K_{1} OF CHEVALLEY GROUPS ARE NILPOTENT

ROOZBEH HAZRAT, NIKOLAI VAVILOV

1. Introduction

Let Φ be a reduced irreducible root system and R be a commutative ring. We consider the corresponding simply connected Chevalley group G = G(Φ, R) and its elementary subgroup E(Φ, R). When rk(Φ) ≥ 2 it is proven by Suslin and Kopeiko [17], [18], [11] for the classical cases and by Taddei [20] for the exceptional cases, that E(Φ, R) is normal in G(Φ, R), so that one can consider the K_{1}-functor modeled on G:

K_{1}(Φ, R) = G(Φ, R)/E(Φ, R),

see references in [15], [25], [13]. Observe that this functor is a generalisation of SK_{1} rather than the usual K_{1}. Namely, SL(l + 1, R) is the Chevalley group of type A_{l} and K_{1}(A_{l}, R) = SK_{1}(l + 1, R).

It is well known that when R is a field, or, more generally a semi-local ring, the functor K_{1}(Φ, R) is trivial, or, in other words, G(Φ, R) = E(Φ, R) (see, for example, [2]). In the stable range, i.e. when rk(Φ) is large with respect to the dimension of R, the functor K_{1}(Φ, R) is abelian. The present paper is an attempt to understand what can be said about K_{1}(Φ, R) in the meta-stable range, when dimension of R is large. There are examples due to van der Kallen and Bak [9], [3] which show that non-stable K_{1}(Φ, R) can be non-abelian, and the natural question is how non-abelian it can be?

In [3], Bak, developed a beautiful localisation-completion method which allowed him to prove that SK_{1}(n, R) is nilpotent, and, more generally, K_{1}(n, R) is nilpotent-by-abelian when Bass-Serre dimension δ(R) of the ground ring R is finite. Recall that a group H is called nilpotent-by-abelian, if it has a normal subgroup F such that F is nilpotent and H/F is abelian. This clearly implies that H is a solvable group.

In [7] the first author uses the same method to extend this result to nonstable K_{1} of general quadratic groups. Classical Chevalley groups fall into this category and it follows from the results of [7] that K_{1} are nilpotent for Chevalley groups of types C_{l} and D_{l}. In fact, [7] establishes much more general results, namely that certain slightly larger K_{1}-functors are nilpotentby-abelian for a huge class of unitary groups over form rings.

Here we show that the same holds for all Chevalley groups. More precisely, the main result of the present work is a construction of a descending central series in the Chevalley group, indexed by the Bass-Serre dimension of the factor-rings of the ground ring. In the case of finite-dimensional rings this leads to the following theorem, which we prove in Section 7.

Date: February 16, 2001.

Theorem. Let Φ be a reduced irreducible root system of rank ≥ 2 and R be a commutative ring such that its Bass-Serre dimension δ(R) is finite. Then for any Chevalley group G(Φ, R) of type Φ over R the quotient G(Φ, R)/E(Φ, R) is nilpotent-by-abelian. In particular K_{1}(Φ, R) is nilpotent of class at most δ(R) + 1.

A special case of this result pertaining to the case when R = ℝ^{X} or ℂ^{X} is the ring of all continuous real or complex-valued functions on a finitedimensional topological space X was stated by Vaserstein in [24], Theorem 7. This theorem is accompanied by the following proof, which we reproduce verbatim: “Proof, using the Bruhat decomposition is the same as for GL_{n}(A).”

Our principal tool is the localisation-completion method of [3] and [7], and we refer the reader to these papers and [4] for more background information and details. Since this method is not as popular as some other techniques and the body of our paper consists of calculations, we take our breath for the moment and explain what really goes on here and how this method stands to other major methods which are used to attack similar problems. To avoid some additional technical complications and present ideas in their simplest form, assume for the time being that G(Φ, R) is simply connected.

Informally the theorem above may be viewed as an extremely strong form of normality of E(Φ, R) in G(Φ, R). In fact, normality asserts that for any elementary generator x_{α}(a) and any g ∈ G one has [x_{α}(a), g] ∈ E(Φ, R). On the other hand our theorem asserts that for finite-dimensional rings something terribly much stronger occurs, namely [… [[g_{1}, g_{2}], g_{3}], …, g_{m}] ∈ E(Φ, R) for any sufficiently long sequence g_{1}, g_{2}, …, g_{m} ∈ G. Of course, in view of the fact that E(Φ, R) is perfect this implies normality of E(Φ, R).

Now for arbitrary^{1} commutative^{2} rings we are aware of five major noticeably different ways to prove such results:

• Suslin’s direct factorisation method [17], [18], [11], [8];

• Suslin’s factorisation and patching method [22], [10], [5];

• Quillen–Suslin–Vaserstein’s localisation and patching method, [17], [23], [20], [19];

• Bak’s localisation-completion method [3], [7], [4];

• Stepanov–Vavilov–Plotkin’s decomposition of unipotents [25], [27], [16], [26].

Suslin’s first method and decomposition of unipotents are based on reduction to groups of smaller rank over the same ring. On the other hand, localisation and patching and localisation-completion are based on reduction to groups of the same type over rings of smaller dimension. Of course, here too one has to invoke reduction to a smaller rank at some stage, but the only such reduction there is, occurs at the level of zero-dimensional rings, is classically known and remains invisible to the reader. For example, the only reduction to groups of smaller rank which is ever used in the present paper, appears under the disguise of Gauß decomposition over semi-local rings. Suslin’s second method combines reduction in dimension and rank. Sometimes these methods are simultaneously used in the same proof, as in [14], which brings into action the combined force of localisation-completion and decomposition of unipotents to obtain length bounds which would be beyond reach of either of these methods individually.

1There are, of course, many further methods, which only work for some classes of rings, most notably, methods using stability conditions, as developed by Bass, Bak, Dennis, van der Kallen, Stein, Suslin, Vaserstein, and others. There are some further methods, which use topological or metric properties of R, or other similar structures. We do not try to survey such methods here.

2There are numerous generalisations and ramifications of these methods for noncommutative rings, including the very powerful Golubchik-Mikhalev non-commutative localisation methods, see references in [25], [5], [16], which we do not discuss here, since we are only interested in commutative rings.

Decomposition of unipotents is a generalisation of Suslin’s first method. It is very powerful and extremely straightforward at the same time. When it can be applied, it usually gives by far the best results algorithmically. What you should expect to get in our problem, would be an explicit polynomial formula, expressing [… [[g_{1}, g_{2}], g_{3}], …, g_{m}] as a product of elementary root unipotents x_{α}(a) with parameters a depending polynomially on the matrix entries of g_{i}’s in a faithful representation of G. Now everybody, who has seen, how such a formula looks like for the commutator [x_{α}(a), g] with one general matrix g ∈ G for the classical groups in vector representations, [11], [12], [25], [16], or for the groups of types E_{6} or E_{7} in micro-weight representations, [25], [27], [26], would immediately recognise, that writing a similar formula for our problem was not an option.

The other three methods are very similar in spirit, they are all based on localisations and partitions of 1 in the ground ring. The real difference is in how they address zero divisors. The relation between Suslin’s second method and localisation and patching is exactly the same as the relation between Suslin’s and Quillen’s solutions of Serre’s problem. Both methods are well documented in the existing literature. This is especially true for localisation and patching which was used in dozens of papers published by Suslin, Kopeiko, Tulenbaev, Abe, Vorst, Vaserstein, Taddei, Li Fuan, and many others. The key feature of localisation and patching is throwing in independent variables to fight zero divisors and then applying Quillen’s theorem [17].

In [3] Bak proposed yet another version of localisation, which does not require passage to a polynomial ring, but operates in R itself. The characteristic feature of this method is reduction to Noetherian rings, where one can easily control the behaviour of zero divisors. The main idea of the method, once you get it, is exceedingly simple. Unfortunately, [3] doesn’t constitute an easy reading, since it throws in all possible technical complications simultaneously: non-commutativity, explicit bounds for lengths, second localisation, completion, and more. The crux of the method is buried somewhere in the proof of Lemma 4.11 in the depth of Section 4. And as the final blow, which was especially frustrating for the second author, the notation in [3] fails to clearly distinguish between an element of g ∈ G and its images under localisations – and at some point in the proof one has to look at the images of g in four different localisations.

We believe that the method introduced in [3] is so natural and important, that it deserves a much better publicity, and one of our broader intentions in writing this paper was to give it the credit it truly deserves. To explain the essence of the method, below we reproduce what we believe is the shortest existing proof for the normality of E(Φ, R) in G(Φ, R). In what follows we denote by F_{M} : R ⟶ R_{M} the localisation homomorphism modulo a maximal ideal and by F_{s} : R ⟶ R_{s} the localisation homomorphism with respect to s ∈ R.

We wish to prove that for any g ∈ G, any α ∈ Φ and any a ∈ R one has x = gx_{α}(a)g^{−1} ∈ E(Φ, R). As typical for localisation proofs, we use partitions of 1. In other words, we have to pick up b_{1}, …, b_{r} ∈ R such that 1 = b_{1} + … + b_{r} and each of gx_{α}(b_{i}a)g^{−1} already lies in E(Φ, R). Of course, the difference between various localisation methods is in how one chooses such a partition. The following paragraph is a friendly takeover of a theme of Bak, [3], Lemma 4.11.

Since the functors G(Φ, −) and E(Φ, −) commute with direct limits and R is a direct limit of its finitely generated subrings, we can from the very start reduce to the case when R is Noetherian. Fix a maximal ideal M ∈ Max(R). Since for local rings E coincides with G, one has F_{M}(g) ∈ E(Φ, R_{M}). Since R_{M} is the direct limit of R_{t}, t ∈ R \ M, there exists such an s ∈ R \ M, that F_{s}(g) ∈ E(Φ, R_{s}). We will search for a b_{i} of the form s^{l} for a sufficiently large exponent l. Set y = gx_{α}(s^{l}a)g^{−1}. The ring R_{s} being Noetherian, for a large power of s, say for s^{n}, the restriction of F_{s} to the principal congruence subgroup G(Φ, R, s^{n}R) is injective. Since F_{s}(g) ∈ E(Φ, R_{s}), by the Chevalley commutator formula there exists a higher power of s, say, s^{l}, l ≥ n, such that F_{s}(y) = F_{s}(g)F_{t}(x_{α}(s^{l}a))F_{s}(g)^{−1} can be expressed as a product ∏ x_{β_{i}}(F_{s}(s^{n}c_{i})), i = 1, …, m. Take the product z = ∏ x_{β_{i}}(s^{n}c_{i}), i = 1, …, m. By the very definition z ∈ E(Φ, R) and F_{s}(y) = F_{s}(z). On the other hand, since G(Φ, R, s^{n}R) is normal in G(Φ, R), one has y, z ∈ G(Φ, R, s^{n}R). Thus y = z ∈ E(Φ, R). Since s^{l} ∉ M and the same works for all maximal ideals, we get the desired partition.

There was no completion so far, only localisation, but this is not the end of the story. Bak’s method was developed to prove much stronger results, than normality, and that’s how completion enters the stage. For suppose we have to prove that x = [h, g] lies in E(Φ, R) for two general matrices h and g. Actually, this is exactly what we have to verify in the proof of the above Theorem, not over R itself though, but over its factor-rings. This cannot be easily done by a single localisation. The main idea in the proof of [3] Theorem 4.1, embodied in Theorem 4.16, is to use localisation in an element t, to prove that there exists an element s such that F_{s}(h) ∈ E(Φ, R_{s}), whereas g ∈ E(Φ, R)G(Φ, R, s^{m}R) for an arbitrarily large power m. In other words, the element h becomes elementary after localisation in s, whereas g becomes elementary after s-completion, which explains the name localisation-completion. Then we can argue exactly as above, by the second localisation in the element s. Namely, F_{s}(h) is elementary, and, taking a sufficiently large power s^{m} in the congruence for g, we can guarantee that even with all the denominators in F_{s}(h), enough s’s survive for F_{s}(x) to be in the image of E(Φ, R, s^{n}R) for an n such that restriction of F_{s} to G(Φ, R, s^{n}R) is injective.

Of course our actual proof is technically somewhat more demanding. In general there is an extra toral factor to take care of^{3} and one has to fiddle with the Chevalley commutator formula a bit to convince herself that she still has large powers of both s and t in the numerator, after all conjugations. But these are details, all the ideas are already there. The paper [7] uses essentially the same ideas, but there are some further technical moments, like non-commutativity, non-triviality of the involution and the form parameter which make calculations in [7] much harder that the ones of the present paper.

The details of our calculations look slightly differently for the non-sym-plectic and the symplectic case, i.e. when Φ ≠ C_{l} or Φ = C_{l} respectively (recall that B_{2} = C_{2} is symplectic!) and for symplectic case the analysis of the simply-connected group is somewhat easier than the analysis of the adjoint group. By skipping symplectic case altogether we could both spare a page or two of calculations and obtain somewhat better bounds in some of the auxiliary results. Unfortunately this couldn’t have been done, if we wish to have our theorem for all groups. In fact, the paper [7] by the first author supplies all the details for the more general case of unitary groups over a form ring. However it does so only in vector representations. Symplectic groups G_{sc}(C_{l}, R) = Sp(2l, R) are obtained as a special case from the general quadratic setting of [7] when the involution is trivial, λ = −1 and Λ = R. However the adjoint symplectic case G_{ad}(C_{l}, R) = PGSp(2l, R) requires some extra care.

The rest of the paper is organised as follows. In § 2 we introduce some notation and in §§ 3,4 prove several easy lemmas on commutators. In § 5 we prove a patching result, which in particular provides a shorter proof of Taddei’s theorem. In § 6 we introduce the last important ingredient of the proof, completion. Finally, main results are established in § 7.

The main stimulus to undertake this work was a discussion with Benson Farb, who mentioned that solvability of the K_{1} functor would have very important geometric consequences. We would like to thank Alexei Stepanov for careful reading of our original manuscript and some very illuminating comments. The second author gratefully acknowledges support of the RFFI projects 98-01-00825 and 00–01–00441.

2. Preliminaries

2.1. Let us fix some notation. Let R be a commutative ring with 1, S be a multiplicative system in R and S^{−1}R be the corresponding localisation. We will mostly use localisation with respect to the two following types of multiplicative systems. If s ∈ R and the multiplicative system S coincides with ⟨s⟩ = {1, s, s^{2}, … } we usually write ⟨s⟩^{−1}R = R_{s}. If M ∈ Max(R) is a maximal ideal in R, and S = R \ M, we usually write (R \ M)^{−1}R = R_{M}. We denote by F_{S} : R ⟶ S^{−1}R the canonical ring homomorphism called the localisation homomorphism. For the two special cases mentioned above, we write F_{s} : R ⟶ R_{s} and F_{M} : R ⟶ R_{M}, respectively. When we write an element as a fraction, like a/s or \frac{a}{s} we always think of it as an element of some localisation S^{−1}R, where s ∈ S. If s were actually invertible in R, we would have written as^{−1} instead.

3Actually, there is exactly one case, when the toral factor plays a role: long roots in adjoint symplectic groups.

2.2. Let as above Φ be a reduced irreducible root system, P, Q(Φ) ≤ P ≤ P (Φ) be a lattice between the root lattice and the weight lattice. We denote by G = G_{P} (Φ, R) the Chevalley group of type (Φ, P) over R, by T = T_{P} (Φ, R) – a split maximal torus and by E = E_{P} (Φ, R) the corresponding (absolute) elementary subgroup. Usually P does not play role in our calculations and we suppress it in the notation. The elementary group E = E(Φ, R) is generated by all root unipotents x_{α}(a), α ∈ Φ, a ∈ R, elementary with respect to T. The subgroup E being normal in G means exactly that E does not depend on the choice of T.

For any Φ, P assignments R ⟶ X(Φ, R), where X = G, T, E, define functors from commutative rings to groups, i.e. to a ring homomorphism ϕ : R_{1} ⟶ R_{2} there corresponds a natural group homomorphism X(ϕ) : X(Φ, R_{1}) ⟶ X(Φ, R_{2}), which we usually denote by the same letter ϕ, rather than by their official names G(ϕ), T (ϕ) and E(ϕ). For E this is obvious, it is enough to define ϕ on elementary generators by ϕ(x_{α}(a)) = x_{α}(ϕ(a)), whereas G and T are by the very construction affine group schemes, i.e. representable functors from rings to groups. In fact E is a subfunctor of G in the sense that the restriction of G(ϕ) to E(Φ, R) coincides with E(ϕ). In particular, if S is a multiplicative system in R, the localisation homomorphism F_{S} : G(Φ, R) ⟶ G(Φ, S^{−1}R) maps E(Φ, R) inside E(Φ, S^{−1}R).

2.3. The property of these functors which will be crucial for what follows is that they commute with direct limits. In other words, if R = \underrightarrow{lim} R_{i}, where \{R_{i}\}_{i∈I} is an inductive system of rings, then X(Φ, \underrightarrow{lim} R_{i}) = \underrightarrow{lim} X(Φ, R_{i}). We will use this property in the following two situations. First, let R_{i} be the inductive system of all finitely generated subrings of R with respect to the embeddings. Then X(Φ, R) = \underrightarrow{lim} X(Φ, R_{i}), which reduces most of the proofs to the case of Noetherian rings. Second, let S be a multiplicative system in R and R_{s}, s ∈ S, the inductive system with respect to the localisation homomorphisms: F_{t} : R_{s} ⟶ R_{st}. Then X(Φ, S^{−1}R) = \underrightarrow{lim} X(Φ, R_{s}), which allows to reduce localisation in any multiplicative system to localisation in one element.

2.4. Let 𝔞 be an additive subgroup of R. Then E(Φ, 𝔞) denotes the subgroup of E(Φ, R) generated by all elementary root unipotents x_{α}(t) where α ∈ Φ and t ∈ 𝔞. Further, let L denote a nonnegative integer and let E^{L}(Φ, 𝔞) denote the subset of E(Φ, 𝔞) consisting of all products of L or fewer elementary root unipotents x_{α}(t), where α ∈ Φ and t ∈ 𝔞. Thus E^{1}(Φ, 𝔞) is the set of all x_{α}(t), α ∈ Φ, t ∈ 𝔞. When 𝔞 ⊴ R is a proper ideal in R, the group E(Φ, 𝔞) shouldn’t be confused with the (relative) elementary group E(Φ, R, 𝔞) of level 𝔞. By definition E(Φ, R, 𝔞) is the normal closure of E(Φ, 𝔞) in E(Φ, R). In general E(Φ, R, 𝔞) is not generated by elementary transvections of level 𝔞. We use the following easy fact on the interrelation of E(Φ, 𝔞) with the relative elementary groups, see, for example, [21], Proposition 2.

Lemma 2.5. Suppose 𝔞 ⊴ R is an ideal in R. In the case Φ ≠ C_{l} one has E(Φ, 𝔞) ≥ E(Φ, R, 𝔞^{2}). In the case Φ = C_{l} one has E(Φ, 𝔞) ≥ E(Φ, R, ((2)+ 𝔞)𝔞^{2}).

2.6. If 𝔞 ⊴ R is an ideal in R, then we denote by G(Φ, R, 𝔞) the principal congruence subgroup of level 𝔞 in G(Φ, R), i.e. the kernel of the reduction homomorphism modulo 𝔞: G(Φ, R) ⟶ G(Φ, R/𝔞). Clearly, E(Φ, 𝔞) ≤ G(Φ, R, 𝔞). Further, set T (Φ, R, 𝔞) = T (Φ, R) ∩ G(Φ, R, 𝔞). Fix an ordering on Φ, let Φ^{+} and Φ^{−} be the corresponding sets of positive and negative roots, respectively. As usual, we set

U(Φ, 𝔞) = ⟨x_{α}(a), a ∈ 𝔞, α ∈ Φ^{+}⟩
U^{−}(Φ, 𝔞) = ⟨x_{α}(a), a ∈ 𝔞, α ∈ Φ^{−}⟩

Obviously, U(Φ, 𝔞), U^{−}(Φ, 𝔞) ≤ E(Φ, 𝔞).

Our reduction to groups of smaller rank is based on the following version of Gauß decomposition, see [1], Corollary 3.3 and [2], Proposition 2.3.

Lemma 2.7. If 𝔞 is an ideal of R contained in the Jacobson radical, then we have

G(Φ, R, 𝔞) = U(Φ, 𝔞)T (Φ, R, 𝔞)U^{−}(Φ, 𝔞).

2.8. Let R^{∗} be the multiplicative group of the ring R. For α ∈ Φ and a ∈ R^{∗} one sets w_{α}(a) = x_{α}(a)x_{−α}(a^{−1})x_{α}(a) and h_{α}(a) = w_{α}(a)w_{α}(−1). Let H(Φ, R) be the subgroup of T (Φ, R) generated by all h_{α}(a):

H(Φ, R) = ⟨h_{α}(a), a ∈ R^{∗}, α ∈ Φ⟩

The following formula (see [1], Section 2.2)

h_{α}(a) = x_{−α}(a^{−1} − 1)x_{α}(1)x_{−α}(a − 1)x_{α}(−1)x_{α}(1 − a^{−1})

shows that h_{α}(a) ∈ E(Φ, R, 𝔞) if a ≡ 1 (mod 𝔞). In particular, H(Φ, R) = T (Φ, R) ∩ E(Φ, R).

It is shown in [2] that Lemma 2.7 immediately implies

Lemma 2.9. Let R be semi-local. Then G(Φ, R) = E(Φ, R)T (Φ, R). In particular,

G(Φ, R)/E(Φ, R) = T (Φ, R)/H(Φ, R)

is abelian and if G(Φ, R) is simply-connected, G(Φ, R) = E(Φ, R).

Lemma 2.10. If 𝔞 is an ideal of local ring R then

G(Φ, R, 𝔞) = T (Φ, R, 𝔞)E(Φ, 𝔞).

Proof. If 𝔞 = R, the conclusion follows from Lemma 2.9. If 𝔞 is a proper ideal, it is contained in the Jacobson radical and we can apply Lemma 2.7. Since T (Φ, R, 𝔞) normalises U(Φ, 𝔞) and both U(Φ, 𝔞) and U^{−}(Φ, 𝔞) are contained in E(Φ, 𝔞), the left hand side is contained in the right hand side. The inverse inclusion is obvious.

2.11. If a and b are elements of a group, we write ^{a}b = aba^{−1} and [a, b] = aba^{−1}b^{−1}. In the sequel we make heavy use of the following commutator formulae: [a, bc] = [a, b]^{b}[a, c] and [ab, c] = ^{a}[b, c][a, c]. Most of the calculations in the present paper are based on the Chevalley commutator formula

[x_{α}(s), x_{β}(t)] = ∏_{iα+jβ∈Φ} x_{iα+jβ}(N_{αβij}s^{i}t^{j}),

where N_{αβij} are the structure constants which do not depend on s and t (but for Φ = G_{2} may depend on the order of the roots in the product on the right hand side). The following observation was made by Chevalley himself: let α − pβ, …, α − β, α, α + β, …, α + qβ be the α-series of roots through β, then N_{αβ11} = ±(p + 1) and N_{αβ12} = ±(p + 1)(p + 2)/2.

Let i_{Φ} be the largest integer which may appear as i in a root iα + jβ ∈ Φ for all α, β ∈ Φ. Obviously i_{Φ} = 1, 2 or 3, depending on whether Φ is simply laced, doubly laced or triply laced. The following result makes the proof for Φ ≠ C_{l} slightly easier than for the symplectic case. Recall that A_{1} = C_{1} and B_{2} = C_{2} so that root systems of types A_{1} and B_{2} are symplectic. All roots of A_{1} are long.

Lemma 2.12. Let β ∈ Φ and either Φ ≠ C_{l} or β is short. Then there exist two roots γ, δ ∈ Φ such that β = γ + δ and N_{γδ11} = 1. If Φ = C_{l}, l ≥ 2, and β is long, then there exist two roots γ, δ ∈ Φ such that either β = γ + 2δ and N_{γδ12} = 1, or β = 2γ + δ and N_{γδ21} = 1.

Proof. If β is long and Φ ≠ C_{l}, then β can be embedded into a root system of type A_{2} consisting of long roots. Take any two roots γ, δ ∈ Φ such that γ + δ = β. Now let β be short. Then β can be embedded into a root system of type B_{2} and G_{2}. Let γ be a short root and δ be a long root such that γ + δ = β. Finally if β is long and Φ = C_{l}, let γ be a long root and δ be a short root such that γ + 2δ = β. In all cases γ − δ is not a root and thus N_{γδ1i} = ±1, where i = 1 in the generic case and i = 2 in the exceptional case. If N_{γδ1i} = −1 switch γ and δ.

Throughout the paper the letters k, l, m, n, p, q, r, K, L are used to denote non-negative integers, a, b, c, d, s, t denote elements of the ground ring R, α, β, γ, δ denote roots in Φ and g, h, x, y, z, u, v denote elements of the Chevalley group G(Φ, R).

3. First localisation

In this and next section we prove some technical results on conjugation

calculas of Chevalley groups. If t ∈ R, let \frac{t}{s^{k}} R denote the additive subgroup

of R_{s} consisting of all quotients \frac{ta}{s^{k}}, where a ∈ R. All calculations in the present section take place in E(Φ, R_{s}). Thus, when we write something like E(Φ, s^{p}t^{q}R), or x_{α}(s^{p}a), what we really mean, is E(Φ, F_{s}(s^{p}t^{q}R)), or x_{α}(F_{s}(s^{p}a)), respectively, but we suppress F_{s} in our notation. However this shouldn’t lead to a confusion since in this section we never refer to elements or subgroups of G(Φ, R). Starting from Section 5, where elements of G(Φ, R) and several of its localisations may appear in the same formula, we always explicitly cite the corresponding localisation homomorphisms.

Lemma 3.1. If p and k are given, there is a q such that

E^{1}(Φ, \frac{1}{s^{k}} R)E(Φ, s^{q}t^{3}R) ⊆ E(Φ, s^{p}tR).

Proof. Since by definition E(Φ, s^{q}t^{3}R) is generated by x_{β}(s^{q}t^{3}b), b ∈ R, it suffices to show that there is a q such that

x_{α}(\frac{a}{s^{k}} )x_{β}(s^{q}t^{3}b) ∈ E(Φ, s^{p}tR).

for any x_{α}(\frac{a}{s^{k}} ) ∈ E^{1}(Φ, \frac{1}{s^{k}} R) and any x_{β}(s^{q}t^{3}b) ∈ E(Φ, s^{q}t^{3}R).

Case 1. Let α ≠ −β and set q ≥ i_{Φ}k + p. By the Chevalley commutator formula,

x_{α}(\frac{a}{s^{k}} )x_{β}(s^{q}tb)x_{α}( − \frac{a}{s^{k}} ) = x_{β}(s^{q}tb) ∏_{iα+jβ∈Φ} x_{iα+jβ}(N_{αβij}(\frac{a}{s^{k}} )^{i}(s^{q}tb)^{j})

and a quick inspection shows that the right hand side of the above equality is in E(Φ, s^{p}tR). Case 2. Let α = −β and one of the following holds: β is short or Φ ≠ C_{l}. By Lemma 2.1 there exist roots γ and δ such that γ + δ = β and N_{γδ11} = 1. We set q = 2(i_{Φ}k + p) and decompose x_{β}(s^{q}t^{2}b) as follows:

x_{β}(s^{q}t^{2}b) = [x_{γ}(s^{q/2}t), x_{δ}(s^{q/2}tb)] ∏ x_{iγ+jδ}(−N_{γδij}(s^{q/2}t)^{i}(s^{q/2}tb)^{j}),

where the product on the right hand side is taken over all roots iγ + jδ ≠ β. Conjugating this expression by x_{α}(\frac{a}{s^{k}} ) we get

x_{α}(\frac{a}{s^{k}} )x_{β}(s^{q}t^{2}b) = [x_{α}(\frac{a}{s^{k}} )x_{γ}(s^{q/2}t), x_{α}(\frac{a}{s^{k}})x_{δ}(s^{q/2}tb)] ×

∏ x_{α}(\frac{a}{s^{k}} )x_{iγ+jδ}( − N_{γδij}(s^{q/2}t)^{i}(s^{q/2}tb)^{j}).

Obviously γ, δ and all the roots iγ + jδ ≠ β are distinct from −α and now Case 1 shows that each term is in E(Φ, s^{p}tR). Case 3. Let Φ = C_{l} and α = −β be a long root. By Lemma 2.1 there exist roots γ and δ such that either γ + 2δ = β and N_{γδ12} = 1, or 2γ + δ = β and N_{γδ21} = 1. We look at the first case, the second case is similar (alternatively, if N_{γδ12} = −1, one could change the sign of x_{γ}(b) in the following formula by x_{γ}(−b)). We set q = 3(i_{Φ}k + p) and decompose x_{β}(s^{q}t^{3}b) as follows:

x_{β}(s^{q}t^{3}b) = [x_{γ}(s^{q/3}tb), x_{δ}(s^{q/3}t)]x_{γ+δ}(−N_{γδ11}s^{2q/3}t^{2}b),

Again conjugating this expression by x_{α}(\frac{a}{s^{k}} ) and applying Case 1, we see that each term on the right hand side is in E(Φ, s^{p}tR). This completes the proof.

The following result immediately follows from Lemma 3.1 by an easy induction on K. In its proof we denote by f^{∘K} the K-th iteration of the function f, namely f^{∘1} = f and f^{∘n} = f ∘ f^{∘n−1} where n ≥ 2.

Lemma 3.2. If p, n, k and K are given, there are q and l such that

E^{K}(Φ, \frac{1}{s^{k}} R)E(Φ, s^{q}t^{l}R) ⊆ E(Φ, s^{p}t^{n}R).

Proof. Consider the function f(x) = 3(i_{Φ}k +x) which appeared in the proof of Lemma 3.1. Clearly l = 3^{K}n and q = f^{∘K}(p) satisfy the desired inclusion.

4. Second localisation

Here we fix two elements s, t ∈ R and consider localisation R_{st} ≡ (R_{s})_{t} ≡ (R_{t})_{s}.

Lemma 4.1. Let p, q, k, m are given. Then there are l and n such that

[E^{1}(Φ, \frac{t^{l}}{s^{k}} R), E^{1}(Φ, \frac{s^{n}}{t^{m}} R)] ⊆ E(Φ, s^{p}t^{q}R).

Proof. The proof follows the same pattern as in Lemma 3.1. Let x_{α}(\frac{t^{l}a}{s^{k}}) ∈

E^{1}(Φ, \frac{t^{l}}{s^{k}} R) and x_{β}(\frac{s^{n}b}{t^{m}}) ∈ E^{1}(Φ, \frac{s^{n}}{t^{m}} R).

Case 1. Let α ≠ −β. Then

[x_{α}(\frac{t^{l}}{s^{k}} a), x_{β}(\frac{s^{n}}{t^{m}} b)] = ∏_{i,j>0} x_{iα+jβ}((\frac{t^{l}}{s^{k}} a)^{i}(\frac{s^{n}}{t^{m}} b)^{j}).

Let l ≥ i_{Φ}m + q and n ≥ i_{Φ}k + p. Clearly all factors on the right hand side of the above formula are in E(Φ, s^{p}t^{q}R). Case 2. Let α = −β and one of the following holds: β is short or Φ ≠ C_{l}. By Lemma 2.1 there are roots γ and δ such that γ + δ = β and N_{γδ11} = 1. Increasing n, if necessary, we can assume that n is even. Thus we can

decompose x_{β}(\frac{s^{n}}{t^{m}} b) as follows

x_{β}(\frac{s^{n}b}{t^{m}}) = [x_{γ}(s^{n/2}), x_{δ}(\frac{s^{n/2}b}{t^{m}})] ∏ x_{iγ+jδ}( − N_{γδij}(s^{n/2})^{i}(\frac{s^{n/2}}{t^{m}} b)^{j}),

where the product on the right hand side is taken over all roots iγ + jδ ≠ β. Next, we consider the commutator formula

[x, [y, z] ∏_{i=1}^{t} u_{i}] = [x, y] ^{y}[x, z] ^{yz}[x, y^{−1}] ^{yzy^{−1}}[x, z^{−1}] ∏_{i=1}^{t}^{[y,z] ∏_{j=1}^{i−1} u_{j}}[x, u_{i}],

and plug into this formula x_{α}(\frac{t^{l}}{s^{k}} a) instead of x, x_{γ}(s^{n/2}) instead of y,

x_{δ}(\frac{s^{n/2}b}{t^{m}}) instead of z and the remaining factors on the right hand side of

the expression of x_{β}(\frac{s^{n}b}{t^{m}}) instead of u_{i}. There are not more than 4 factors in the right hand side of the Chevalley commutator formula anyway, one of them is discarded from the very start, and in the conjugation we discard at least one more. This means that the maximum length K of the exponent in the elementary unipotents is at most 6. Now Lemma 3.2 and Case 1 imply that l ≥ f^{∘6}(q) + mi_{Φ}^{2} and n ≥ 2(ki_{Φ} + 3^{6}p) satisfy the required condition.

Case 3. Let Φ = C_{l} and α = −β be a long root. By Lemma 2.1 there exist roots γ and δ such that either γ + 2δ = β and N_{γδ12} = 1, or 2γ + δ = β and N_{γδ21} = 1. As in the proof of Lemma 3.1 we lose nothing by looking at the first case. Increasing n if necessary we can assume that n is divisible by

3 and decompose x_{β}(\frac{s^{n}b}{t^{m}}) as follows:

x_{β}(\frac{s^{n}b}{t^{m}}) = [x_{γ}(\frac{s^{n/3}b}{t^{m}}), x_{δ}(s^{n/3})]x_{γ+δ}( − N_{γδ11} \frac{s^{2n/3}b}{t^{m}}).

Repeating the same arguments as in Case 2, and observing that now the maximum length K of the exponent in the elementary unipotents is at most 4, we see that Lemma 3.2 and Case 1 imply that l ≥ f^{∘4}(q) + mi_{Φ} and n ≥ 3(ki_{Φ} + 3^{4}p) satisfy the required condition. Now comparing Case 1, Case 2 and Case 3, it is clear that a bound, for example, l ≥ f^{∘6}(q) + mi_{Φ}^{2} and n ≥ 3(ki_{Φ} + 3^{6}p) satisfy the lemma.

Combining Lemma 4.1 and commutator formulae, we get the main result of this section.

Theorem 4.2. Let p, q, k, m and K, L are given. Then there are l and n such that

[E^{K}(Φ, \frac{t^{l}}{s^{k}} R), E^{L}(Φ, \frac{s^{n}}{t^{m}} R)] ⊆ E(Φ, s^{p}t^{q}R).

Proof. The proof follows from Lemma 4.1 by an easy induction.

5. Patching

Fix an element s ∈ R, s ≠ 0. In general if R has zero divisors, the group homomorphism F_{s} : G(Φ, R) ⟶ G(Φ, R_{s}) induced by the localisation homomorphism R ⟶ R_{s} is not injective. There are several methods to circumvent this difficulty. Our approach is based on the following observation, [3], Lemma 4.10.

Lemma 5.1. Suppose R is Noetherian and s ∈ R. Then there exists a natural number k such that the homomorphism F_{s} : G(Φ, R, s^{k}R) ⟶ G(Φ, R_{s}) is injective.

Proof. The homomorphism F_{s} : G(Φ, R, s^{k}R) ⟶ G(Φ, R_{s}) is injective whenever F_{s} : s^{k}R ⟶ R_{s} is injective. For i ≥ 0, let 𝔞_{i} = Ann_{R}(s^{i}) be the annihilator of s^{i} in R. Since R is Noetherian, there exists a k such that 𝔞_{k} = 𝔞_{k+1} = . . .. If s^{k}a vanishes in R_{s}, then s^{i}s^{k}a = 0 for some i. But since 𝔞_{k+i} = 𝔞_{k}, already s^{k}a = 0 and thus s^{k}R injects in R_{s}.

Now we are all set to start a localisation and patching procedure. As in Section 3 a fraction of the form \frac{a}{s^{k}} is considered as an element of the localisation R_{s}, unless specified otherwise.

Lemma 5.2. Fix an element s ∈ R, s ≠ 0. Then for any k and q, there exists an r such that for any a ∈ R, g ∈ G(Φ, R, s^{r}R) and any maximal ideal M of R, there exist an element t ∈ R \ M, and an l such that

[x_{α}(\frac{t^{l}a}{s^{k}}), F_{s}(g)] ∈ E(Φ, F_{s}(s^{q}R)) ⊆ G(Φ, R_{s}).

Proof. By 2.3 one has G(Φ, R) = \underrightarrow{lim} G(Φ, R_{i}), where the limit is taken over all finitely generated subrings of R. Thus without loss of generality one may assume that R is Noetherian (replace R by the ring generated by a, s and the matrix entries of g in a faithful polynomial representation).

Let M be a maximal ideal of R. Then R_{M} is a local ring and thus by Lemma 2.10 F_{M}(g) ∈ G(Φ, R_{M}) can be decomposed as F_{M}(g) = uh where h is an element of T (Φ, R_{M}, s^{r}R_{M}), and u ∈ E(Φ, s^{r}R_{M}) ≤ G(Φ, R_{M}). But since G(Φ, R_{M}) = \underrightarrow{lim} G(Φ, R_{t}), over all t ∈ R \ M, and the same holds for E(Φ, s^{r}R_{M}), T (Φ, R_{M}, s^{r}R_{M}), etc., we can find an element t ∈ R \ M such that already F_{t}(g) can be factored as F_{t}(g) = uh where h ∈ T (Φ, R_{t}, s^{r}R_{t}) and u ∈ E(Φ, R_{t}, s^{r}R_{t}).

Now since R is assumed to be Noetherian, R_{s} is also Noetherian and by Lemma 5.1 there exists an n such that the canonical homomorphism

F_{t} : G(Φ, R_{s}, t^{n}R_{s}) ⟶ G(Φ, R_{st})

is injective. Let l > n. Since x_{α}(\frac{t^{l}a}{s^{k}}) ∈ G(Φ, R_{s}, t^{n}R_{s}), and G(Φ, R_{s}, t^{n}R_{s})

is normal in G(Φ, R_{s}), we have

x = [x_{α}(\frac{t^{l}a}{s^{k}}), F_{s}(g)] ∈ G(Φ, R_{s}, t^{n}R_{s}).

Consider the image F_{t}(x) ∈ G(Φ, R_{st}) of x under localisation with respect

to t. Since F_{t} is a homomorphism, one has F_{t}(x) = [F_{t}(x_{α}(\frac{t^{l}a}{s^{k}})), F_{st}(g)]. Now F_{st}(g) can be factored as F_{st}(g) = F_{s}(u)F_{s}(h) ∈ G(Φ, R_{st}). It follows that

F_{t}(x) = [F_{t}(x_{α}(\frac{t^{l}a}{s^{k}})), F_{s}(u)F_{s}(h)] =

[F_{t}(x_{α}(\frac{t^{l}a}{s^{k}})), F_{s}(u)] ^{F_{s}(u)}[F_{s}(x_{α}(\frac{t^{l}a}{s^{k}})), F_{s}(h)].

For all cases apart from the case of a long root α in the adjoint symplectic group of type C_{l} one could choose a decomposition F_{t}(g) = uh such that h commutes with x_{α}(∗). Therefore

F_{t}(x) = [F_{t}(x_{α}(\frac{t^{l}a}{s^{k}})), F_{s}(u)]

Now by Theorem 4.2, we can choose a suitable r and l such that F_{t}(x) ∈ E(Φ, F_{st}(s^{q}t^{n}R)) ≤ G(Φ, R_{st}).

We are left with the case of adjoint symplectic...

This means that F_{t}(x) can be presented as a product of elementary transvections of the form

F_{t}(x) = x_{α_{1}}(F_{st}(s^{q}t^{n}a_{1})) … x_{α_{m}}(F_{st}(s^{q}t^{n}a_{m}))

for some a_{1}, …, a_{m} ∈ R. Form the product of elementary root unipotents

y = x_{α_{1}}(F_{s}(s^{q}t^{n}a_{1})) … x_{α_{m}}(F_{s}(s^{q}t^{n}a_{m})) ∈ E(Φ, F_{s}(s^{q}R)) ∩ G(Φ, R_{s}, t^{n}R_{s}).

Clearly F_{t}(y) = F_{t}(x) and since both x and y belong to G(Φ, R_{s}, t^{n}R_{s}) and by the very choice of n the restriction of F_{t} to G(Φ, R_{s}, t^{n}R_{s}) is injective, it follows that x = y ∈ E(Φ, s^{q}R). This completes the proof.

Remark. For all cases apart from the case of a long root α in the adjoint symplectic group of type C_{l} one could choose a decomposition F_{t}(g) = hu such that h commutes with x_{α}(∗).

The following result is a broad generalisation of the normality of the elementary subgroup.

Theorem 5.3. Fix an element s ∈ R, s ≠ 0. Then for any p , K and k there exists an r such that

[E^{K}(Φ, \frac{1}{s^{k}} R), F_{s}(G(Φ, R, s^{r}R))] ⊆ E(Φ, F_{s}(s^{p}R)) ≤ G(Φ, R_{s}).

Proof. We shall show that there is an r such that

[E^{1}(Φ, \frac{1}{s^{k}} R), F_{s}(G(Φ, R, s^{r}R))] ⊆ E(Φ, F_{s}(s^{q}R)),

where q = f^{∘K−1}(p) and f(x) = 3(i_{Φ}k + x) as in Lemma 3.2. Then by the commutator formulae in 2.11,

[E^{K}(Φ, \frac{1}{s^{k}} R), F_{s}(G(Φ, R, s^{r}R))] ⊆ E^{K−1}(Φ, \frac{1}{s^{k}} R)E(Φ, F_{s}(s^{q}R)),

and by Lemma 3.2

E^{K−1}(Φ, \frac{1}{s^{k}} R)E(Φ, F_{s}(s^{q}R)) ⊆ E(Φ, F_{s}(s^{p}R)),

which proves the theorem.

Therefore let x_{α}(\frac{a}{s^{k}} ) ∈ E^{1}(Φ, \frac{1}{s^{k}} R) and F_{s}(g) ∈ F_{s}(G(Φ, R, s^{r}R)). By

Lemma 5.2, for k and 3(i_{Φ}k + q), there is a r such that for every maximal ideal M ∈ Max(R) there exists an element t_{M} ∈ A\M and a natural number l_{M} such that

[x_{α}(\frac{t_{M}^{l_{M}} a}{s^{k}}), F_{s}(g)] ∈ E(Φ, F_{s}(s^{3(i_{Φ}k+q)}R)).(5.3.1)

Since the set {t_{M}^{l_{M}} |M ∈ Max(R)} is not contained in any maximal ideal of R, there exists its finite set {t_{1}^{l_{1}}, ⋯, t_{r}^{l_{r}}} which generates R as an ideal. Choose x_{1}, …, x_{r} ∈ R such that x_{1}t_{1}^{l_{1}} + … + x_{r}t_{r}^{l_{r}} = 1. Then

[x_{α}(\frac{a}{s^{k}} ), F_{s}(g)] = [x_{α}(\frac{x_{1}t_{1}^{l_{1}}a}{s^{k}}) … x_{α}(\frac{x_{r}t_{r}^{l_{r}}a}{s^{k}}), F_{s}(g)].

Using the commutator formula in 2.11, Lemma 3.1 and (5.3.1) we see that

[x_{α}(\frac{a}{s^{k}} ), F_{s}(g)] ∈ E(Φ, F_{s}(s^{q}R)),

which concludes the proof.

In particular the contents of the present section gives a slightly shorter proof of Taddei’s result [20]. In fact only a small fraction of our arguments would be necessary to prove this result:

Corollary 5.4. Assume rk(Φ) ≥ 2. Then E(Φ, R) is a normal subgroup of G(Φ, R).

Proof. Set s = 1 in the above theorem.

6. Completion

In the present section we describe the last important ingredient of the proof of the main theorem. Let s ∈ R. Recall that the s-completion R̂_{s} of the ring R is usually defined as the following inverse limit:

R̂_{s} = \underleftarrow{lim} R/s^{n}R, n ∈ ℕ.

However this definition is not quite compatible with our purposes. Namely, as always, to control zero divisors, we have to reduce to Noetherian rings first. However if R = \underrightarrow{lim} R_{i} is a direct limit of Noetherian rings, the canonical homomorphism \underrightarrow{lim}(R̂_{i})_{s} ⟶ R̂_{s} is in general neither surjective, nor injective. This forces us to modify the definition of completion as follows:

R̃_{s} = \underrightarrow{lim}(R̂_{i})_{s},

where the limit is taken over all finitely generated subgrings R_{i} of R which contain s. Let us denote by F̃_{s} the canonical homomorphism R ⟶ R̃_{s}. For the case, when R is Noetherian F̃_{s} = F̂_{s} coincides with the inverse limit of reduction homomorphisms π_{s^{n}} : R ⟶ R/s^{n}R

Theorem 6.1. Let R be a commutative ring, Φ an irreducible root system of rank ≥ 2. Then

[F_{s}^{−1}(E(Φ, R_{s})), F̃_{s}^{−1}(E(Φ, R̃_{s})] ⊆ E(Φ, R).

Proof. Let R_{i} be the inductive system of all finitely generated subrings of R, containing s. By 2.3 one has

F_{s}^{−1}(E(Φ, R_{s})) = \underrightarrow{lim} F_{s}^{−1}(E(Φ, (R_{i})_{s})),
F̃_{s}^{−1}(E(Φ, R̃_{s})) = \underrightarrow{lim} F̂_{s}^{−1}(E(Φ, (R̂_{i})_{s})),

and the proof reduces to the case when R is Noetherian, as in Lemma 5.2.

Let x ∈ F_{s}^{−1}(E(Φ, R_{s})) and y ∈ F̂_{s}^{−1}(E(Φ, (R̂_{i})_{s})). By definition the condition on x means that F_{s}(x) ∈ E^{K}(Φ, 1/s^{k}A) for some k and K. On the other hand the condition on y means that π_{s^{n}}(y) ∈ E(Φ, R/s^{n}R) for all n, or, what is the same, y ∈ E(Φ, R)G(Φ, R, s^{n}R). In other words, for any n we can present y as a product y = uz, u ∈ E(Φ, R) and z ∈ G(Φ, R, s^{n}R).

As in the proof of Lemma 5.2 we can choose p such that the restriction of the localisation homomorphism F_{s} to the principal congruence subgroup G(Φ, R, s^{p}R) is injective. Now for k, K and p choose q as in Theorem 5.3. Now [x, y] = [x, u]^{u}[x, z]. The first commutator belongs to E(Φ, R) together with u since E(Φ, R) is normal. Thus it remains only to prove that [x, z] ∈ E(Φ, R). By Theorem 5.3 the F_{s}([x, z]) ∈ E(Φ, F_{s}(s^{q})). On the other hand, since G(Φ, R, s^{q}R) is normal, [x, z] ∈ G(Φ, R, s^{q}R) exactly as the proof of Lemma 5.2 we can conclude that [x, z] ∈ E(Φ, s^{q}R).

7. Main theorem

To introduce the main new concept of the paper, we have to recall the notion of Bass-Serre dimension of a ring. Let X be a topological space. The dimension of X is the length n of the longest chain X_{0} ⫋ X_{1} ⫋ … ⫋ X_{n} of nonempty closed irreducible subsets X_{i} of X, ([6], § III). Define δ(X) to be the smallest nonnegative integer d such that X is a finite union of irreducible Noetherian subspaces of dimension ≤ d. If there is no such d, then by definition δ(X) = ∞. Let R be a commutative ring. Let Max(R) be the maximal spectrum of the ring R, endowed with Zariski topology. Then the Bass-Serre dimension of R is δ(R) = δ(Max(R)). It is easy to see that δ(R) = 0 if and only if R is a semi-local ring.

The following ‘induction lemma’ (see [3], Lemma 4.17) is the main instrument to conduct induction on dimension.

Lemma 7.1. Suppose δ(R) is finite and Max(R) = X_{1} ∪ … ∪ X_{r} be a decomposition into a union of irreducible Noetherian subspaces of dimension ≤ δ(R). If s ∈ R is such that for each k, 1 ≤ k ≤ r, the element s is not contained in some member of X_{k}, then δ(R̃_{s}) < δ(R).

Now we are all set to state and prove our principal result.

Definition 7.2. Let R be a commutative ring, Φ an irreducible root system of rank ≥ 2. Define

S^{d}G(Φ, R) = ⋂_{{R→A \atop δ(A)≤d}} Ker(G(Φ, R) ⟶ G(Φ, A)/E(Φ, A)).

Theorem 7.3. Let R be a commutative ring, Φ an irreducible root system of rank ≥ 2 and G(Φ, R) the Chevalley group of Φ with coefficients in R. Then G(Φ, R)/S^{0}G(Φ, R) is abelian, the sequence

S^{0}G(Φ, R) ≥ S^{1}G(Φ, R) ≥ S^{2}G(Φ, R) ≥ ⋯

is a descending central series in S^{0}G(Φ, R) and S^{d}G(Φ, R) = E(Φ, R) whenever δ(R) = d.

Proof. By 2.9, if A is a semi-local ring, then G(Φ, A)/E(Φ, A) is abelian. Since δ(A) = 0 if and only if A is semi-local, one sees that the following homomorphism is injective

G(Φ, R)/S^{0}G(Φ, R) ⟶ ∏_{δ(A)=0} G(Φ, A)/E(Φ, A).

Thus it follows that G(Φ, R)/S^{0}G(Φ, R) is an abelian group.

For the main part of the theorem, we proceed by induction on δ(R). The theorem holds for zero dimensional rings. It suffices to show that for any x ∈ S^{0}G(Φ, R) and y ∈ S^{d}G(Φ, R), the commutator [x, y] ∈ S^{d+1}G(Φ, R). Since

G(Φ, R)/S^{d+1}G(Φ, R) ⟶ ∏_{{R→A \atop δ(A)≤n+1}} G(Φ, A)/E(Φ, A)

is a monomorphism, it is enough to prove the theorem for rings of dimension d + 1. Hence S^{d+1}G(Φ, R) = E(Φ, R).

Let X_{1}∪…∪X_{r} be a decomposition of Max(R) into irreducible Noetherian subspaces of dimension ≤ δ(R). For any 1 ≤ i ≤ r, let M_{i} ∈ X_{i}. Take the multiplicative set S = R \ (M_{1} ∪ ⋯ ∪ M_{r}). Since S^{−1}R is a semi-local ring, δ(\underrightarrow{lim} R_{s}) = δ(S^{−1}R) = 0, where the limit is taken over all s ∈ S. Therefore there exists an element s ∈ S such that the F_{s}(x) ∈ E(Φ, R_{s}). Thus x ∈ F_{s}^{−1}(E(Φ, R_{s})). On the other hand by Lemma 7.1 for any s ∈ S, δ(R̃_{s}) < δ(R). Thus F̃_{s}(y) ∈ E(Φ, R̃_{s}). Now by Theorem 6.1 one has [x, y] ∈ E(Φ, R). This shows that S^{0}G(Φ, R) ≥ S^{1}G(Φ, R) ≥ ⋯ is a descending central series. The fact that S^{d}G(Φ, R) = E(Φ, R) whenever δ(R) = d is immediate from the definition of S^{d}G(Φ, R).

Corollary 7.4. Let rk(Φ) ≥ 2 and R be a finite-dimensional ring. Then the quotient G(Φ, R)/E(Φ, R) is nilpotent-by-abelian. In particular it is solvable.

Proof. The corollary is an immediate consequence of Theorem 7.3.

Corollary 7.5. Let rk(Φ) ≥ 2 and R be a finite-dimensional ring. Then K_{1}(Φ, R) is nilpotent.

Proof. Since G is simply connected, for any semi-local ring R, G(Φ, R) = E(Φ, R) and thus G(Φ, R) = S^{0}G(Φ, R). Now the corollary follows from Theorem 7.3.

References

[1] E. Abe, Chevalley groups over local rings, Tôhoku Math. J. 21 (1969), 474–494.

[2] E. Abe, K. Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. 28 (1), (1976), 185–198.

[3] A. Bak, Non-abelian K-Theory: The nilpotent class of K_{1} and general stability, K-Theory 4 (1991), 363–397.

[4] A. Bak, N. Vavilov, Normality for elementary subgroup functors, Math. Proc. Cambridge Philos. Soc. 118 (1), (1995), 35–47.

[5] A. Bak, N. Vavilov, Structure of hyperbolic Unitary groups I: Elementary subgroups, Algebra Coll. 7 (1), (2000), 159–196.

[6] H. Bass, Algebraic K-Theory, Benjamin, New York, 1968.

[7] R. Hazrat, Dimension theory and non-stable K_{1} of quadratic module, To appear in K-Theory.

[8] A. J. Hahn, O. T. O’Meara, The Classical Groups and K-Theory, Springer-Verlag, Berlin, 1989.

[9] W. van der Kallen, A module structure on certain orbit sets of unimodular rows, J. Pure Appl. Algebra, 57 (3) (1989), 281–316.

[10] S. Khlebutin, Elementary subgroups of linear groups over rings, Ph. D. Thesis, Moscow State Univ., 1987 (in Russian).

[11] V. Kopeiko, The stabilisation of symplectic groups over a polynomial ring, Math. USSR Sbornik, 34 (1978), 655–669.

[12] H. Park, C. Woodburn, An algorithmic proof of Suslin’s stability theorem for polynomial rings, J. Algebra 178 (1) (1995), 277–298.

[13] E. Plotkin, On the stability of the K_{1}-functor for Chevalley groups of type E_{7}, J. Algebra, 210 (1), (1998), 67–95.

[14] A. Sivatsky, A. Stepanov, On the word length of commutators in GL_{n}(R), K-Theory, 17, (1999), 295–302.

[15] M. R. Stein, Stability theorems for K_{1}, K_{2} and related functors modeled on Chevalley groups, Japan. J. Math., 4 (1), (1978), 77–108.

[16] A. Stepanov, N. Vavilov, Decomposition of transvections: a theme with variations, K-Theory, 19, (2000), 109–153.

[17] A. Suslin, On the structure of the general linear group over polynomial rings, Soviet Math. Izv., 41 (2), (1977), 503–516.

[18] A. Suslin, V. Kopeiko, Quadratic modules and orthogonal groups over polynomial rings, J. Sov. Math., 20 (6), (1985), 2665–2691.

[19] K. Suzuki, Normality of the elementary subgroups of twisted Chevalley groups over commutative rings, J. Algebra 175 (2), (1995), 526–536.

[20] G. Taddei, Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau, Contemp. Math., 55 (II), (1986), 693–710.

[21] J. Tits, Systèmes générateurs de groupes de congruence, C. R. Acad. Sci. Paris, Sér A, 283, (1976), 693–695.

[22] M. Tulenbaev, Schur multiplier of the group of elementary matrices of finite order, J. Sov. Math., 17 (4), (1981), 2062–2067.

[23] L. Vaserstein, On normal subgroups of GL_{n} over a ring, in Lecture Notes Math., Vol. 854, Springer, 1981, pp.456–465.

[24] L. Vaserstein, Unstable K_{1}-theory of topological spaces is nilpotent, Contemp. Math., 126 (1992), 193–196.

[25] N. Vavilov, Structure of Chevalley groups over commutative rings, Non-associative algebras and related topics (Hiroshima, 1990 ), World Sci. Publishing, London et al., 1991, 219–335.

[26] N. Vavilov, A third look at weight diagrams, Rend. Sem. Math. Univ. Padova, 104 (2), (2000), 1–50.

[27] N. Vavilov, E. Plotkin, Chevalley groups over commutative rings I: Elementary calculations, Acta Applic. Math., 45, (1996), 73–113.

Department of Mathematics, University of Bielefeld, P. O. Box 100131, 33501 Bielefeld, Germany

   E-mail address: rhazrat@mathematik.uni-bielefeld.de

Department of Mathematics and Mechanics, Saint-Petersburg State University, Saint-Petersburg 198904, Russia and Department of Mathematics, Northwestern University, Evanston, IL 60208, USA

   E-mail address: vavilov@math.nwu.edu