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 GLn and En for n ≥ 1), [GLn (A), En (A, I)] = En (A, I),

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

which was the key to define stable K1 and relative stable K1 groups. 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: 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 localization 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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R. 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 −1 R and S −1 A 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 hsi = {1, s, s2 , . . .}. The localization with respect to mulitplicative system hsi is usually denoted by Rs and 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 Rm and Am . For a multiplicative system S, the canonical localization map with respect to S is denoted by θS : R → S −1 R. For the special cases mentioned above, we write θs : R → Rs and 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 Ai of 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 /Ri of A such that each Ai is module finite over Ri and such that lim Ri = R and lim Ai = A. −→ −→ (2) There is a direct system of subalgebras Ai /Ri of A such that each Ai is module finite over Ri and each Ri is finitely generated as a Z-algebra and such that lim 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 EnK (I) to denote the subset of En (I), which can be represented as the product K elementary matrices. EnK (I) is clearly not a group. We have the following relations among elementary matrices which will be used in the paper: (E1) ei,j (a)ei,j (b) = ei,j (a + b). (E2) [ei,j (a), ek,l (b)] = 1 if i 6= l, j 6= k. (E3) [ei,j (a), ej,k (b)] = ei,k (ab) if i 6= 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 Ai be an inductive system of rings, and A = lim Ai . Then −→ GLn (A) = GLn (lim Ai ) ∼ GLn (Ai ) and En (lim Ai ) ∼ E (A ). = lim = lim −→ −→ −→ −→ n i

GENERALIZED COMMUTATOR FORMULAS

3

Also, if J is an ideal of A, then there are ideals Ji of Ai such that J = lim Ji and −→ ∼ 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, Rs with 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 −1 R) = lim F (Rs ). −→ 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). 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, tl A) → 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, x y = xyx−1 denotes the left x-conjugate of y. Let [x, y] = xyx−1 y −1 denote 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 = x [y, x−1 ]−1 , z =

y

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

−1

(C4) [x,y z] =y [y x, z]; (C5) [y x, z] =y [x,y

−1

z]. 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 (tl A, tl I) to be a subgroup of En (A, tl I) generated by e

ei,j (tl α)

for all

α ∈ I, e ∈ En (tl A) and 1 ≤ i, j ≤ n.

Here by tl I, we are considering the image of t ∈ R in A under the algebra structure homomorphism. It is clear that tl I is also a two-sided ideal of A. For any element α ∈ A, we use En (tl A, tl α) to denote the subgroup generated by e

ei,j (tl α)

for all

e ∈ En (tl A) and i, j ≤ n.

From the definition, it is clear that En (tl A, tl I) is normalized by En (tl A). This will be used throughout out the calculations. Also, by Lemma 3, both En (tl A, tl I) and En (tl A, tl α) 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 En1 (tl α) or ei,j (tl α) what we mean is En1 (θt (tl α)) or ei,j (θt (tl α)), respectively.

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R. 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 1( A ) En tm

En1 (tp α) ⊆ En (tl A, tl hαi).

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

ρ =ei,j (a/t

1

A

ei0 ,j 0 (t2q α) ∈ En ( tm ) En1 (t2q α)

where a ∈ A.

We need to consider four cases: • If i 6= j 0 and j 6= i0 then by (E2), ρ = ei0 ,j 0 (t2q α). • if i 6= j 0 , and j = i0 , then by (E3), ρ = ei,j 0 (t2q−m aα)ei0 ,j 0 (t2q α). • if i = j 0 and j 6= i0 then by (E3), ρ = ei0 ,j 0 (t2q α)ei0 ,j (−t2q−m αa). In all the cases above, by choosing 2q ≥ l + m, we then have ρ ∈ En (tl A, tl hαi). • if i = j 0 and j = i0 , choose h 6= i, j. Then ρ =

ei,j (a/tm )

ej,i (t2q α)

=

ei,j (a/tm )

[ej,h (tq ), eh,i (tq α)]

(a/tm )

m

= [ei,j ej,h (tq ),ei,j (a/t ) eh,i (tq α)] i h = ei,h (tq−m a)ej,h (tq ), eh,i (tq α)eh,j (−tq−m αa) , {z } | {z } | e1

e2

where e2 , by definition, belongs to En (tq−m hαi), and e1 ∈ En (tq−m A). Hence, [e1 , e2 ] ∈ En (tq−m A, tq−m hαi). Now in all the cases, choosing q ≥ l + m guarantees that ρ ∈ En (tl A, tl hαi).



The following lemma is a direct consequence of Lemma 5 by using the equation c (ab) = (ca)(c b). 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 1( A ) En tm

EnK (tp α) ⊆ En (tl A, tl hαi).

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 1( A ) En tm

En (tp A, tp α) ⊆ En (tl A, tl hαi).

Proof. By definition, En (tp A, tp I) is generated by all elements of the form eei,j (tp α)e−1

where e ∈ En (tp A), α ∈ I.

Therefore, we have the following typical element a ei0 ,j 0 ( tm )

1

A

(eei,j (tp α)e−1 ) ∈ En ( tm ) En (tp A, tp α).

GENERALIZED COMMUTATOR FORMULAS

5

Denote ei0 ,j 0 ( tam ) by x. Then a ei0 ,j 0 ( tm )

(eei,j (tp α)e−1 )

= (x e)(x ei,j (tp α))(x e−1 )  1 A  1( A ) En tm En (tp A) En ( tm ) 1 p ∈ En (t α) by Lemma 6 and Lemma 5 one can choose a suitable p such that En (tl A,tl A)

⊆

En (tl A, tl hαi) = En (tl A, tl hαi),

as En (tl A, tl hαi) is normalized by En (tl A) = En (tl A, tl A). This finishes the proof.



The following lemma is an easy application of Lemma 7. The proof is left to reader. 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 1( c ) En tm

[En (tp A, tp hai), En (tp A, tp hbi)] ⊆ [En (tl A, tl hai), En (tl A, tl hbi)].

(2)

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 b [En1 (tp a), En1 ( m )] ⊆ [En (tl A, tl hai), En (tl A, tl hbi)]. (3) t b b b Proof. Let ei,j (tp a) ∈ En1 (tp a), ei0 ,j 0 ( tm ) ∈ En1 ( tm ) and ρ = [ei,j (tp a), ei0 ,j 0 ( tm )]. Set p = 2q for some integer q. We need to consider several cases similar to Lemma 5:

• If i 6= j 0 and j 6= i0 then by (E2), ρ = 1. • if i 6= j 0 , and j = i0 , then by (E3), ρ = ei,j 0 (t2q−m ab) which is in [En (tl A, tl hai), En (tl A, tl hbi)] for 2q ≥ 2l + m by using (E3) again. b b • if i = j 0 and j 6= i0 then by (E3), ρ = [ei,j (t2q a), ei0 ,i ( tm )] = [ei0 ,i ( tm ), ei,j (t2q a)]−1 . Then by (E3) 2q−m l l l l ρ = ei0 ,j (−t ab) which is in [En (t A, t hai), En (t A, t hbi)] for 2q ≥ 2l + m by using (E3) again. b • if i = j 0 and j = i0 , then ρ = [ei,j (t2q a), ej,i ( tm )]. Choose an h 6= i, j and consider eh,j (−tq )

ρ

b [ei,j (t2q a), ej,i ( m )] t h b i q = eh,j (−t ) [ei,h (−tq a), eh,j (tq )]−1 , ej,i ( m ) t       By the Hall-Witt identity x [y, x−1 ]−1 , z = y x, [z, y −1 ]−1 z y, [x, z −1 ]−1 ie ( b ) h i h b b q −1 j,i tm ei,h (−tq a) q = ei,h (−tq a), [eh,j (−tq ), ej,i (− m )]−1 eh,j (−t ), [ej,i ( m ), ei,h (t a)] t t h i h i b ej,i (− tm ) q q−m ei,h (−tq a) q q−m ei,h (−t a), eh,i (−t b) . = eh,j (−t ), ej,h (−t ab) | {z }| {z } =

eh,j

(−tq )

e1

e2

But q a)

h

eh,j (−tq ), ej,h (−tq−m ab)

q a)

h

eh,j (−tq ), [ej,i (−tb

e1 =ei,h (−t

=ei,h (−t

q−m c 2

i

a), ei,h (−tq−m−b

q−m c 2

b)

i

(4)

6

R. HAZRAT AND Z. ZHANG l l l l When b q−m 2 c > l, since En (t A, t hai) and En (t A, t hbi), and so

[En (tl A, tl hai), En (tl A, tl hbi)] are normalized by En (tl A), it follows e1 ∈ [En (tl A, tl hai), En (tl A, tl hbi)]. Now, h ii b e2 = ej,i (− tm ) ei,h (−tq a), eh,i (−tq−m b) h i b b ej,i (− tm ) = ei,h (−tq a), ej,i (− tm ) eh,i (−tq−m b) By invoking Lemma 5 twice, and by choosing a large enough integer q we have, b ej,i (− tm )

ei,h (−tq a) ∈ En (tl hai),

b ej,i (− tm )

eh,i (−tq−m b) ∈ En (tl A, tl hbi).

Hence e2 ∈ [En (tl A, tl hai), En (tl A, tl hbi)]. Thus b q eh,j (−tq ) ρ =eh,j (−t ) [ei,j (t2q a), ej,i ( m )] ∈ [En (tl A, tl hai), En (tl A, tl hbi)]. t Again since these groups are normalized by En (tl A), this immediately implies ρ ∈ [En (tl A, tl hai), En (tl A, tl hbi)]. The proof is now complete by choosing a large enough q which satisfy all the cases above.



Lemma 10. Suppose A is a module finite R-algebra, I and J are two-sided ideals of A and t ∈ R. If m, l are given, there is an integer p such that J 1 p [En (t A) En1 (tp I), En1 ( m )] ⊆ [En (tl A, tl I), En (tl A, tl J)]. t Proof. Let hei,j (tp a)

i β i hEn1 (tp A) 1 p 1 J ) ∈ ) E (t I), E ( n n m tm t 0 00 0 00 for α ∈ I, β ∈ J, a ∈ A and some positive integers i, i , i , j, j , j . Then hei,j (tp a) β i ei0 ,j 0 (tp α), ei00 ,j 00 ( m ) t h β i p p = ei0 ,j 0 (t α)[ei0 ,j 0 (−t α), ei,j (tp a)], ei00 ,j 00 ( m ) t By identity (C2) ei0 ,j 0 (tp α) h β i β = [ei0 ,j 0 (−tp α), ei,j (tp a)], ei00 ,j 00 ( m ) [ei0 ,j 0 (tp α), ei00 ,j 00 ( m )]. t t By Lemma 9, we may choose a large enough p such that β [ei0 ,j 0 (tp α), ei00 ,j 00 ( m )] ∈ [En (tl A, tl I), En (tl A, tl J)]. t We claim that for a suitable p, h β i ei0 ,j 0 (tp α) [ei0 ,j 0 (−tp α), ei,j (tp a)], ei00 ,j 00 ( m ) ∈ [En (tl A, tl I), En (tl A, tl J)]. t Set h p β i e = ei0 ,j 0 (t α) [ei0 ,j 0 (−tp α), ei,j (tp a)], ei00 ,j 00 ( m ) . t  −1    z  x y −1 −1 By the Hall-Witt identity [x , y], z = x, [y , z] y, [z , x] h β i p e = ei,j (t a) ei0 ,j 0 (tp α), [ei,j (−tp a), ei00 ,j 00 ( m )] × t h i β β ei00 ,j 00 ( tm ) ei,j (tp a), [ei00 ,j 00 (− m ), ei0 ,j 0 (tp α)] t ei0 ,j 0 (tp α), ei00 ,j 00 (

GENERALIZED COMMUTATOR FORMULAS

7

By Lemma 9, a large enough choice of p implies that [ei00 ,j 00 (− tβm ), ei0 ,j 0 (tp α)] belongs to [En (tl1 A, tl1 I), En (tl1 A, tl1 J)] for a given l1 . Thus h i β ei,j (tp a), [ei00 ,j 00 (− m ), ei0 ,j 0 (tp α)] ∈ [En (tl1 A, tl1 I), En (tl1 A, tl1 J)]. t Hence, by Lemma 8, there is a sufficient large p such that h i β ei00 ,j 00 ( tβ m) ei,j (tp a), [ei00 ,j 00 (− m ), ei0 ,j 0 (tp α)] ∈ [En (tl A, tl I), En (tl A, tl J)]. t In the same manner, by Lemma 9, one can choose a large enough p such that [ei,j (−tp a), ei00 ,j 00 ( tβm )] belongs to En (tl A, tl J). Since En (tl A, tl I) and En (tl A, tl J) are normalized by En (tp A) for p ≥ l, h β i ei,j (tp a) ei0 ,j 0 (tp α), [ei,j (−tp a), ei00 ,j 00 ( m )] ∈ [En (tl A, tl I), En (tl A, tl J)]. t By choosing the largest p appeared in the cases above, it follows that e belongs to [En (tl A, tl I), En (tl A, tl J)], which proves the claim.  Lemma 11. Suppose A is a module finite R-algebra, I and J are two-sided ideals of A and t ∈ R. If m, l are given, there is an integer p such that J [En (tp A, tp I), En1 ( m )] ⊆ [En (tl A, tl I), En (tl A, tl J)]. t Proof. Since En (tp A, tp I) is generated by all elements of the form ei,j (tp a)

ei0 ,j 0 (tp α)

where a ∈ A, α ∈ I,

the lemma follows, by identity (C2) and Lemma 10, and the fact that En (tl A, tl I) and En (tl A, tl J) are normalized by En (tp A) for p ≥ l.  Lemma 12. Suppose A is a module finite R-algebra, I and J are two-sided ideals of A and t ∈ R. If m, l are given, there is an integer p such that J 1 A [En1 (tp I),En ( tm ) En1 ( m )] ⊆ [En (tl A, tl I), En (tl A, tl J)]. t Proof. Let a ∈ A. Then h h 1 a J i J i 1 a 1 a En1 (tp I),En ( tm ) En1 ( m ) =En ( tm ) En (− tm ) En1 (tp I), En1 ( m ) . t t 0 By Lemma 5, for any give p , there exists a sufficient large p such that 1 (− a ) En tm

0

(5)

0

En1 (tp I) ⊆ En (tp A, tp I).

Hence, h i a J i En1 ( m ) p0 p0 1 J t ) ⊆ E (t ) . A, t I), E ( n n tm tm 00 0 By Lemma 11, for any give integer p , a sufficient large p satisfies the equation h i J i h 0 0 00 00 00 00 En (tp A, tp I), En1 ( m ) ⊆ En (tp A, tp I), En1 (tp A, tp J) . t Thus h h i J i 1 a 00 00 00 1 a 00 En1 (tp I),En ( tm ) En1 ( m ) ⊆En ( tm ) En (tp A, tp I), En1 (tp A, tp J) . t By Lemma 8, for the given l, we may find a large enough p00 such that i h i h 00 00 00 00 1( a ) En tm En (tp A, tp I), En1 (tp A, tp J) ⊆ En (tl A, tl I), En1 (tl A, tl J) . h

1 ( a ) E 1 (− a ) En n tm tm

This finishes our proof.

En1 (tp I), En1 (



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R. HAZRAT AND Z. ZHANG

Lemma 13. Suppose A is a module finite R-algebra, I and J are two-sided ideals of A and t ∈ R. If m, l, K are given, there is an integer p such that [En1 (tp I), EnK (

A J , )] ⊆ [En (tl A, tl I), En (tl A, tl J)]. tm tm

Proof. The lemma follows from Lemma 12 and Lemma 8 by an easy induction.



So far all the calculations took place in the group En (At ). In the important Lemma 14, using the fact that for a suitable positive integer l, the restriction of θt to GLn (A, tl A) induces an injective homomorphism θt : GLn (A, tl A) → GLn (At ), by Lemma 3, we are able to “pull back” the elements into the group GLn (A). Lemma 14. Suppose A is a quasi-finite R-algebra, I and J are two-sided ideals of A and m a maximal ideal of R. For any g ∈ GLn (A, J), there is a t ∈ R\m and an integer p, such that [e, g] ∈ [En (A, I), En (A, J)] for any e ∈ En1 (tp I). Proof. The functors En and GLn commute with direct limits. By Proposition 2 and §1.4, one reduces the proof to the case A is finite over R and R is Noetherian. For any maximal ideal m of R, the ring Am contains Jm as an ideal. Consider the natural homomorphism θm : A → Am which induces a homomprphism (call it again θm ) in the level of general linear groups, θm : GL(A) → GL(Am ). Therefore, for g ∈ GLn (A, J), θm (g) ∈ GLn (Am , Jm ). Since Am is module finite over the local ring Rm , Am is semilocal [6, III(2.5), (2.11)], therefore its stable rank is 1. It follows that GLn (Am , Jm ) = En (Am , Jm )GL1 (Am , Jm ) (see [12, Th. 4.2.5]). So θm (g) can be decomposited as θm (g) = εh, where ε ∈ En (Am , Jm ) and h is a diagonal matrix all of whose diagonal coefficients are 1, except possibly the k-th diagonal coefficient, and k can be chosen arbitarily. By (1.4), we may reduce the problem to the case At with t ∈ R\m. Namely θt (g) is a product of ε and h, where ε ∈ En (At , Jt ), and h is a diagonal matrix with only one non-trivial diagonal entry which lies in At . Therefore ε is a product of the elementary matrices, thus one has ε ∈ EnK (

A J , ). tm tm

Let e ∈ En1 (tp I). We choose h such that it commutes with θt (e). By Lemma 13, for any given l, there is a large enough p such that [θt (e), θt (g)] = [θt (e), ε] ∈ [En (tl A, tl I), En (tl A, tl J)].

(6)

Since e ∈ En (tp I) ⊆ GLn (A, tl A) and GLn (A, tl A) is normal in GLn (A), it follows [e, g] ∈ GLn (A, tl A). On the other hand, using (6), one can find an x of the form [En (tl A, tl I), En (tl A, tl J)] in E(A) such that θt (x) = [θt (e), θt (g)]. Since for suitable l, the restriction of θt to GLn (A, tl A) is injective by Lemma 3, it follows [e, g] = x and thus [e, g] ∈ [En (A, I), En (A, J)].  3. Main result We prove the main theorem in this section. Theorem 15. Let A be a quasi-finite R-algebra and I, J be two-sided ideals of A. Then for n ≥ 3, [En (A, I), GLn (A, J)] = [En (A, I), En (A, J)].

(7)

Proof. The functors En and GLn commute with direct limits. By Proposition 2 and §1.4, one reduces the proof to the case A is finite over R and R is Noetherian. First we claim that [En1 (I), GLn (A, J)] ⊆ [En (A, I), En (A, J)]. (8)

GENERALIZED COMMUTATOR FORMULAS

9

Let ei0 ,j 0 (α) ∈ En1 (I), and g ∈ GLn (A, J). For any maximal ideal mi C R, choose a ti ∈ R\mi and a positive integer pi according to Lemma 14. Since the collection of all tpi i is not contained in any maximal ideal, we may find a finite number of ti and xi ∈ R such that X p ti i xi = 1. i

We have, X p Y ei0 ,j 0 (α) = ei0 ,j 0 (( ti i xi )α) = ei0 ,j 0 (tpi i xi α). i

i

By Lemma 14, it follows immediately that for each i, [ei0 ,j 0 (tpi i xi α), g] ∈ [En (tli A, tli I), En (tli A, tli J)] ⊆ [En (A, I), En (A, J)].

(9)

A direct computation using (9) and Formula (C2) and the fact that En (A, I) and En (A, J) are normal in En (A), shows that Y [ei0 ,j 0 (α), g] = [ ei0 ,j 0 (tpi i xi α), g] ∈ [En (A, I), En (A, J)]. i

This proves our claim. Now, since En (A, I) is by definition the normal closure of En (I) in En (A), and GLn (A, J) is normal in GLn (A), using (8) along with Formula (C2), one gets [En (A, I), GLn (A, J)] ⊆ [En (A, I), En (A, J)]. The inclusion of (7) in the other direction is clear. Therefore [En (A, I), GLn (A, J)] = [En (A, I), En (A, J)].



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