NK_1 of Bak's unitary group over Graded Rings
aa r X i v : . [ m a t h . K T ] J a n NK OF BAK’S UNITARY GROUP OVER GRADED RINGS
RABEYA BASU AND KUNTAL CHAKRABORTY
Abstract:
For an associative ring R with identity, we study the absence of k -torsion inNK GQ( R ); Bass nil-groups for the general quadratic or Bak’s unitary groups. By using agraded version of Quillen–Suslin theory we deduce an analog for the graded rings. Key words: General linear groups, Elementary subgroups, Quadratic forms, Higman lin-earisation, k -torsion, Whitehead group - K .1. Introduction
Let R be an associative ring with identity element 1. When R is commutative, we defineSK ( R ) as the kernel of the determinant map from the Whitehead group K ( R ) to thegroup of units of R . The Bass nil-group NK ( R ) = ker(K ( R [ X ]) → K ( R )); X = 0. i.e. , the subgroup consisting of elements [ α ( X )] ∈ K ( R [ X ]) such that [ α (0)] = [I]. HenceK ( R [ X ]) ∼ = NK ( R ) ⊕ K ( R ). The aim of this paper is to study some properties of Bassnil-groups NK for the general quadratic groups or Bak’s unitary groups.It is well-known that for many rings, e.g. if R is regular Noetherian, Dedekind domain,or any ring with finite global dimension, the group NK ( R ) is trivial. On the other hand,if it is non-trivial, then it is not finitely generated as a group. e.g. if G is a non-trivialfinite group, the group ring Z G is not regular. In many such cases, it is difficult to computeNK ( Z [ G ]). In [13], D.R. Harmon proved the triviality of this group when G is finite groupof square free order. C. Weibel, in [20], has shown the non-triviality of this group for G = Z / ⊕ Z / Z /
4, and D . Some more results are known for finite abelian groups from thework of R.D. Martin; cf. [16]. It is also known ( cf. [12]) that for a general finite group G ,NK ( R [ G ]) is a torsion group for the group ring R [ G ]. In fact, for trivial NK ( R ), everyelement of finite order of NK ( R [ G ]) is some power of the cardinality of G . For R = Z , thisis a result of Weibel. In particular, if G is a finite p -group ( p a prime), then every elementof NK ( Z [ G ]) has p -primary order. In [17], J. Stienstra showed that NK ( R ) is a W( R )-module, where W( R ) is the ring of big Witt vectors ( cf. [11] and [19]). Consequently, in ([18], § k is a unit in R , then SK ( R [ X ]) has no k -torsion, when R is a commutative local ring. Note that if R is a commutative local ring then SK ( R [ X ])coincides with NK ( R ); indeed, if R is a local ring then SL n ( R ) = E n ( R ) for all n > α ( X ) by α ( X ) α (0) − and assume that [ α (0)] = [I]. In [7], thefirst author extended Weibel’s result for arbitrary associative rings. In this paper we provethe analog result for λ -unitary Bass nil-groups, viz. NK GQ λ ( R, Λ), where ( R, Λ) is theform ring as introduced by A. Bak in [1]. The main ingredient for our proof is an analog ofHigman linearisation (for a subclass of Bak’s unitary group) due to V. Kopeiko; cf. [15]. Forthe general linear groups, Higman linearisation ( cf. [6]) allows us to show that NK ( R ) has aunipotent representative. The same result is not true in general for the unitary nil-groups. Research by the first author was supported by SERB-MATRICS grant for the financial year 2020–2021. And, research by the second author was supported by IISER (Pune) post-doctoral researchgrant.Corresponding Author: [email protected], [email protected]. opeiko’s results in [15] explain a complete description of the elements of NK GQ λ ( R, Λ)that have (unitary) unipotent representatives. Followings are the main results in this article.
Theorem 1.1.
Let [ α ( X )] = (cid:2) (cid:18) A ( X ) B ( X ) C ( X ) D ( X ) (cid:19) (cid:3) ∈ NK GQ λ ( R, Λ) with A ( X ) ∈ GL r ( R [ X ]) for some r ∈ N . Then [ α ( X )] has no k -torsion if kR = R . And, an analog for the graded rings:
Theorem 1.2.
Let R = R ⊕ R ⊕ . . . be a graded ring. Let k be a unit in R . Let N = N + N + · · · + N r ∈ M r ( R ) be a nilpotent matrix, and I denote the identity matrix. If [(I + N )] k = [I] in K GQ λ ( R, Λ) , then [I + N ] = [I + N ] . In the proof of 1.2, we have used a graded version of Quillen–Suslin’s local-global principlefor Bak’s unitary group over graded rings. This unify and generalize the results proved in[5], [7], [9], and [10].
Theorem 1.3. (Graded local-global principle)
Let R = R ⊕ R ⊕ R ⊕ · · · be agraded ring with identity . Let α ∈ GQ(2 n, R, Λ) be such that α ≡ I n (mod R + ) . If α m ∈ EQ(2 n, R m , Λ m ) , for every maximal ideal m ∈ Max(C(R )) , then α ∈ EQ(2 n, R, Λ) . Preliminaries
Let R be an associative ring with identity element 1. Let M( n, R ) denote the additivegroup of n × n matrices, and GL( n, R ) denote the multiplicative group of n × n invertiblematrices. Let e ij be the matrix with 1 in the ij -th position and 0’s elsewhere. The elementarysubgroup of GL( n, R ) plays a key role in classical algebraic K-theory. We recall, Definition 2.1. Elementary Group E( n, R ) : The subgroup of all matrices in GL( n, R )generated by { E ij ( λ ) : λ ∈ R, i = j } , where E ij ( λ ) = I n + λe ij , and e ij is the matrix with 1in the ij -position and 0’s elsewhere. Definition 2.2.
For α ∈ M( r, R ) and β ∈ M( s, R ), the matrix α ⊥ β denotes its embeddingin M( r + s, R ) (here r and s are even integers in the non-linear cases), given by α ⊥ β = (cid:18) α β (cid:19) . There is an infinite counterpart: Identifying each matrix α ∈ GL( n, R ) with the large matrix( α ⊥ { } ) gives an embedding of GL( n, R ) into GL( n + 1 , R ). Let GL( R ) = ∞ ∪ n =1 GL( n, R ),and E( R ) = ∞ ∪ n =1 E( n, R ) be the corresponding infinite linear groups.As a consequence of classical Whitehead Lemma ( cf. [3]) due to A. Suslin, one gets[GL( R ) , GL( R )] = E( R ) . Definition 2.3.
The quotient groupK ( R ) = GL( R )[GL( R ) , GL( R )] = GL( R )E( R )is called the Whitehead group of the ring R . For α ∈ GL( n, R ), let [ α ] denote itsequivalence class in K ( R ).In the similar manner we define K group for the other types of classical groups; viz. , thesymplectic Whitehead group K Sp( R ) and the orthogonal Whitehead group K O( R ).This paper explores a uniform framework for classical type groups over graded structures.Let us begin by recalling the concept of form rings and form parameter as introduced by A.Bak in [1]. This allows us to give a uniform definition for classical type groups. efinition 2.4. (Form rings): L et R be an associative ring with identity, and with aninvolution − : R → R , a a . Let λ ∈ C ( R ) = the center of R , with the property λλ = 1 .We define two additive subgroups of R , viz. Λ max = { a ∈ R | a = − λa } and Λ min = { a − λa | a ∈ R } . O ne checks that for any x ∈ R , Λ max and Λ min are closed under the conjugation operation a xax . A λ -form parameter on R is an additive subgroup Λ of R such that Λ min ⊆ Λ ⊆ Λ max ,and x Λ x ⊆ Λ for all x ∈ R . i.e. , a subgroup between two additive groups which is also closedunder the conjugation operation. A pair ( R, Λ) is called a form ring.To define Bak’s unitary group or the general quadratic group, we fix a central element λ ∈ R with λλ = 1, and then consider the form ψ n = (cid:18) n λ I n (cid:19) . For more details, see [7], and [8].
Bak’s Unitary or General Quadratic Groups GQ : GQ(2 n, R,
Λ) = { σ ∈ GL(2 n, R, Λ) | σψ n σ = ψ n } . Elementary Quadratic Matrices :
Let ρ be the permutation, defined by ρ ( i ) = n + i for i = 1 , . . . , n . For a ∈ R , and 1 ≤ i, j ≤ n , we define qε ij ( a ) = I n + ae ij − ae ρ ( j ) ρ ( i ) for i = j , qr ij ( a ) = (cid:26) I n + ae iρ ( j ) − λae jρ ( i ) for i = j I n + ae ρ ( i ) j for i = jql ij ( a ) = (cid:26) I n + ae ρ ( i ) j − λae ρ ( j ) i for i = j I n + ae ρ ( i ) j for i = j (Note that for the second and third type of elementary matrices, if i = j , then we get a = − λa , and hence it forces that a ∈ Λ max ( R ). One checks that these above matricesbelong to GQ(2n , R , Λ); cf. [1]. n -th Elementary Quadratic Group EQ(2 n, R,
Λ):The subgroup generated by qε ij ( a ) , qr ij ( a )and ql ij ( a ), for a ∈ R and 1 ≤ i, j ≤ n . Foruniformity we denote the elementary generators of EQ(2 n, R, Λ) by η ij ( ∗ ). Stabilization map:
There are standard embeddings:GQ(2 n, R, Λ) −→ GQ(2 n + 2 , R, Λ)given by (cid:18) a bc d (cid:19) a b
00 1 0 0 c d
00 0 0 1 . Hence we have GQ( R, Λ) = lim −→ GQ(2 n, R,
Λ).It is clear that the stabilization map takes generators of EQ(2 n, R,
Λ) to the generatorsof EQ(2( n + 1) , R, Λ). Hence we haveEQ( R, Λ) = lim −→ EQ(2 n, R, Λ) here are standard formulas for the commutators between quadratic elementary matrices.For details, we refer [1] (Lemma 3.16). In later sections there are repeated use of thoserelations. The analogue of the Whitehead Lemma for the general quadratic groups ( cf. [1])due to Bak allows us to write:[GQ( R, Λ) , GQ( R, Λ)] = [EQ( R, Λ) , EQ( R, Λ)] = EQ( R, Λ) . Hence we define the
Whitehead group of the general quadratic groupK GQ = GQ( R, Λ)EQ( R, Λ) . And, the Whitehead group at the level m K ,m GQ = GQ m ( R, Λ)EQ m ( R, Λ) , where m = 2 n in the non-linear cases.Let ( R, Λ) be a form ring. We extend the involution of R to the ring R [ X ] of polynomialsby setting X = X . As a result we obtain a form ring ( R [ X ] , Λ[ X ]). Definition 2.5.
The kernel of the group homomorphismK GQ( R [ X ] , Λ[ X ]) → K GQ( R, Λ)induced from the form ring homomorphism ( R [ X ] , Λ[ X ]) → ( R, Λ) : X GQ( R, Λ). We often say it as Bass nilpotent unitary K -group of R , or just unitarynil-group.From the definition it follows thatK GQ( R [ X ] , Λ[ X ]) = K GQ( R, Λ) ⊕ NK GQ( R, Λ) . In this context, we will use following two types of localizations, mainly over graded ring R = R ⊕ R ⊕ R ⊕ · · · .(1) Principal localization: for a non-nilpotent, non-zero divisor s in R with s = s , weconsider the multiplicative subgroup S = { , s, s , . . . } , and denote localized formring by ( R s , Λ s ).(2) Maximal localization: for a maximal ideal m ∈ Max(R ), we take the multiplicativesubgroup S = R − m , and denote the localized form ring by ( R m , Λ m ). Blanket assumption : We always assume that 2 n ≥ Swan–Weibel’s homotopy trick ”, which is the mainingredient to handle the graded case. Let R = R ⊕ R ⊕ R ⊕ · · · be a graded ring. Anelement a ∈ R will be denoted by a = a + a + a + · · · , where a i ∈ R i for each i , and allbut finitely many a i are zero. Let R + = R ⊕ R ⊕ · · · . Graded structure of R induces agraded structure on M n ( R ) (ring of n × n matrices). Definition 2.6.
Let a ∈ R be a fixed element. We fix an element b = b + b + · · · in R and define a ring homomorphism ǫ : R → R [ X ] given by ǫ ( b ) = ǫ ( b + b + · · · ) = b + b X + b X + · · · + b i X i + · · · . Then we evaluate the polynomial ǫ ( b )( X ) at X = a and denote the image by b + ( a ) i.e. , b + ( a ) = ǫ ( b )( a ). Note that (cid:0) b + ( x ) (cid:1) + ( y ) = b + ( xy ). Observe, b = b + (0). We shall use thisfact frequently.The above ring homomorphism ǫ induces a group homomorphism at the GL(2 n, R ) levelfor every n ≥ i.e. , for α ∈ GL(2 n, R ) we get a map ǫ : GL(2 n, R, Λ) → GL(2 n, R [ X ] , Λ[ X ]) defined by α = α ⊕ α ⊕ α ⊕ · · · 7→ α ⊕ α X ⊕ α X · · · , where α i ∈ M(2 n, R i ). As above for a ∈ R , we define α + ( a ) as α + ( a ) = ǫ ( α )( a ) . otation 2.7. By GQ(2 n, R [ X ] , Λ[ X ] , ( X )) we shall mean the group of all quadraticmatrices over R [ X ], which are I n modulo ( X ). Also if R is a graded ring, then byGQ(2 n, R, Λ , ( R + )) we shall mean the group of all quadratic matrices over R which areI n modulo R + .The following lemma highlights very crucial fact which we use (repeatedly) in the proofof “Dilation Lemma”. Lemma 2.8.
Let R be a Noetherian ring and s ∈ R . Then there exists a natural num-ber k such that the homomorphism GQ(2 n, R, Λ , s k R ) → GQ(2 n, R s , Λ s ) ( induced by local-ization homomorphism R → R s ) is injective. Moreover, it follows that the induced map EQ(2 n, R, Λ , s k R ) → EQ(2 n, R s , Λ s ) is injective. For the proof of the above lemma we refer [14], Lemma 5.1. Recall that any module finitering R is direct limit of its finitely generated subrings. Also, G( R, Λ) = lim −→ G( R i , Λ i ), wherethe limit is taken over all finitely generated subring of R . Thus, one may assume that C ( R )is Noetherian. Hence one may consider module finite (form) rings ( R, Λ) with identity.Now we recall few technical definitions and useful lemmas.
Definition 2.9.
A row ( a , a , . . . , a n ) ∈ R n is said to be unimodular if there exists( b , b , . . . , b n ) ∈ R n such that P ni =1 a i b i = 1. The set of all unimodular rows of length n is denoted by Um n ( R ).For any column vector v ∈ ( R n ) t we define the row vector e v = v t ψ n . Definition 2.10.
We define the map M : ( R n ) t × ( R n ) t → M (2 n, R ) and the innerproduct h , i as follows: M ( v, w ) = v. e w − λ w. e v h v, w i = e v.w Note that the elementary generators of the groups EQ(2 n, R,
Λ) are of the form I n + M ( ∗ , ∗ ) for suitably chosen standard basis vectors. Lemma 2.11. ( cf. [1]) The group
E(2 n, R, Λ) is perfect for n ≥ , i.e., [EQ(2 n, R, Λ) , EQ(2 n, R,
Λ)] = EQ(2 n, R, Λ) . Lemma 2.12.
For all elementary generators of
GQ(2 n, R, Λ) we have the following splittingproperty: for all x, y ∈ R , η ij ( x + y ) = η ij ( x ) η ij ( y ) . Proof : See pg. 43-44, Lemma 3.16, [1].
Lemma 2.13.
Let G be a group, and a i , b i ∈ G , for i = 1 , , . . . , n . Then for r i = Π ij =1 a j ,we have Π ni =1 r i b i r − i Π ni =1 a i . Lemma 2.14.
The group
GQ(2 n, R, Λ , R + ) ∩ EQ(2 n, R, Λ) generated by the elements of thetype εη ij ( ∗ ) ε − , where ε ∈ EQ(2 n, R, Λ) and ∗ ∈ R + . Proof : Let α ∈ GQ(2 n, R, Λ , R + ) ∩ EQ(2 n, R,
Λ). Then we can write α = Π rk =1 η i k j k ( a k )for some element a k ∈ R , k = 1 , . . . , r . We can write a k as a k = ( a ) k + ( a + ) k for some( a ) k ∈ R and ( a + ) k ∈ R + . Using Lemma 2.12, we can write α as, α = Π rk =1 ( η i k j k ( a ) k )( η i k j k ( a + ) k ) . Let ǫ t = Π tk =1 η i k j k (( a ) k ) for 1 ≤ t ≤ r . By the Lemma 2.13, we have α = (cid:0) Π rk =1 ǫ k η i k j k (( a + ) k ) ǫ − k (cid:1) (Π rk =1 η i k j k (( a ) k )) . Let us write A = Π rk =1 ǫ k η i k j k (( a + ) k ) ǫ − k and B = Π rk =1 η i k j k (( a ) k ). Hence α = AB .Let ‘over-line’ denotes the quotient ring modulo R + . Now going modulo R + , we have = AB = ¯ A ¯ B = I n ¯ B = I n , the last equality holds as α ∈ GQ(2 n, R, Λ , R + ). Hence, B = I n . Since the entries of B are in R , it follows that B = I n . Therefore it follows that α = Π rk =1 ǫ k η i k j k (( a + ) k ) ǫ − k . ✷ Quillen–Suslin Theory for Bak’s Group over Graded Rings
Local–Global Principle.Lemma 3.1.
Let ( R, Λ) be a form ring and v ∈ EQ(2 n, R, Λ) e . Let w ∈ R n be a columnvector such that h v, w i = 0 . Then I n + M ( v, w ) ∈ EQ(2 n, R, Λ) . Proof : Let v = εe . Then we have I n + M ( v, w ) = ε (I n + M ( e , w )) ε − , where w = ε − w . Since h e , w i = h v, w i = 0, we have w T = ( w , . . . , w n − , , . . . , w n ).Therefore, since λ ¯ λ = 1, we haveI n + M ( v, w ) = Y ≤ j ≤ n ≤ i ≤ n − εql in ( − ¯ λw n + i ) qε jn ( − ¯ λw j ) ql − nn ( ∗ ) ε − Lemma 3.2.
Let R be a graded ring. Let α ∈ EQ(2 n, R, Λ) . Then for every a ∈ R onegets α + ( a ) ∈ EQ(2 n, R, Λ) . Proof : Let α = Π tk =1 (I n + aM ( e i k , e j k )) , where a ∈ R and t ≥
1. Hence for b ∈ R ,we have α + ( b ) = Π tk =1 (I n + a + ( b ) M ( e i k , e j k )). Now taking v = e i and v = a + ( b ) e j wehave h v, w i = 0 and I n + M ( v, w ) = I n + a + ( b ) M ( e i , e j )) which belongs to EQ(2 n, R, Λ)by Corollary 3.1. Hence we have α + ( b ) ∈ EQ(2 n, R,
Λ) for b ∈ R . ✷ Lemma 3.3. (Graded Dilation Lemma)
Let α ∈ GQ(2 n, R, Λ) with α + (0) = I n and α s ∈ EQ(2 n, R s , Λ s ) for some non-zero-divisor s ∈ R . Then there exists β ∈ EQ(2 n, R, Λ) such that β + s ( b ) = α + s ( b ) for some b = s l and l ≫ . Proof : Since α s ∈ EQ(2 n, R s , Λ s ) with ( α ) s = I n , then α s = γ , where γ ii = 1 + g ii where g ii ∈ ( R + ) s and γ ij = g ij for i = j , where g ij ∈ ( R + ) s . Choose l large enoughsuch that every denominator of g ij for all i, j divides s l . Then by Lemma 3.2, we have α + s ( s l ) ∈ EQ(2 n, R s , Λ s ). As all denominator is cleared then α + s ( s l ) permits a naturalpullback. Hence we have α + ( s l ) ∈ EQ(2 n, R, Λ) . Call this pullback as β . ✷ Lemma 3.4.
Let α s ∈ EQ(2 n, R s , Λ s ) with α + s (0) = I n . Then one gets α + s ( b + d ) α + s ( d ) − ∈ EQ(2 n, R, Λ) for some s, d ∈ R and b = s l , l ≫ . Proof : Since α s ∈ EQ(2 n, R s , Λ s ), we have α + s ( X ) ∈ EQ(2 n, R s [ X ] , Λ s [ X ]). Let β + ( X ) = α + ( X + d ) α + ( d ) − , where d ∈ R . Then we have β + s ( X ) ∈ EQ(2 n, R s [ X ] , Λ s [ X ])and β + (0) = I n . Hence by Lemma 3.3, we have, there exists b = s l , l ≫
0, such that β + ( bX ) ∈ EQ(2 n, R [ X ] , Λ[ X ]). Putting X = 1, we get our desired result. ✷ Proof of Theorem 1.3 – Graded Local-Global Principle:
Since α m ∈ EQ(2 n, R m , Λ m ) for all m ∈ Max( C ( R )), for each m there exists s ∈ C ( R ) \ m such that α s ∈ EQ(2 n, R s , Λ s ). Using Noetherian property we can consider a finite cover of C ( R ), say s + · · · + s r = 1. From Lemma 3.3, we have α + ( b i ) ∈ EQ(2 n, R,
Λ) for some b i = s l i i with b + · · · + b r = 1. Now consider α s s ...s r , which is the image of α in R s s ...s r . y Lemma 2.8, α α s s ...s r is injective. Hence we can perform our calculation in R s s ...s r and then pull it back to R . α s s ...s r = α + s s ...s r ( b + b + · · · + b r )=(( α s ) s s ... ) + ( b + · · · + b r )(( α s ) s s ... ) + ( b + · · · + b r ) − . . . (( α s i ) s ... ˆ s i ...s r ) + ( b i + · · · + b r )(( α s i ) s ... ˆ s i ...s r ) + ( b i +1 + · · · + b r ) − (( α s r ) s s ...s r − ) + ( b r )(( α s r ) s s ...s r − ) + (0) − Observe that (( α s i ) s ... ˆ s i ...s r ) + ( b i + · · · + b r )(( α s i ) s ... ˆ s i ...s r ) + ( b i +1 + · · · + b r ) − ∈ EQ(2 n, R,
Λ)due to Lemma 3.4 (here ˆ s i means we omit s i in the product s . . . ˆ s i . . . s r ), and hence α s s ...s r ∈ EQ(2 n, R s ...s r , Λ s ...s r ). This proves α ∈ EQ(2 n, R,
Λ). ✷ Normality and Local–Global.
Next we are going to show that if K is a commutativering with identity and R is an associative K -algebra such that R is finite as a left K -module, then the normality criterion of elementary subgroup is equivalent to the Local-Globalprinciple for quadratic group. (One can also consider R as a right K -algebra.) Lemma 3.5. ( Bass; cf. [4])
Let A be an associative B -algebra such that A is finite as a left B -module and B be a commutative local ring with identity. Then A is semilocal. Theorem 3.6. ( cf. [7]) Let A be a semilocal ring ( not necessarily commutative ) with invo-lution. Let v ∈ Um n ( A ) .Then v ∈ e EQ(2 n, A ) . In other words the group EQ(2 n, A ) actstransitively on Um n ( A ) . Before proving the next theorem we need to recall a theorem from [7]:
Theorem 3.7. ( Local-Global Principle ) Let A be an associative B -algebra such that A is finite as a left B -module and B be a commutative ring with identity.. If α ( X ) ∈ GQ(2 n, A [ X ] , Λ[ X ]) , α (0) = I and α m ( X ) ∈ EQ(2 n, A m [ X ] , Λ m [ X ]) for every maximalideal m ∈ Max( B ) , then α ∈ EQ(2 n, A [ X ] , Λ[ X ]) . Theorem 3.8.
Let K be a commutative ring with unity and R = ⊕ ∞ i =0 R i be a graded K -algebra such that R is finite as a left K -module. Then for n ≥ the following are equivalent: (1) EQ(2 n, R, Λ) is a normal subgroup of GQ(2 n, R, Λ) . (2) If α ∈ GQ(2 n, R, Λ) with α + (0) = I n and α m ∈ EQ(2 n, R m , Λ m ) for every maximalideal m ∈ Max( K ) , then α ∈ EQ(2 n, R, Λ) . Proof : (1) ⇒ (2) We have proved the Lemma 3.1 for any form ring with identity andshown that the local-global principle is a consequence of Lemma 3.1. So, the result is truein particular if we have EQ(2 n, R, Λ) is a normal subgroup of GQ(2 n, R,
Λ).(2) ⇒ (1) Since polynomial rings are special case of graded rings, the result follows byusing the Theorem 3.7. Let α ∈ EQ(2 n, R,
Λ) and β ∈ GQ(2 n, R,
Λ). Then we have α canbe written as product of the matrices of the form (I n + βM ( ∗ , ∗ ) β − ), with h∗ , ∗ i = 0where ∗ and ∗ are suitably chosen basis vectors. Let v = β ∗ . Then we can write βαβ − as a product of the matrices of the form I n + M ( v, w ) for some w ∈ R n . We must showthat each I n + M ( v, w ) ∈ EQ(2 n, R,
Λ).Consider γ = I n + M ( v, w ). Then γ + (0) = I n . By Lemma 3.5 we have the ring S − R is semilocal where S = K \ m , and m ∈ Max( K ). Since v ∈ Um n ( R ), then by Theorem3.6, we have v ∈ EQ(2 n, S − R, S − Λ) e . Therefore by applying Lemma 3.1 to the ring( S − R, S − Λ), we have γ m ∈ EQ(2 n, R m , Λ m ) for every maximal ideal m ∈ Max( K ). Henceby hypothesis we have γ ∈ EQ(2 n, R,
Λ). This completes the proof. ✷ Remark 3.9.
We conclude that the local-global principle for the elementary subgroups andtheir normality properties are equivalent. . Bass Nil Group NK GQ(R)In this section recall some basic definitions and properties of the representatives ofNK GQ(R). We represent any element of M n ( R ) as (cid:18) a bc d (cid:19) , where a, b, c, d ∈ M n ( R ).For α = (cid:18) a bc d (cid:19) we call (cid:0) a b (cid:1) the upper half of α . Let ( R, λ,
Λ) be a form ring. By setting¯Λ = { ¯ a : a ∈ Λ } we get another form ring ( R, ¯ λ, ¯Λ). We can extend the involution of R toM n ( R ) by setting ( a ij ) ∗ = ( a ji ). Definition 4.1.
Let (
R, λ,
Λ) be a form ring. A matrix α = ( a ij ) ∈ M n ( R ) is said to beΛ-Hermitian if α = − λα ∗ and all the diagonal entries of α are contained in Λ. A matrix β ∈ M n ( R ) is said to be ¯Λ-Hermitian if β = − ¯ λβ ∗ and all the diagonal entries of β arecontained in ¯Λ. Remark 4.2.
A matrix α ∈ M n ( R ) is Λ-Hermitian if and only if α ∗ is ¯Λ-Hermitian. Theset of all Λ-Hermitian matrices forms a group under matrix multiplication. Lemma 4.3. [15, Example 2]
Let β ∈ GL n ( R ) be a Λ -Hermitian matrix. Then the matrix α ∗ βα is Λ -Hermitian for every α ∈ GL n ( R ) . Definition 4.4.
Let α = (cid:18) a bc d (cid:19) ∈ M n ( R ) be a matrix. Then α is said to be a Λ-quadraticmatrix if one of the following equivalent conditions holds:(1) α ∈ GQ(2 n, R,
Λ) and the diagonal entries of the matrices a ∗ c, b ∗ d are in Λ,(2) a ∗ d + λc ∗ d = I n and the matrices a ∗ c, b ∗ d are Λ-Hermitian,(3) α ∈ GQ(2 n, R,
Λ) and the diagonal entries of the matrices ab ∗ , cd ∗ are in Λ,(4) ad ∗ + λbc ∗ = I n and the matrices ab ∗ , cd ∗ are Λ-Hermitian. Remark 4.5.
The set of all Λ-quadratic matrices of order 2 n forms a group called Λ-quadratic group. We denote this group by GQ λ (2 n, R, Λ). If 2 ∈ R ∗ , then we have Λ min =Λ max . In this case notions of quadratic groups and notions of Λ-quadratic groups coincides.Also this happens when Λ = Λ max . Hence quadratic groups are special cases of Λ-quadraticgroups. Other classical groups appear as Λ-quadratic groups in the following way. Let R bea commutative ring with trivial involution. ThenGQ λ (2 n, R, Λ) = ( Sp n ( R ) , if λ = − max = R O n ( R ) , if λ = 1 and Λ = Λ min = 0And for general linear group GL n ( R ), we have, GL n ( R ) = GQ (2 n, H ( R ) , Λ = Λ max ), where H ( R ) denotes the ring R ⊕ R op with R op is the opposite ring of R and the involution on H ( R ) is defined by ( x, y ) = ( y, x ). Thus the study of Λ-quadratic matrices unifies the studyof quadratic matrices.We recall following results from [15]. Lemma 4.6.
Let α = (cid:18) a d (cid:19) ∈ M n ( R ) . Then α ∈ GQ λ (2 n, R, Λ) if and only if a ∈ GL n ( R ) and d = ( a ∗ ) − . Proof : Let α ∈ GQ λ (2 n, R, Λ). In view of (2) of Definition 4.4, we have, a ∗ d = I n .Hence a is invertible and d = ( a ∗ ) − . Converse holds by (2) of Definition 4.4. ✷ Definition 4.7.
Let α ∈ GL n ( R ) be a matrix. A matrix of the form (cid:18) α
00 ( α ∗ ) − (cid:19) isdenoted by H ( α ) and is said to be hyperbolic. Remark 4.8.
In a similar way we can show that matrices of the form T ( β ) := (cid:18) I n β n (cid:19) is Λ-quadratic matrix if and only if β is ¯Λ-Hermitian. And the matrix of the form T ( γ ) := (cid:18) I n γ I n (cid:19) is Λ-quadratic matrix if and only if γ is Λ-Hermitian. ikewise in the quadratic case we can define the notion of Λ-elementary quadratic groupsin the following way: Definition 4.9.
The Λ-elementary quadratic group is denoted by EQ λ (2 n, R, Λ) and definedby the group generated by 2 n × n matrices of the form H ( α ) where α ∈ E n ( R ), T ( β ) and β is ¯Λ-Hermitian and T ( γ ) is γ Λ-Hermitian.
Lemma 4.10.
Let A = (cid:18) α β δ (cid:19) ∈ M n ( R ) . Then A ∈ GQ λ (2 n, R, Λ) if and only if α ∈ GL n ( R ) , δ = ( α ∗ ) − and α − β is ¯Λ -Hermitian. In this case A ≡ H ( α ) (mod EQ λ (2 n, R, Λ)) . Proof : Let A ∈ GQ λ (2 n, R, Λ). Then by (4) of Definition 4.4, we have αδ ∗ = I n and αβ ∗ is Λ-Hermitian. Hence α is invertible and δ = ( α ∗ ) − . For α − β , we get( α − β ) ∗ = β ∗ ( α − ) ∗ = α − ( αβ ∗ )( α − ) ∗ , which is Λ-Hermitian by Lemma 4.3. Hence α − β is ¯Λ-Hermitian. Conversely, the conditionon A will fulfill the condition (4) of Definition 4.4. Hence A is Λ-quadratic. Since α − β is¯Λ-Hermitian, T ( − α − β ) ∈ EQ λ (2 n, R, Λ)and AT ( α − β ) = H ( α ). Thus A ≡ H ( α ) (mod EQ λ (2 n, R, Λ)). ✷ A similar proof will prove the following:
Lemma 4.11.
Let B = (cid:18) α γ δ (cid:19) ∈ M n ( R ) . Then B ∈ GQ λ (2 n, R, Λ) if and only if α ∈ GL n ( R ) , δ = ( α ∗ ) − and γ is Λ -Hermitian. In this case B ≡ H ( α ) (mod EQ λ (2 n, R, Λ)) . Lemma 4.12.
Let α = (cid:18) a bc d (cid:19) ∈ GQ λ (2 n, R, Λ) . Then α ≡ H ( a ) (mod EQ λ (4 n, R, Λ)) if a ∈ GL n ( R ) . Moreover, if a ∈ E n ( R ) , then α ≡ H ( a ) (mod EQ λ (2 n, R, Λ)) . Proof : By same argument as given in Lemma 4.10, we have a − b is Λ-Hermitian. Hence T ( − a − b ) ∈ EQ λ (2 n, R, Λ), and consequently αT ( − a − b ) = (cid:18) a c d ′ (cid:19) ∈ GQ λ (2 n, R, Λ)for some d ′ ∈ GL n ( R ). Hence by Lemma 4.11, we get αT ( − a − b ) ≡ H ( a ) (mod EQ λ (2 n, R, Λ)) . Hence α ≡ H ( a ) (mod EQ λ (2 n, R, Λ)). ✷ Definition 4.13.
Let α = (cid:18) a b c d (cid:19) ∈ M r ( R ), β = (cid:18) a b c d (cid:19) ∈ M s ( R ). As before, wedefine α ⊥ β , and consider an embeddingGQ λ (2n , R , Λ) → GQ λ (2n + 2 , R , Λ) , α α ⊥ I . We denote GQ λ ( R, Λ) = ∞ ∪ n =1 GQ λ (2 n, R, Λ) and EQ λ ( R, Λ) = ∞ ∪ n =1 EQ λ (2 n, R, Λ).In view of quadratic analog of Whitehead Lemma, we have the group EQ λ ( R, Λ) coincideswith the commutator of GQ λ ( R, Λ). Therefore the groupK GQ λ ( R, Λ) := GQ λ ( R, Λ)EQ λ ( R, Λ)is well-defined. The class of a matrix α ∈ GQ λ ( R, Λ) in the group K GQ λ ( R, Λ) is denotedby [ α ]. In this way we obtain a K -functor K GQ λ acting form the category of form ringsto the category of abelian groups. emark 4.14. Likewise in the quadratic case, the kernel of the group homomorphismK GQ λ ( R [ X ] , Λ[ X ]) → K GQ λ ( R, Λ)induced from the form ring homomorphism ( R [ X ] , Λ[ X ]) → ( R, Λ); X GQ λ ( R, Λ). Since the Λ-quadratic groups are subclass of the quadratic groups, theLocal-global principle holds for Λ-quadratic groups. We use this throughout for the nextsection. 5.
Absence of torsion in NK GQ λ ( R, Λ)In this section we give the proof of Theorem 1.1 and Theorem 1.2. In [6], the proof ofthe theorem for the linear case is based on two key results, viz. the Higman linearisation,and a lemma on polynomial identity in the truncated polynomial rings. Here we recall thelemma with its proof to highlight its connection with the big Witt vectors. Recently, in [15],V. Kopeiko deduced an analog of Higman linearisation process for a subclass of the generalquadratic groups.
Definition 5.1.
For a associative ring R with unity we consider the truncated polynomialring R t = R [ X ]( X t +1 ) . Lemma 5.2. ( cf. [6] , Lemma 4.1 ) Let P ( X ) ∈ R [ X ] be any polynomial. Then the followingidentity holds in the ring R t :(1 + X r P ( X )) = (1 + X r P (0))(1 + X r +1 Q ( X )) , where r > and Q ( X ) ∈ R [ X ] , with deg( Q ( X )) < t − r . Proof : Let us write P ( X ) = a + a X + · · · + a t X t . Then we can write P ( X ) = P (0) + XP ′ ( X ) for some P ′ ( X ) ∈ R [ X ]. Now, in R t (1 + X r P ( X ))(1 + X r P (0)) − = (1 + X r P (0) + X r +1 P ′ ( X ))(1 + X r P (0)) − = 1 + X r +1 P ′ ( X )(1 − X r P (0) + X r ( P (0)) − · · · )= 1 + X r +1 Q ( X )where Q ( X ) ∈ R [ X ] with deg( Q ( X )) < t − r . Hence the lemma follows. ✷ Remark.
Iterating the above process we can write for any polynomial P ( X ) ∈ R [ X ],(1 + XP ( X )) = Π ti =1 (1 + a i X i )in R t , for some a i ∈ R . By ascending induction it will follow that the a i ’s are uniquelydetermined. In fact, if R is commutative then a i ’s are the i -th component of the ghostvector corresponding to the big Witt vector of (1 + XP ( X )) ∈ W( R ) = (1 + XR [[ X ]]) × . Fordetails see ([11], § I).
Lemma 5.3.
Let R be a ring with /k ∈ R and P ( X ) ∈ R [ X ] . Assume P (0) lies in thecenter of R . Then (1 + X r P ( X )) k r = 1 ⇒ (1 + X r P ( X )) = (1 + X r +1 Q ( X )) in the ring R t for some r > and Q ( X ) ∈ R [ X ] with deg( Q ( X )) < t − r . Following result is due to V. Kopeiko, cf. [15].
Proposition 5.4. (Higman linearisation)
Let ( R, Λ) be a form ring. Then, every elementof the group NK GQ λ ( R, Λ) has a representative of the form [ a ; b, c ] n = (cid:18) I r − aX bX − cX n I r + a ∗ X + · · · + ( a ∗ ) n X n (cid:19) ∈ GQ λ (2 r, R [ X ] , Λ[ X ]) for some positive integers r and n , where a, b, c ∈ M r ( R ) satisfy the following conditions: (1) the matrices b and ab are Hermitian and also ab = ba ∗ , the matrices c and ca are Hermitian and also ca = a ∗ c , (3) bc = a n +1 and cb = ( a ∗ ) n +1 . Corollary 5.5.
Let [ α ] ∈ NK GQ λ ( R, Λ) has the representation [ a ; b, c ] n for some a, b, c ∈ M n ( R ) according to Proposition 5.4. Then [ α ] = [ H (I r − aX )] in NK GQ λ ( R, Λ) if (I r − aX ) ∈ GL r ( R ) . Proof : By Lemma 4.12 we have [ a ; b, c ] n ≡ H (I r − aX ) (mod EQ λ (2 r, R [ X ] , Λ[ X ])).Hence we have [ α ] = [ H (I r − aX )] in NK GQ λ ( R, Λ). ✷ Proof of Theorem 1.1:
By the Theorem 5.4, we have [ α ] = [[ a ; b, c ] n ] for some a, b, c ∈ M s ( R ) and for some naturalnumbers n and s . Note that in the Step 1 of the Proposition 5.4, the invertibility of thefirst corner of the matrix α will not be changed during the linearisation process. Also theinvertibility of the first corner is preserved in the remaining steps of the Proposition 5.4.Therefore since the first corner matrix A ( X ) ∈ GL r ( R [ X ]), then we have (I s − aX ) ∈ GL s ( R [ X ]). By Corollary 5.5, we have [ α ] = [ H (I s − aX )]. Now let [ α ] be a k -torsion. Thenwe have [ H (I r − aX )] is a k -torsion. Since (I r − aX ) is invertible, it follows that a is nilpotent.Let a t +1 = 0. Since [(I r − aX )] k = [I] in K GQ λ ( R [ X ] , Λ[ X ]), then by arguing as given in[7], we have [I r − aX ] = [ I ] in K GQ λ ( R [ X ] , Λ[ X ]). This completes the proof. ✷ Proof of Theorem 1.2 – (Graded Version):
Consider the ring homomorphism f : R → R [ X ] defined by f ( a + a + . . . ) = a + a X + . . . . Then [(I + N ) k ] = [I] ⇒ f ([I + N ] k ) = [ f (I + N )] k = [I] ⇒ [(I + N + N X + · · · + N r X r )] k = [I] . Let m be a maximal ideal in R . By Theorem 1.1, we have[(I + N + N X + · · · + N r X r )] = [I]in NK GQ λ (( R ) m , Λ m ). Hence by using the local-global principle we conclude[(I + N )] = [I + N ]in NK GQ λ ( R, Λ), as required. ✷ Acknowledgment:
We thank Sergey Sinchuk and V. Kopeiko for many useful discussions.
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