Page 1

GENERALIZED COMMUTATOR FORMULAS

R. HAZRAT AND Z. ZHANG

Abstract. Let A be an algebra which is a direct limit of module finite algebras over a commutative ring R

with 1. Let I,J be two-sided ideals of A, GLn(A,I) the principal congruence subgroup of level I in GLn(A),

and let En(A,I) be the relative elementary subgroup of level I. Using Bak’s localization-patching method,

we prove the following commutator formula

[En(A,I),GLn(A,J)] = [En(A,I),En(A,J)],

which is a generalization of the standard commutator formular. This answers a problem posed by A. Stepanov

and N. Vavilov.

Introduction

Let A be an associative ring with 1, GLn(A) the general linear group of degree n over A, and let En(A)

be its elementary subgroup. For a two-sided ideal I of A, we denote the principal congruence subgroup of

level I by GLn(A,I) and the relative elementary subgroup of level I by En(A,I) (see 1.3). The following

two well known formulas were first obtained by H. Bass [7] in the stable level (i.e., the union of GLnand

Enfor n ≥ 1),

[GLn(A),En(A,I)] = En(A,I),

which was the key to define stable K1and relative stable K1groups.

Later, A. Suslin, L. Vaserstein, Z. Borevich, and N. Vavilov [20, 21, 24] proved that, in the case

of commutative rings, these formulas are valid for n ≥ 3, thus paved the way to define non-stable K1

groups. At the same time, V. Gerasimov [8] gave counter-examples of rings in which En(R) is nontivially

distinguished as a free factor in GLn(R). In this case, En(R) is not normal in GLn(R), and thus the above

commutator formulas fail to be valid in general.

A natural generalization of these formulas was first considered by A. Mason [16, 17, 18]. For two

ideals I and J of A, one would like to establish a relation between mutual commutator subgroups of the

congruence subgroups and the elementary subgroups of level I and J, respectively. Recently, A. Stepanov

and N. Vavilov [23] obtained the following theorem by giving a slick proof using the decomposition of

unipotents:

[En(A),GLn(A,I)] = En(A,I),

Theorem 1. Let R be a commutative ring and n ≥ 3. Then any two ideals I and J of R satisfy the

equality

[En(R,I),GLn(R,J)] = [En(R,I),En(R,J)].

(1)

The method of their proof would not extend to classical groups, specially, exceptional groups. For this

reason, they ask in [23, Problem 2], to find a localization proof of this commutator formula.

In this paper, using a powerful localization-patching method employed by Bak in [2], we give a local-

ization proof of the commutator formula (1) for the quasi-finite algebras, which are defined as a direct

limit of module finite algebras, and therefore as a special case, we obtain the case of commutative rings,

i.e., Theorem 1. To do this, we need to modify conjugation calculus which is used in the literature in

order to establish our result. The localization approach will pave the way to use the same techniques to

establish the generalized commutator formulas in more complex settings, such as general quadratic groups

The first author acknowledges the support of EPSRC first grant scheme EP/D03695X/1. The second author acknowledges

the support of NSFC (Grant 10971011). The authors thank Nikolai Vavilov for suggesting the topic of the paper to them.

1

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2R. HAZRAT AND Z. ZHANG

and Chevalley groups. These shall be established in a sequel to this paper. For the recent work on relative

structure of classical groups see [3, 13, 14, 15, 20].

1. Preliminaries

In this section, we fix some notations. At same time, we list some preliminary results concerning the

localization-patching method without proofs. We refer to Bak’s original paper [2] or Hazrat and Vavilov’s

technically simplified version [14] for details.

1.1. Let R be a commutative ring with 1, S a multiplicative closed system in R and A an R-algebra. Then

S−1R and S−1A denote the corresponding localization. In the current paper, we mostly use localization

with respect to the following two types of multiplicative systems.

1.) For any s ∈ R, the mulitiplicative system generated by s is defined as

?s? = {1,s,s2,...}.

The localization with respect to mulitplicative system ?s? is usually denoted by Rsand As. Note that, for

any α ∈ As, there exists an integer n and an element a ∈ A such that α = a/sn.

2.) If m is a maximal ideal of R, and S = R\m a multiplicative system, then we denote the localization

with respect to S by Rmand Am.

For a multiplicative system S, the canonical localization map with respect to S is denoted by θS: R →

S−1R. For the special cases mentioned above, we write θs: R → Rsand θM: R → RM, respectively.

1.2. An R-algebra A is called module finite over R, if A is finitely generated as an R-module. An R-algebra

A is called quasi-finite over R if there is a direct system of module finite R-subalgebras Aiof A such that

lim

− →Ai= A.

Proposition 2. An R-algebra A is quasi-finite over R if and only if it satisfies the following equivalent

conditions:

(1) There is a direct system of subalgebras Ai/Riof A such that each Aiis module finite over Riand

such that lim

− →Ri= R and lim

(2) There is a direct system of subalgebras Ai/Riof A such that each Aiis module finite over Riand

each Riis finitely generated as a Z-algebra and such that lim

− →Ai= A.

− →Ri= R and lim

− →Ai= A.

1.3. For any associative ring A, GLn(A) denotes the general linear group of A, and En(A) denotes the

elementary subgroup of GLn(A). Let I be any two-sided ideal of A. If ρI denotes the natural ring

homomorphism A → A/I, then ρI induces a group homomorphism, denoted also by ρI, ρI: GLn(A) →

GLn(A/I). The congruence subgroup of level I is defined as GLn(A,I) = ker(ρI: GLn(A) → GLn(A/I)).

The elementary subgroup of level I is, by definition, the subgroup generated by all elementary matrices

ei,j(α) with α ∈ I. The normal closure of En(I) in En(A), the relative elementary subgroup of level I,

is denoted by En(A,I). We use EK

product K elementary matrices. EK

We have the following relations among elementary matrices which will be used in the paper:

n(I) to denote the subset of En(I), which can be represented as the

n(I) is clearly not a group.

(E1) ei,j(a)ei,j(b) = ei,j(a + b).

(E2) [ei,j(a),ek,l(b)] = 1 if i ?= l,j ?= k.

(E3) [ei,j(a),ej,k(b)] = ei,k(ab) if i ?= k.

1.4. GLn and En define two functors from the category of associative rings to the category of groups.

These functors commute with direct limits. In another words, let Aibe an inductive system of rings, and

A = lim

− →Ai. Then

GLn(A) = GLn(lim

− →Ai)∼= lim

− →GLn(Ai)and

En(lim

− →Ai)∼= lim

− →En(Ai).

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GENERALIZED COMMUTATOR FORMULAS3

Also, if J is an ideal of A, then there are ideals Jiof Aisuch that J = lim

− →Jiand

GLn(A,J) = GLn(lim

− →Ai,lim

− →Ji)∼= lim

− →GLn(Ai,Ji).

By Proposition 2 and the above observation, we may reduce some of our problems to the case of the

module finite algebras over the Noetherian rings.

If S be a multiplicative system in R, Rswith s ∈ S is a inductive system with respect to the localization

map : θt: Rs→ Rst. If F is a functor commuting with direct limits, then

F(S−1R) = lim

This allows us to reduce our problems in any localization to the localization in one element. Starting from

Section 2, we will be working in the ring At. However, eventually we need to return to the ring A. The

following Lemma provide a way to “pull back” elements from GLn(At) to GLn(A).

− →F(Rs).

Lemma 3. Let R be a commutative Noetherian ring and let A be a module finite R-algebra. Then for any

t ∈ R, there exists a positive integer l such that the homomorphism

θt: GLn(A,tlA) → GLn(At)

is injective.

Proof. See [2, Lemma 4.10] or [14, Lemma 5.1].

?

1.5. Let G be a group. For any x,y ∈ G,xy = xyx−1denotes the left x-conjugate of y. Let [x,y] =

xyx−1y−1denote the commutator of x and y. The following formulas will be used frequently,

(C1) [x,yz] = [x,y](y[x,z]);

(C2) [xy,z] = (x[y,z])[x,z];

(C3)x?[x−1,y],z?

(C4) [x,yz] =y[y−1x,z];

(C5) [yx,z] =y[x,y−1z].

=

x?[y,x−1]−1,z?

=

y?x,[y−1,z]?z?y,[z−1,x]?(the Hall-Witt identity);

2. Localization and patching

In this section we prove some technical results needed for employing the localization and patching

method. Throughout the section we assume n > 2 for any general linear group GLn. We start with the

following definition:

Definition 4. Let A be an R-algebra, I a two-sided ideal of A, t ∈ R, and l a positive integer. Define

En(tlA,tlI) to be a subgroup of En(A,tlI) generated by

eei,j(tlα)for all

α ∈ I,e ∈ En(tlA) and 1 ≤ i,j ≤ n.

Here by tlI, we are considering the image of t ∈ R in A under the algebra structure homomorphism. It is

clear that tlI is also a two-sided ideal of A.

For any element α ∈ A, we use En(tlA,tlα) to denote the subgroup generated by

eei,j(tlα)for all

e ∈ En(tlA) and i,j ≤ n.

From the definition, it is clear that En(tlA,tlI) is normalized by En(tlA). This will be used throughout

out the calculations. Also, by Lemma 3, both En(tlA,tlI) and En(tlA,tlα) are embedded in GLn(At) for

a sufficient large integer l. This fact will be used in Lemma 14.

Starting from Lemma 5 to Lemma 13, all the calculations take place in E(At), for t ∈ R. Thus, when

we write something like E1

n(tlα) or ei,j(tlα) what we mean is E1

n(θt(tlα)) or ei,j(θt(tlα)), respectively.

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4R. HAZRAT AND Z. ZHANG

Lemma 5. Suppose A is a module finite R-algebra. For any given t ∈ R and positive integers l,m, there

exists a sufficient large integer p such that

E1

n(A

tm)E1

n(tpα) ⊆ En(tlA,tl?α?).

Proof. Set p = 2q for some integer q. Suppose that

ρ =ei,j(a/tm)ei?,j?(t2qα) ∈E1

n(A

tm)E1

n(t2qα)where

a ∈ A.

We need to consider four cases:

• If i ?= j?and j ?= i?then by (E2), ρ = ei?,j?(t2qα).

• if i ?= j?, and j = i?, then by (E3), ρ = ei,j?(t2q−maα)ei?,j?(t2qα).

• if i = j?and j ?= i?then by (E3), ρ = ei?,j?(t2qα)ei?,j(−t2q−mαa).

In all the cases above, by choosing 2q ≥ l + m, we then have ρ ∈ En(tlA,tl?α?).

• if i = j?and j = i?, choose h ?= i,j. Then

ρ

=

=

=[ei,j(a/tm)ej,h(tq),ei,j(a/tm)eh,i(tqα)]

?

e1

ei,j(a/tm)ej,i(t2qα)

ei,j(a/tm)[ej,h(tq),eh,i(tqα)]

=

ei,h(tq−ma)ej,h(tq)

?

???

,eh,i(tqα)eh,j(−tq−mαa)

????

e2

?

,

where e2, by definition, belongs to En(tq−m?α?), and e1∈ En(tq−mA). Hence,

[e1,e2] ∈ En(tq−mA,tq−m?α?).

Now in all the cases, choosing q ≥ l + m guarantees that ρ ∈ En(tlA,tl?α?).

?

The following lemma is a direct consequence of Lemma 5 by using the equationc(ab) = (ca)(cb).

Lemma 6. Suppose A is a module finite R-algebra. For any given t ∈ R and positive integers l,m,K,

there exists a sufficient large integer p such that

E1

n(A

tm)EK

n(tpα) ⊆ En(tlA,tl?α?).

Lemma 7. Suppose A is a module finite R-algebra. For any given t ∈ R and positive integers l, there

exists a sufficient large integer p such that

E1

n(A

tm)En(tpA,tpα) ⊆ En(tlA,tl?α?).

Proof. By definition, En(tpA,tpI) is generated by all elements of the form

eei,j(tpα)e−1

where e ∈ En(tpA),α ∈ I.

Therefore, we have the following typical element

ei?,j?(a

tm)(eei,j(tpα)e−1) ∈E1

n(A

tm)En(tpA,tpα).

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GENERALIZED COMMUTATOR FORMULAS5

Denote ei?,j?(a

tm) by x. Then

ei?,j?(a

(xe)(xei,j(tpα))(xe−1)

tm)En(tpA)?

by Lemma 6 and Lemma 5 one can choose a suitable p such that

En(tlA,tlA)En(tlA,tl?α?) = En(tlA,tl?α?),

as En(tlA,tl?α?) is normalized by En(tlA) = En(tlA,tlA). This finishes the proof.

The following lemma is an easy application of Lemma 7. The proof is left to reader.

tm)(eei,j(tpα)e−1)

=

∈

E1

n(A

E1

n(A

tm)E1

n(tpα)

?

⊆

?

Lemma 8. Suppose A is a module finite R-algebra, a,b,c ∈ A and t ∈ R. If m,l are given, there is an

integer p such that

E1

n(

c

tm)[En(tpA,tp?a?),En(tpA,tp?b?)] ⊆ [En(tlA,tl?a?),En(tlA,tl?b?)].

Lemma 9. Suppose A is a module finite R-algebra, a,b ∈ A, and t ∈ R. If m,l are given, there is an

integer p such that

n(b

tm)] ⊆ [En(tlA,tl?a?),En(tlA,tl?b?)].

(2)

[E1

n(tpa),E1

(3)

Proof. Let ei,j(tpa) ∈ E1

q. We need to consider several cases similar to Lemma 5:

n(tpa), ei?,j?(b

tm) ∈ E1

n(b

tm) and ρ = [ei,j(tpa),ei?,j?(b

tm)]. Set p = 2q for some integer

• If i ?= j?and j ?= i?then by (E2), ρ = 1.

• if i ?= j?, and j = i?, then by (E3), ρ = ei,j?(t2q−mab) which is in [En(tlA,tl?a?),En(tlA,tl?b?)] for

2q ≥ 2l + m by using (E3) again.

• if i = j?and j ?= i?then by (E3), ρ = [ei,j(t2qa),ei?,i(b

ρ = ei?,j(−t2q−mab) which is in [En(tlA,tl?a?),En(tlA,tl?b?)] for 2q ≥ 2l + m by using (E3) again.

• if i = j?and j = i?, then ρ = [ei,j(t2qa),ej,i(b

eh,j(−tq)ρ

eh,j(−tq)[ei,j(t2qa),ej,i(b

tm)]

eh,j(−tq)?

By the Hall-Witt identityx?[y,x−1]−1,z?

=

ei,h(−tqa)?

e1

But

e1=ei,h(−tqa)?

eh,j(−tq),[ej,i(−t?q−m

tm)] = [ei?,i(b

tm),ei,j(t2qa)]−1. Then by (E3)

tm)]. Choose an h ?= i,j and consider

=

=[ei,h(−tqa),eh,j(tq)]−1,ej,i(b

tm)

?

=

y?x,[z,y−1]−1?z?y,[x,z−1]−1?

ei,h(−tqa),[eh,j(−tq),ej,i(−b

ei,h(−tqa),eh,i(−tq−mb)

??

?

?a),ei,h(−tq−m−?q−m

ei,h(−tqa)?

eh,j(−tq),[ej,i(b

eh,j(−tq),ej,h(−tq−mab)

??

tm),ei,h(tqa)]−1?ej,i(

b

tm)?

tm)]−1?

=

?

??

ej,i(−

?

b

tm)??

?

e2

.

(4)

eh,j(−tq),ej,h(−tq−mab)

=ei,h(−tqa)?

22

?b)

?