∈I}A) and (f : (k_{A}, A) ⟶ (k_{A′}, A′)) ⟼

(⊕_{i∈ℐ}f : (k_{A}, ⊕_{i∈ℐ}A) ⟶ (k_{A′}, ⊕_{i∈ℐ}A′)), and Φ(ℐ) : alg (ℐ) ⟶ alg, (k_{A}, ⊕_{i∈ℐ}A_{i}) ⟼

(k_{A}, A_{i_{0}}) (i_{0} ∈ ℐ_{0}, A_{i_{0}} = A_{i}) and (⊕_{i∈ℐ}f_{i} : (k_{A}, ⊕_{i∈ℐ}A_{i}) ⟶ (k_{A′}, ⊕_{i∈ℐ}A′_{i})) ⟼ (f_{i_{0}} :

(k_{A}, A_{i_{0}}) ⟶ (k_{A′}, A′_{i_{0}})) (f_{i_{0}} = f_{i}). For n ∈ ℤ^{≥2}, let ℐ(n) = {(i, j)| 1 ≤ i, j ≤ n, i ≠ j}. Let alg (n) =alg (ℐ(n)), Ψ(n) = Ψ(ℐ(n)) and Φ(n) = Φ(ℐ(n)). Let GL_{n} : alg ⟶ G G G denote the usual general linear functor. Composing GL_{n} with Φ(n), we get a functor alg (n) ⟶ G G G, which we also denote by GL_{n}. Thus, if A ∈alg then

GL_{n}(A) = GL_{n}(Ψ(n)(A)).

If A ∈alg, a ∈ A, and (i, j) ∈ ℐ(n), let ϵ_{ij}^{Ψ(n)(A)}(a) denote the n × nelementary matrix whose (i, j)’th coefficient is a. Clearly, the assignment a ⟼ ϵ_{ij}^{Ψ(n)(A)}(a) defines a group embedding ϵ_{ij}^{Ψ(n)(A)} : A ⟶ GL_{n}(A) and these embeddings taken over all objects Ψ(n)(A) of alg (n) define a natural transformation ϵ_{ij} : λ_{ij} ⟶ GL_{n}. Thus, the functor GL_{n} :alg (n) ⟶ G G G is an alg(n)-group functor in the sense of (2.1). Clearly, the usual elementary subgroup E_{n}(A) ⊆ GL_{n}(A) is just the subgroup generated by the subgroups image (ϵ_{ij}^{Ψ(n)(A)}) where (i, j) ranges over the elements of ℐ(n).

If A ∈alg, let 𝕄_{n}(A) denote the set of all n×n-matrices with coefficients in A. Let τ^{A} : GL_{n}(A) ⟶ 𝕄_{n}(A), σ ↦ σ − I, where I denotes the identity n × n-matrix. The functions τ^{A} define a natural transformation τ : GL_{n} ⟶ 𝕄_{n} of functors defined on alg. Using τ, one shows easily that the functor GL_{n} on alg (n) satisfies (2.7.1). Moreover, if n ≥ 3, the standard commutator relations (cf. [B4, (2.2)]) for elementary matrices show that GL_{n} satisfies (3.1.1) and (3.1.2). Suppose now that A is module finite over k_{A}. Let 𝔭 denote a maximal ideal of k_{A} and let S_{𝔭} = k_{A} \ 𝔭. Let σ ∈ GL_{n}(A) and ‘σ’ its image in GL_{n}(S_{𝔭}^{−1}A). Since S_{𝔭}^{−1}A is semilocal, it is well known (cf. [B4, (4.9) (b)] with s = 1) that ‘σ’= δ_{1}ϵ_{1} where δ_{1} is a diagonal matrix and ϵ_{1} a product of elementary matrices. It follows that for some s ∈ S_{𝔭}, the image “σ” of σ in GL_{n}(< s >^{−1} A) has a product decomposition “σ ”= δϵ where δ is a diagonal matrix and ϵ a product of elementary matrices. Since δ is diagonal, one deduces easily that it has the property ascribed to it in (3.1.3). Thus, we have shown the following:

Lemma 3.3 The functor GL_{n} on alg (n) is an alg (n)-group functor satisfying (2.7.1). Furthermore, if n ≥ 3 then GL_{n} satisfies (3.1) on objects A which are module finite over k_{A}.

Theorem 3.4 Let GL_{n} denote the general linear group functor on associative algebras A with identity. If n ≥ 3 and A is quasi-Noetherian then E_{n}(A) ◁ GL_{n}(A).

Proof This follows immediately from (3.2) and (3.3). Q.E.D.

Corollary 3.5 (Suslin [S], [Tu]) Let GL_{n} denote the general linear group functor on associative algebras A with identity. If n ≥ 3 and A is module finite over k_{A} then E_{n}(A) ◁ GL_{n}(A).

Proof One shows easily that A is quasi-Noetherian. The corollary follows now from (3.4). Q.E.D.

Following procedures similar to those used in the proof above of Suslin’s theorem, one can prove the analog of his theorem for the general quadratic functor GQ_{2n}. We do this next.

Let A denote an associative ring with identity and involution a ⟼ \overline{a}. Thus, \overline{ab} = \overline{b}\overline{a} and a^{=} = a for all a, b ∈ A. Let λ ∈ center (A) such that λ\overline{λ} = 1. Let min^{λ}(A) = \{a − λ\overline{a}| a ∈ A\} and max^{λ}(A) = \{a ∈ A | a = −λ\overline{a}\}. Let Λ denote a form parameter on A. By definition, Λ is an additive subgroup of A such that min^{λ}(A) ⊆ Λ ⊆ max^{λ}(A) and aΛ\overline{a} ⊆ Λ for all a ∈ A. The triple (A, λ, Λ) is called a form ring and is frequently abbreviated by (A, Λ). A form algebra is a 4-tuple (k_{A}, A, λ, Λ) where (k_{A}, A) ∈ alg and (A, λ, Λ) is a form ring such that the trivial involution on k_{A} is compatible with that on A and k_{A}Λ ⊆ Λ. We shall frequently abbreviate the form algebra (k_{A}, A, λ, Λ) by (A, Λ). A morphism (k_{A}, A, λ, Λ) ⟶ (k_{A′}, A′, λ′, A′) of form algebras is a morphism (g, f) : (k_{A}, A) ⟶ (k_{A′}, A′) of algebras such that f preserves the involution, f(λ) = λ′ and f(Λ) ⊆ Λ′. Let form alg denote the category of all form algebras and morphisms of form algebras.

If n ∈ ℤ^{≥2}, let ℐ(n, n) = {(i, j) | 1 ≤ i, j ≤ n, i ≠ j}. Define the subsets ℐ^{ℓ,+}(n, n) = {(i, n + i) | 1 ≤ i ≤ n}, ℐ^{ℓ,−}(n, n) = {(n + i, i) | 1 ≤ i ≤ n}, ℐ^{ℓ}(n, n) = ℐ^{ℓ,+}(n, n) ∪ ℐ^{ℓ,−}(n, n) and ℐ^{s}(n, n) = ℐ(n, n) \ ℐ^{ℓ}(n, n). The superscript ℓ (resp. s) is derived from the terminology long (resp. short) root in the theory of Chevalley and classical groups. Let form alg (n) denote the subcategory of M M M(ℐ(n, n)) whose objects are all (k_{M}, M) such that for some form algebra (k_{A}, A, λ, Λ),

(3.6.1)

k_{M} = k_{A}

M_{ij} = \begin{cases}A if (i, j) ∈ ℐ^{s}(n, n) \\ Λ if (i, j) ∈ ℐ^{ℓ}(n, n). \end{cases}

A morphism f = ⊕_{(i,j)∈ℐ(n,n)}f_{i,j} : (k_{M}, M) ⟶ (k_{M′}, M′) is any morphism f in

M M M(ℐ(n, n)) such that the following holds:

(3.6.2) For each (i, j) and (i′, j′) ∈ ℐ^{s}(n, n), f_{i,j} = f_{i′,j′} and the map f_{i,j} : A ⟶ A′ defines a morphism (k_{A}, A, λ, Λ) ⟶ (k_{A′}, A′, λ′, Λ′) of form algebras.

(3.6.3) For each (i, j) ∈ ℐ^{ℓ}(n, n), f_{i,j} = f_{i′,j′}|_{Λ} for any (i′, j′) ∈ ℐ^{s}(n, n).

There are mutually inverse functors Ψ(n) : form alg ⟶ form alg (n) and ϕ(n) : form alg (n) ⟶ form alg which are analogous to those between alg and alg (n).

Let (A, Λ) ∈ form alg. If σ is an n × n-matrix (a_{kℓ}) with coefficients a_{kℓ} ∈ A, let \overline{σ} = transpose (\overline{a}_{kℓ}). Let GQ_{2n}(A, Λ) denote the general quadratic group [B3, §3]of rank 2n over (A, Λ). By definition, GQ_{2n}(A, Λ) =

\{(\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈ GL_{n}(A) | (\begin{matrix}α & β \\ γ & δ \end{matrix})^{−1} = (\begin{matrix}\overline{γ} & λ\overline{β} \\ \overline{λ}\overline{γ} & \overline{α} \end{matrix}), diagonal coefficients of \overline{γ}α

and \overline{δ}β lie in Λ\}. The elementary matrices of GQ_{2n}(A, Λ) are described in detail in [B3, §3]. If M = Ψ(n)(A, Λ), (i, j) ∈ ℐ(n, n), and a ∈ M_{ij} then the description of elementary matrices in [B3, p. 27 - 29] makes it clear that there is precisely one elementary matrix ϵ_{ij}^{Ψ(n)(A,Λ)}(a) corresponding to the data above. The assignment a ⟼ ϵ_{ij}^{Ψ(n)(A,Λ)}(a) clearly defines an embedding M_{ij} ⟶ GQ_{2n}(A, Λ) of groups. As usual, we define EQ_{2n}(A, Λ) to be the subgroup of GQ_{2n}(A, Λ) generated by the images of the embeddings above where (i, j) ranges over ℐ(n, n). Composing the functor GQ_{2n} : form alg ⟶ G G G with the functor ϕ(n) : form alg (n) ⟶ form alg, we get a functor from form alg (n) to G G G, which we also denote by GQ_{2n}. Evidently,

GQ_{2n}(A, Λ) = GQ_{2n}(Ψ(n)(A, Λ))

and the embeddings ϵ_{ij}^{Ψ(n)(A,Λ)} : λ_{ij}(Ψ(n)(A, Λ)) ⟶ GQ_{2n}(A, Λ) where (i, j) ∈ ℐ(n, n) is fixed and Ψ(n)(A, Λ) ranges over the objects of form alg (n) define a natural transformation ϵ_{ij} : λ_{ij} ⟶ GQ_{2n}. This shows that the functor GQ_{2n} on form alg (n) is a form alg (n)-group functor.

If (A, Λ) ∈ form alg, let 𝕄_{2n}(A) denote the set of all n × n-matrices with coefficients in A. Let τ^{A,Λ} : GQ_{2n}(A, Λ) ⟶ 𝕄_{2n}(A), σ ⟼ σ−I, where I denotes the 2n×2n-identity matrix. The functions τ^{A,Λ} where (A, Λ) ranges over form alg define a natural transformation τ : GQ_{2n} ⟶ 𝕄_{2n} of functors defined on form alg. Using τ, one shows easily that the functor GQ_{2n} on form alg (n) satisfies (2.7.1). Moreover, if n ≥ 3, the standard commutator relations [B3, (3.16)] for the elementary matrices on GQ_{2n}(A, Λ) show that GQ_{2n} satisfies (3.1.1) and (3.1.2) over all objects of form alg (n). And, if n = 2 and the form algebra (A, Λ) has the property that the subgroup AΛ + ΛA of A, generated additively by all products ax and xa where a ∈ A and x ∈ Λ, equals A then again by [B3, (3.16)], GQ_{2n}(Ψ(n)(A, Λ)) satisfies (3.1.1) and (3.1.2). Furthermore, for any n ≥ 2 and any (A, Λ) ∈ form alg such that A is module finite over k_{A}, GQ_{2n}(Ψ(n)(A, Λ)) satisfies (3.1.3). One checks this as follows. Let 𝔭 be a maximal ideal of k_{A} and let S_{𝔭} = (k_{A} \ 𝔭). Let σ ∈ GQ_{2n}(A, Λ). If ‘σ’ denotes the image of σ in GQ_{2n}(S_{𝔭}^{−1}A, S_{𝔭}^{−1}Λ) then by [B1, III (4.9)] (cf. also [Bs2, III (3.11)]), one can write ‘σ’= ρτ where ρ ∈ GQ_{2}(S_{𝔭}^{−1}A, S_{𝔭}^{−1}Λ) and τ ∈ EQ_{2n}(S_{𝔭}^{−1}A, S_{𝔭}^{−1}Λ). Thus, for some s ∈ S_{𝔭}, one can write the image “σ” of σ in GQ_{2n}(< s >^{−1} A, < s >^{−1} Λ) as a product “σ”= δϵ where δϵGQ_{2}(< s >^{−1} A, < s >^{−1} Λ) and ϵ ∈ EQ_{2n}(< s >^{−1} A, < s >^{−1} Λ). It is straightforward to check that δ satisfies (3.1.3). Thus, we have shown the following:

Lemma 3.7 The functor GQ_{2n} on form alg (n) is a form alg (n)-group functor satisfying (2.7.1). Furthermore, if (A, Λ) ∈ form alg is such that A is module finite over k_{A} (resp. A is module finite over k_{A} and A = AΛ+ΛA) and n ≥ 3 (resp. n ≥ 2) then GQ_{2n} satisfies (3.1) on Ψ(n)(A, Λ).

Definition 3.8 A form algebra (A, Λ) is called quasi-Noetherian if it satisfies one of the following equivalent conditions:

(i) (A, Λ) is a direct limit of form algebras (A′, Λ′) such that A′ is Noetherian over k_{A′}.

(ii) Ψ(n)(A, Λ) is quasi-Noetherian in form alg (n) for any n > 1.

Theorem 3.9 Let GQ_{2n} denote the general quadratic functor on form algebras (A, Λ). Suppose that (A, Λ) is quasi-Noetherian. If either n ≥ 3 or n = 2 and A = AΛ + ΛA then EQ_{2n}(A, Λ) ◁ GQ_{2n}(A, Λ).

Proof By definition, Ψ(n)(A, Λ) is a direct limit of objects Ψ(n)(A′, Λ′) such that A′ is Noetherian over k_{A′}. Moreover, if the condition A = AΛ + ΛA is satisfied then we can arrange that each (A′, Λ′) above has the property that A′ = A′Λ′ +Λ′A′. The assertion of the theorem follows now from (3.2), (3.7), and the fact that GQ_{2n} commutes with direct limits. Q.E.D.

Corollary 3.10 Let GQ_{2n} denote the general quadratic functor on form algebras. Suppose that (A, Λ) is a form algebra such that A is module finite over k_{A}. If either n ≥ 3 or n = 2 and A = AΛ + ΛA then EQ_{2n}(A, Λ) ◁ GQ_{2n}(A, Λ).

Proof One shows easily that (A, Λ) is quasi-Noetherian. The corollary follows now from (3.9). Q.E.D.

Remark 3.10.1 There is a slight generalization of the general quadratic group, given in [B3, §13], for which the conclusions of (3.9) and (3.10) are also valid. The generalization is based on replacing the notion of form ring above by a pair (A, Λ) where A is a ring with antiautomorphism a ⟼ \overline{a} (by definition, \overline{ab} = \overline{b}\overline{a} for all a ∈ A) such that for some fixed element λ ∈ A, \overline{a} = λa\overline{λ} for all a ∈ A and where Λ is a form parameter in the sense defined above. One can then define a generalized form algebra to be a triple (k_{A}, A, Λ) where (A, Λ) is as above, (k_{A}, A) is an algebra over k_{A}, the antiautomorphism a ⟼ \overline{a} on A is compatible with the trivial involution on k_{A}, and k_{A}Λ ⊂ Λ. The notion quasi-Noetherian for (k_{A}, A, Λ) is defined as in (3.8). The general quadratic group GQ_{2n}(A, Λ) and its elementary subgroup EQ_{2n}(A, Λ) are defined as for ordinary form rings above. This done, the conclusions in (3.9) and (3.10) remain valid. Checking details is straightforward.

The next result for Chevalley groups overlaps with (3.5) and (3.10).

Theorem 3.11 (Taddei [T2]) Let Φ be a reduced, irreducible root system of rank ≧ 2. Let R denote a commutative ring (which we view as an algebra over itself). Let G(Φ, R) denote the (untwisted) Chevalley group of Φ with coefficients in R and let E(Φ, R) denote its elementary subgroup. Then E(Φ, R) ◁ G(Φ, R).

Proof Let the index set ℐ = Φ. Let as agree to denote the elements of ℐ by small Greek letters α, β, …, as is customary when dealing with root systems. Let Λ Λ Λ denote the subcategory of M M M(ℐ) whose objects are all pairs (k, ⊕_{α∈ℐ}k_{α}) where k is a commutative ring and k_{α} = k for all α ∈ ℐ. A

morphism (g, f) : (k, ⊕k_{α}) ⟶ (k′, ⊕k′_{α}) is a pair (g, f) where g : k ⟶ k′ is a ring homomorphism and f : ⊕k_{α} ⟶ ⊕k′_{α} a degree 0 map such that f|_{k_{α}} = g for all α ∈ ℐ. The assignment k ⟼ (k, ⊕k_{α}) defines an isomorphism ((commutative rings)) ⟶ Λ Λ Λ of categories. The Chevalley functor G(Φ, −) : ((commutative rings)) ⟶ G G G defines a Λ Λ Λ-group G : Λ Λ Λ ⟶ G G G such that G(k, ⊕k_{α}) = G(Φ, k) and for each α ∈ ℐ, ϵ_{α} : λ_{α} ⟶ G(Φ,_{−}) is the natural transformation with the property that if a ∈ k then ϵ_{α}^{k}(a) ∈ G(Φ, k) is the usual elementary transformation defined by the root α ∈ ℐ(= Φ) and the element a ∈ k.

We want to apply now (3.2) to deduce that E(Φ, R) ◁ G(Φ, R). So, we must check that all the hypotheses of (3.2) are satisfied. It is well known that there is a natural number n and an embedding G(Φ, R) ↣ GL_{n}(R) which is functorial in R. Using the set theoretic map GL_{n}(R) ⟶ 𝕄_{n}(R), σ ⟼ σ−I (where I is the n × n-identity matrix), one deduces easily that condition (2.7.1) is satisfied. Clearly, any commutative ring R is a direct limit of Noetherian subrings of itself and the functor G(Φ,_{−}) commutes with direct limits. Thus, to complete the proof of the theorem, it suffices to show that condition (3.1) is satisfied.

Since ℐ is a root system Φ, the negation −α of an element α ∈ ℐ is defined. Define ℐ(α) = ℐ \ {−α}. Condition (3.1.1) follows directly from the standard commutator formula [C,(5.2)]

[ϵ_{α}(a), ϵ_{β}(b)] = ∏_{\begin{matrix}iα+jβ∈ℐ \\ i,j∈ℕ \end{matrix}} ϵ_{iα+jβ}(N_{α,β,i,j}a^{i}b^{j})

where β ∈ ℐ(α). It is an easy exercise to check that the hypotheses on Φ guarentee that given γ ∈ ℐ, there are roots α, β ∈ ℐ, β ∈ ℐ(α), α, β ∈ ℐ(γ) such that α + β = γ and N_{α,β,1,1} = 1. Rewriting the commutator formula above as

ϵ_{γ}(ab) = [ϵ_{α}(a), ϵ_{β}(b)]( ∏_{\begin{matrix}iα+jβ∈ℐ \\ i,j∈ℕ \\ (i,j)≠(1,1) \end{matrix}} ϵ_{iα+jβ}(−N_{α,β,i,j}a^{i}b^{j})),

one deduces easily that (3.1.2) holds. Let 𝔭 denote a maximal ideal of R and let S_{𝔭} = R \ 𝔭. Let σ ∈ G(Φ, R) and ‘σ ’ its image in G(Φ, S_{𝔭}^{−1}R). Since S_{𝔭}^{−1}R is a local ring, it follows from a result of Matsumoto [M, (4.4)] in the simply connected case and Abe-Suzuki [AS] in general that ‘σ ’= δ_{1}ϵ_{1} where δ_{1} is a diagonal matrix, i.e. an element of the split maximal torus of G(Φ, S_{𝔭}^{−1}R), and ϵ_{1} a product of elementary transformations. It follows that for some element s ∈ S_{𝔭}, the image “σ” of σ in G(Φ, < s >^{−1} R) has a product decomposition “σ ”= δϵ where δ is a diagonal matrix and ϵ a product of elementary transformations. Since δ is diagonal, one deduces easily that it has the property ascribed to it in (3.1.3). Q.E.D.

Let Φ be a reduced, irreducible root system of type A_{ℓ}, D_{ℓ}, or E_{6} and rank ≧ 2. After fixing a system of fundamental roots for Φ, we have the notion [St, ], [A, ], [P, ] of a canonical automorphsim ϱ of Φ. Except for D_{4}, there is exactly one such automorphism and it has order 2. In the case of D_{4}, there are two such automorphisms, one of order 3 and the other of order 3. Given a pair (Φ, ϱ) and a commutative ring R with a Ring action of < ϱ > (: = the group generated by ϱ), one has the notion ^{ϱ}G(Φ, R) of a twisted Chevalley group with coefficients in R and the notion ^{ϱ}E(Φ, R) of its elementary subgroup.

The next theorem is the analog of the one above, for twisted Chevalley groups of type ≠ A_{2ℓ}, and overlaps also with (3.10).

Theorem 3.12 Let Φ be a completeley reduced, irreducible root system of type A_{2ℓ+1}, D_{ℓ} or E_{6} and rank ≧ 2. Let ϱ be a canonical automorphism of Φ. Let R denote a commutative ring with a ring action of ϱ. (We view R as an algebra over the fixed ring R^{ϱ}). Let ^{ϱ}G(Φ, R) denote the twisted Chevalley group of Φ with coefficients in R and let ^{ϱ}(Φ, R) denote its elementary subgroup. Then E(Φ, R) ◁ G(Φ, R).

Proof Our proof is the same as the one above, except ℐ is set equal to the root system defined by the orbits of the action of ϱ on Φ, the standard commutator formula [C, (5.2) ]is replaced by its twisted analogs [A, §3], [P, ] and the reference to Matsumoto [M, (4.) ] and Abe-Suzuki [AS] are replaced by reference Suzuki [Sz]. We fill in now the details.

The action of ϱ on the enveloping Euclidean space 𝔼 of Φ preserves the usual inner product. If V denotes the subspace of 𝔼 of all elements fixed by the action of ϱ then by [St, ], [A], the orthogonal projection of 𝔼 on V maps the orbit space of the action of ϱ on Φ onto a root system in V. Let ℐ denote this root system. If (Φ, ϱ) = (A_{2ℓ+1,ϱ}), (D_{ℓ}, ϱ), (E_{6}, ϱ) and throughout ϱ has order 2, or (D_{4}, ϱ) and ϱ has order 3 then ℐ = C_{ℓ+1}, B_{ℓ−1}, F_{4}, G_{2}, respectively. (The twisted Chevalley group G(A_{2ℓ+1}, R) is the general quadratic group GQ_{2(ℓ+1)}(R, R^{ϱ}) in (3.10).)

Let k denote a commutative ring with a ring action of ϱ. Let k^{ϱ} = {a ∈ k|ϱ(a) = a}. Let Λ Λ Λ denote the subcategory of M M M(ℐ) whose objects are all pairs (k, ⊕_{α∈ℐ}k_{α}) such that

k_{α} = \begin{cases}k, if α is a short root \\ k^{ϱ}, if α is a long root. \end{cases}

A morphism (g, f) : (k, ⊕_{α∈ℐ}k_{α}) ⟶ (k′, ⊕_{α∈ℐ}k′_{α}) is a pair (g, f) where g : k ⟶

k′ is a ϱ-equivariant, ring homomorphism and f : ⊕k_{α} ⟶ ⊕k′_{α} a degree 0 map such that f|_{k_{α}} = g|_{k_{α}} for all α ∈ ℐ. (Note that each k_{α} ⊆ k.) Let ((comm ϱ-rings)) denote the category of all commutative rings with a ring action of ϱ and all ϱ-equivariant ring homomorphisms. The assignment k ⟼ (k, ⊕k_{α}) defines an isomorphism ((comm ϱ-rings)) ⟶ Λ Λ Λ of categories. The twisted Chevalley functor ^{ϱ}G(Φ, −) : ((comm ϱ-rings)) ⟶ G G G defines a Λ Λ Λ-group G : Λ Λ Λ ⟶ G G G such that G(k, ⊕k_{α}) = ^{ϱ}G(Φ, k) and for each α ∈ ℐ, ϵ_{α} : λ_{α} ⟶ ^{ϱ}G(Φ, −) is a natural transformation with the property that if a ∈ k_{α}(= λ_{α}(⊕k_{β})) then ϵ_{α}^{k}(a) ∈ ^{ϱ}G(Φ, k) is the elementary transformation (which is also referred to in the literature by the expression elementary root unipotent element) (cf. [C,(13.6)] ) defined by the root α ∈ ℐ and the element a ∈ k_{α}.

We want to apply now (3.2) to deduce that ^{ϱ}E(Φ, R)◁^{ϱ} G(Φ, R). So, we must check that all the hypotheses of (3.2) are satisfied. It is well known that there is a natural number n and an embedding ^{ϱ}G(Φ, R) ↣ GL_{n}(R) which is functorial in R. Using the set theoretic map GL_{n}(R) ⟶ 𝕄_{n}(R), σ ⟼ σ−I (where I is the n × n-identity matrix), one deduces easily that condition (2.7.1) is satisfied. Clearly, any commutative ϱ-ring R is a direct limit in ((comm ϱ-rings)) of Noetherian ϱ-subrings of itself and the functor ^{ϱ}G(Φ,_{−}) commutes with direct limits. Thus, to complete the proof of the theorem, it suffices to show that condition (3.1) is satisfied.

Since ℐ is a root system, the negation −α of an element α ∈ ℐ is defined. Define ℐ(α) = ℐ \ {−α}. Condition (3.1.1) follows directly form the twisted analogs [A, §3], [P, ] of the standard commutator formula [C, (5.2)] used in the proof of (3.11). It is an easy exercise to check that the hypotheses on Φ guarente that given γ ∈ ℐ, there are roots α, β ∈ ℐ, α, β ∈ ℐ(γ) such that α + β = γ and N_{α,β,1,1} = 1. Using the analogs above of the standard commutator formula, one can write

ϵ_{γ}(ab) = [ϵ_{α}(a), ϵ_{β}(b)] ∏_{\begin{matrix}iα+jβ∈ℐ \\ i,j∈ℕ \\ (i,j)≠(1,1) \end{matrix}} ϵ_{iα+jβ}(⋆(α, β, i, j, a, b)),

for suitable coefficients ⋆(α, β, i, j, a, b) and deduces that (3.1.2) holds. We leave the details here to the reader. Let 𝔭 denote a maximal ideal of R^{ϱ} and let S_{𝔭} = R^{ϱ} \ 𝔭. Let σ ∈ ^{ϱ}G(Φ, R) and ‘σ’ its image in ^{ϱ}G(Φ, S_{𝔭}^{−1}R). Since S_{𝔭}^{−1}R^{ϱ} is a local ring, it follows from a result of Suzuki [Sz] that ‘σ’ = δ_{1ϵ1} where δ_{1} is a diagonal matrix, i.e. an element of the Cartan subgroup of ^{ϱ}G(Φ, S_{𝔭}^{−1}R), and ϵ_{1} a product of elementary transformations. It follows that for some elements s ∈ S_{𝔭}, the image “σ” of σ in ^{ϱ}G(Φ, < s >^{−1} R) has a product decomposition “σ” = δϵ where δ is a diagonal matrix and ϵ a product of elementary transformations. Since δ is diagonal, one deduces easily that it has the property ascribed to it in (3.1.3). Q.E.D.