aa r X i v : . [ m a t h . K T ] M a y A NOTE ON GENERAL QUADRATIC GROUPS
RABEYA BASU
INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH, PUNE, INDIA
Abstract:
We deduce an analogue of Quillen–Suslin’s local-global principle for thetransvection subgroups of the general quadratic (Bak’s unitary) groups. As an applicationwe revisit the result of Bak–Petrov–Tang on injective stabilization for the K -functor ofthe general quadratic groups. Introduction
In this article we are going to discuss three major problems in algebraic K-theory studied rigorously during 1950’s to 1970’s, viz. stabilization problems forthe K -funtor, Quillen–Suslin’s local-global principle, and Bak’s unitary groupsover form rings.Let us begin with the stabilization problem for the K -functor of the generallinear groups initiated by Bass–Milnor–Serre during mid 60’s. For details, see [6].For a commutative ring R with identity, let GL n ( R ) be the general linear group,and E n ( R ) the subgroup generated by elementary matrices. They had studied thefollowing sequence of mapsGL n ( R )E n ( R ) −→ GL n +1 ( R )E n +1 ( R ) −→ GL n +2 ( R )E n +2 ( R ) −→ · · · · · · = K ( R )and posed the problem that when does the natural map stabilize? That time itwas not known that the elementary subgroup is a normal subgroup in the generallinear group for size n ≥
3. They showed that the above map is surjective for n ≥ d + 1, and injective for n ≥ d + 3, where d is the Krull dimension of R , andconjectured that it must be injective for n ≥ d +2. In 1969, L.N. Vaserstein provedthe conjecture for an associative ring with identity, where d is the stable rank of thering ( cf. [25]). In 1974, he generalized his result for projective modules. After that,in [26], he had studied the orthogonal and the unitary K -functors, and deducedanalogue results for those groups, and showed that for the non-linear cases abovementioned natural map is surjective for n ≥ d + 2 and injectivity for n ≥ d + 4.It was a period when people were working on Serre’s problem on projectivemodules, which states that the finitely generated projective modules over polyno-mial rings over a field are free. If the number of variables n in the polynomial ringis 1, then the affirmative answer follows from a classical result in commutative lgebra. The first non-trivial case n = 2 was proved affirmatively by C.S. Seshadriin 1958. For details see [19]. After 16 years, in 1974, Murthy–Swan–Towber (thenRoitman, Vaserstein et al. , for details cf. [19]) proved the result for algebraicallyclosed fields. The general case was proved by Daniel Quillen and Andrei Suslinindependently in 1976. Suslin proved that for an associative ring A , which is finiteover its center, the elementary subgroup E n ( A ) is a normal subgroup of GL n ( A )for n ≥
3. It was first appeared in a paper by M.S. Tulenbaev, for details see [24].Quillen, in his proof, introduced a localization technique which was one of the mainingredient for the proof of Serre’s problem (now widely known as Quillen–SuslinTheorem). Shortly after the original proof, motivated by Quillen’s idea, Suslinintroduced the following matrix theoretic version of the local-global principle. Weare stating the theorem for a commutative ring with identity, where Max( R ) is themaximum spectrum of the ring R . The statement is true for almost commutativerings, i.e. rings which are finite over its center. Suslin’s Local-Global Principle:
Let R be a commutative ring with iden-tity and α ( X ) ∈ GL n ( R [ X ]) with α (0) = I n . If α m ( X ) ∈ E n ( R m [ X ]) for everymaximal ideal m ∈ Max( R ) , then α ( X ) ∈ E n ( R [ X ]) . In [12], we have deduced an analogue of the above statement for the transvectionsubgroups of the full automorphism groups of global rank at least 1 in the linearcase, and at least 2 for the symplectic and orthogonal cases. In this article wegeneralize the statement for the general quadratic (Bak’s unitary) groups. Themain aim of this article is to show that using this result one can generalize theoremson free modules to the classical modules. We shall discuss such technique forthe stabilization problems for the K -functor. In similar manner one can alsogeneralize results for unstable K -groups ( cf. [5], [12]), local-global principle forthe commutator subgroups ( cf. [10]), and may be for the results on congruencesubgroup problems, etc.Let us briefly discuss the historical aspects of the general quadratic group de-fined with the concept of form ring introduced by Antony Bak in his Ph.D. thesis( cf. [1]) around 1967. For details see [1] and [15].We know that a quadratic form on an R -module M is a map q : M → R suchthat(1) q ( ax ) = a q ( x ), a ∈ R , x ∈ M ,(2) B q : M × M → A defined by B q ( x, y ) = q ( x + y ) − q ( x ) − q ( y ) is bilinear(symmetric).The map B q is called the bilinear form associated to q . If 2 ∈ R ∗ , then for all x ∈ M we get q ( x ) = B q ( x, x ). The pair ( M, q ) is called the quadratic R -module.Suppose B : V × V → K is a bilinear form on a vector space V over a field K .We say B is(1) symmetric if B ( u, v ) = B ( v, u ),(2) anti-symmetric if B ( u, v ) = − B ( v, u ), and(3) symplectic (alternating) if B ( u, u ) = 0 for all u ∈ V . ow, symplectic ⇒ anti-symmetric, and if char( K ) = 2, then symplectic ⇔ anti-symmetric. On the other hand, if char( K ) = 2, then anti-symmetric ⇔ symmetric.Also, if char( K ) = 2, then q is a quadratic form if and only if it is symmetricbilinear form B ( u, v ) = q ( u + v ) − q ( u ) − q ( v ) , as we get B ( u, u ) = q ( u ). Thestudy of Dickson–Dieudonne shows that the cases char( K ) = 2 and char( K ) = 2differs even at the level of definition. During 1950’s–1970’s it was a problem thathow to generalize classical groups by constructing theory which does not dependon the invertibility of 2. In 1967, A. Bak came up with the following:First of all, a classical group should be considered as preserving a pair of forms( B, q ). Secondly, a quadratic form q should take its value not in the ring R , butin its factor group R/ Λ, where Λ is certain additive subgroup of R , the so called form parameter .In his seminal work “K-Theory of Forms”, Bak generalized many results forthe general quadratic groups which were already known for the general linear,symplectic and orthogonal groups. In 2003, Bak–Petrov–Tang proved that theinjective stabilization bound for the K -functor of the general quadratic groupsover form rings is also 2 d + 4, like traditional classical groups of even size, ( cf. [4]).Later Petrov in his Ph.D Thesis ( cf. [21]), and Tang (unpublished) independentlystudied the result for the classical modules. But none of those works are published.In this article we revisit their results for the general quadratic modules, as anapplication of the local-global principle for the transvection subgroups. But, eventhough the local-global principle holds for the module finite rings, our method isapplicable only for the commutative rings. On the other hand, it is possible toapply this method to deduce analogue results for any other kind of classical groups,in particular, for the Bak–Petrov’s groups ( cf. [20]), where we have analogue local-global principle. In this connection, we mention that for the structure of unstableK -groups of the general quadratic groups, we refer [12] and [14]. In both thepapers it has been shown that the unstable K -groups are nilpotent-by-abelianfor n ≥
6, which generalizes the result of Bak for the general linear groups in [2].Following our method one can generalize that result for the module case. i.e. itcan be shown that the unstable K -groups for the (extended) general quadraticmodules with rank ≥ Preliminaries
In this section we shall recall necessary definitions.
Form Rings:
Let − : R → R , defined by a a , be an involution on R , and λ ∈ C ( R ) = center of R such that λλ = 1. We define two additive subgroups of R Λ max = { a ∈ R | a = − λa } & Λ min = { a − λa | a ∈ R } . One checks that Λ max and Λ min are closed under the conjugation operation a xax for any x ∈ R . A λ - form parameter on R is an additive subgroup Λ of R suchthat Λ min ⊆ Λ ⊆ Λ max , and x Λ x ⊆ Λ for all x ∈ R . A pair ( R, Λ) is called a formring . eneral Quadratic Groups: Let V be a right R -module. By GL( V ) we denote the group of all R -linearautomorphisms of V . Throughout the paper we shall consider V as a projective R -module. To define the general quadratic module we need following definitions: Definition 2.1. A sesquilinear form is a map f : V × V → R such that f ( ua, vb ) = af ( u, v ) b for all u, v ∈ V and a, b ∈ R . Definition 2.2.
A Λ- quadratic form on V is a map q : V → R/ Λ such that q ( v ) = f ( v, v ) + Λ. Definition 2.3. An associated λ -Hermitian form is a map h : V × V → R withthe property h ( u, v ) = f ( u, v ) + λf ( v, u ). Definition 2.4. A quadratic module over ( R, Λ) is a triple (
V, h, q ).The λ -Hermitian form h : V × V → R induces a map V → Hom R ( V, R ), givenby v h ( v, − ). We say that V is non-singular if V is a projective R -module andthe Hermitian form h is non-singular. Definition 2.5.
A morphism of quadratic modules over ( R, Λ) is a map µ :( V, q, Λ) → ( V ′ , q ′ , Λ ′ ) such that µ : V → V ′ is R -linear, µ ( λ ) = λ ′ , and µ (Λ) = Λ ′ . General Quadratic (Bak’s Unitary) Groups:
Let (
V, q, h ) be non-singularquadratic module over ( R, Λ). We define the general quadratic group as follows:GQ(
V, q, h ) = { α ∈ GL( V ) | h ( αu, αv ) = h ( u, v ) , q ( αu ) = q ( v ) } . i.e. the group consisting of all automorphisms which fixes the λ -Hermitian formand the Λ-quadratic form. One observes that the traditional classical groups arethe special cases of Bak’s unitary groups. The central concept is “form parameter”due to Bak. Earlier version is due to K. McCrimmon which plays an importantrole in his classification theory of Jordan Algebras. He defined for the wider classof alternative rings (not just associative rings), but for associative rings it is aspecial case of Bak’s concept. For details, see N. Jacobson, Lectures on QuadraticJordan Algebras, TIFR, Bombay 1969, ( cf. [17]) and [16]. Free Case:
Let V be a non-singular free right R -module of rank 2 n withordered basis e , e , . . . , e n , e − n , . . . , e − , e − . Consider the sesquilinear form f : V × V → R defined by f ( u, v ) = u v − + · · · + u n v − n . Let h be the λ -Hermitianform, and q be the Λ-quadratic form defined by f . We get h ( u, v ) = u v − + · · · + u n v − n + λu − n v n + · · · + λu − v ,q ( u ) = Λ + u u − + · · · + u n u − n . Using the above basis we can identify GQ(
V, h, q ) with a subgroup of GL n ( R, Λ)of rank 2 n , say GQ n ( R, Λ). HenceGQ n ( R, Λ) = { σ ∈ GL n ( R, Λ) | σψ n σ = ψ n } , where ψ n = (cid:18) λ I n I n (cid:19) . nitary Transvections EQ(V) : Let (
V, h, q ) be a quadratic module over ( R, Λ).Let u, v ∈ V and a ∈ R be such that f ( u, u ) ∈ Λ, h ( u, v ) = 0 and f ( v, v ) = a modulo Λ. Then we quote the definition of unitary transvection from ( § σ = σ u,a,v : M → M defined by σ ( x ) = x + h ( v, x ) − vλh ( u, x ) − uλah ( u, x ) . Unitary Transvections
EQ(M) in M = V ⊥ H ( P ) : Let P be a projective R -module of rank at least one, and H ( P ) the hyperbolic space. Let x = ( v, p, q ) ∈ M for some v ∈ V , p ∈ P , and q ∈ P ∗ . For any element p ∈ P , w ∈ V and a ∈ A such that a = f ( w , w ) modulo Λ, the above conditions hold, and hence we candefine σ p ,a ,w as follows: σ p ,a ,w ( x ) = x + p h ( w , x ) − w λh ( p , x ) − p λa h ( p , x ) . Elementary Quadratic Matrices: (Free Case)
Let ρ be the permutation, defined by ρ ( i ) = n + i for i = 1 , . . . , n . Let e i denotethe column vector with 1 in the i -th position and 0’s elsewhere. Let e ij be thematrix with 1 in the ij -th position and 0’s elsewhere. 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 ) = ( I n + ae iρ ( j ) − λae jρ ( i ) for i = j I n + ae ρ ( i ) j for i = j,ql ij ( a ) = ( 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 weget a = − λa , and hence it forces that a ∈ Λ max ( R ). One checks that these abovematrices belong to GQ n ( R, Λ); cf. [1].) n-th Elementary Quadratic Group EQ 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 . Stabilization map:
There are standard embeddings:GQ n ( R, Λ) −→ GQ n +2 ( R, Λ)given by (cid:18) a bc d (cid:19) a b c d or a b
00 1 0 0 c d
00 0 0 1 or a b c d
00 0 0 1 . Hence we have GQ( R, Λ) = lim −→ GQ n ( R, Λ).It is clear that the stabilization map takes generators of EQ n ( R, Λ) to thegenerators of EQ n +2 ( R, Λ). ommutator Relations: There are standard formulas for the commutatorsbetween quadratic elementary matrices. For details, we refer [1] (Lemma 3.16). Inlater sections there are repeated use of those relations.
Remark: If M = R n , then under the choice of the above basis we get EQ( M ) =EQ n ( R, Λ). ( cf. proof of Lemma 2.20 in [5]). Hence we haveEQ( R, Λ) = lim −→ EQ n ( R, Λ)Using analogue of the Whitehead Lemma for the general quadratic groups ( cf. [1])due to Bak, one gets[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. For classical modules we replace ( R, Λ) by M = V ⊕ H ( P ).3. Suslin’s Local-Global Principle for Tansvection Subgroups
In this section we prove analogue of Quillen–Suslin’s local-global principle forthe transvection subgroups of the general quadratic groups. We start with thefollowing splitting lemma.
Lemma 3.1. ( cf. pg. 43-44, Lemma 3.16, [1]) Let q ij denote any one of theelementary generator of qε ij , ql ij and qr ij in GQ n ( R, Λ) . Then, for all x, y ∈ R , q ij ( x + y ) = q ij ( x ) q ij ( y ) . We shall need following standard fact.
Lemma 3.2.
Let G be a group, and a i , b i ∈ G , for i = 1 , . . . , n . Then for r i = i Π j =1 a j , we have n Π i =1 a i b i = n Π i =1 r i b i r − i n Π i =1 a i . Notation 3.3.
By GQ n ( R [ X ] , Λ[ X ] , ( X )) we shall mean the group of all invert-ible matrices in GQ n ( R [ X ] , Λ[ X ]) which are I n modulo ( X ). Let Λ[ X ] denotethe λ -form parameter on R [ X ] induced from ( R, Λ), i.e. , λ -form parameter on R [ X ] generated by Λ, i.e. , the smallest form parameter on R [ X ] containing Λ. LetΛ s denote the λ -form parameter on R s induced from ( R, Λ).
Lemma 3.4.
The group GQ n ( R [ X ] , Λ[ X ] , ( X )) ∩ EQ n ( R [ X ] , Λ[ X ]) is generatedby the elements of the types εq ij ( f ( X )) ε − , where ε ∈ EQ n ( R, Λ) , f ( X ) ∈ R [ X ] and q ij ( f ( X )) are congruent to I n modulo ( X ) . roof. Let α ( X ) ∈ EQ n ( R [ X ] , Λ[ X ]) be such that α ( X ) = I n modulo ( X ).Then we can write α ( X ) as a product of elements of the form q ij ( f ( X )), where f ( X ) is a polynomial in R [ X ]. We write each f ( X ) as a sum of a constant term anda polynomial which is identity modulo ( X ). Hence by using the splitting propertydescribed in Lemma 3.1 each elementary generator q ij ( f ( X )) can be written as aproduct of two such elementary generators with the left one defined on R and theright one defined on R [ X ] which is congruent to I n modulo ( X ).Therefore, we can write α ( X ) as a product of elementary generators of the form q ij ( f (0)) q ij ( Xg ( X )) for some g ( X ) ∈ R [ X ] with g (0) ∈ R. Now the result follows by using the identity described in Lemma 3.2. ✷ By repeated application of commutator formulas stated in ([1], pg. 43-44,Lemma 3.16) one gets the following lemma.
Lemma 3.5.
Suppose ϑ is an elementary generator of the general quadratic group GQ n ( R [ X ] , Λ[ X ]) . Let ϑ be congruent to identity modulo ( X m ) , for m > .Then, if we conjugate ϑ with an elementary generator of the general quadraticgroup GQ n ( R, Λ) , we get the final matrix as a product of elementary generatorsof the general quadratic group GQ n ( R [ X ] , Λ[ X ]) each of which is congruent toidentity modulo ( X m ) . Corollary 3.6.
In Lemma 3.5 we can take ϑ as a product of elementary generatorsof the general quadratic group GQ n ( R [ X ] , Λ[ X ]) . Let us recall following useful and well known fact. We use this fact for the proofof dilatation lemma . Lemma 3.7. ( cf., [3] , Lemma 5.1 ) Let A be Noetherian ring and s ∈ A . Let s ∈ A and s = 0 . Then there exists a natural number k such that the homomorphism G( A, s k A, s k Λ) → G( A s , Λ s ) ( induced by the localization homomorphism A → A s ) is injective. We recall that one has any module finite ring R as a direct limit of its finitelygenerated subrings. Also, G( R, Λ) = lim −→ G( R i , Λ i ), where the limit is taken overall finitely generated subring of R . Thus, one may assume that C ( R ) is Noetherian.For the rest of this section we shall consider module finite rings ( R, Λ) with identity.In [12], we have given a proof of the dialation lemma for the general Hermitiangroups. For the general quadratic modules, the proof is similar, in fact easier.We are giving the sketch of the proof, as it is not clearly written any availableliterature. It is almost done in [21] for the odd unitary groups which contains thegeneral quadratic groups. For the hyperbolic unitary groups it is done in [3].
Lemma 3.8. (Dilation Lemma: Free Case)
Let α ( X ) ∈ GQ n ( R [ X ] , Λ[ X ]) ,with α (0) = I n . If α s ( X ) ∈ EQ n ( R s [ X ] , Λ s [ X ]) for some non-nilpotent s ∈ R ,then α ( bX ) ∈ EQ n ( R [ X ] , Λ[ X ]) for b ∈ ( s l )C( R ) , l ≫ . ( Actually, we mean there exists some β ( X ) ∈ EQ n ( R [ X ] , Λ[ X ]) such that β (0) = I n and β s ( X ) = α s ( bX )) . roof. Given that α s ( X ) ∈ EQ n ( R s [ X ] , Λ s [ X ]). Since α (0) = I , us-ing Lemma 3.4 we can write α s ( X ) as a product of the matrices of the form εq ij ( h ( X )) ε − , where ε ∈ EQ n ( R s , Λ s ), h ( X ) ∈ R s [ X ] with q ij ( h ( X )) congruentto I n modulo ( X ). Applying the homomorphism X XT d , for d ≫
0, fromthe polynomial ring R [ X ] to the polynomial ring R [ X, T ], we look on α ( XT d ).Note that R s [ X, T ] ∼ = ( R s [ X ])[ T ]. As C ( R ) is Noetherian, it follows from Lemma3.7 and Corollary 3.6 that over the ring ( R s [ X ])[ T ] we can write α s ( XT d ) as aproduct of elementary generators of the general quadratic group such that eachof those elementary generator is congruent to identity modulo ( T ). Let l be themaximum of the powers occurring in the denominators of those elementary gener-ators. Again, as C ( R ) is Noetherian, by applying the homomorphism T s m T for m ≥ l it follows from Lemma 3.6 that over the ring R [ X, T ] we can write α s ( XT d ) as a product of elementary generators of general quadratic group suchthat each of those elementary generator is congruent to identity modulo ( T ). i.e. there exists some β ( X, T ) ∈ EQ n ( R [ X, T ] , Λ[ X, T ]) such that β (0 ,
0) = I n and β s ( X, T ) = α s ( bXT d ), for some b ∈ ( s l ) C ( R ). Finally, the result follows by putting T = 1. ✷ Lemma 3.9. (Dilation Lemma: Module Case)
Assume M = V ⊕ H ( R ) ,where V is a right R -module, and H ( R ) is the usual hyperbolic space. Let rank of M is n . Let us denote M [ X ] = ( V ⊥ H ( R ))[ X ] . Let s ∈ R be such that V s is free.Let α ( X ) ∈ GQ(M[X]) with α (0) = Id . Suppose α s ( X ) ∈ EQ n ( R s [ X ] , Λ s [ X ]) .Then there exists b α ( X ) ∈ EQ( M [ X ]) and l > such that b α ( X ) localizes to α ( bX ) for some b ∈ ( s l ) and b α (0) = Id . Proof.
Arguing as in the proof of Proposition 3.1 in [5] we can deduce theproof. One observes the repeated use of the commutator formulas stated in pg.43, [1]. ✷ Lemma 3.10. (Local Global Principle for Tansvection Subgroups)
Let M = V ⊥ H ( R ) , where V is as above. Let M [ X ] = ( V ⊥ H ( R ))[ X ] . Let α ( X ) ∈ GQ( M [ X ]) with α (0) = Id . Suppose α m ( X ) ∈ EQ n ( R m [ X ] , Λ m [ X ]) for everymaximal ideal m in R . Then α ( X ) ∈ EQ( M [ X ]) . Proof.
Since α m ( X ) ∈ EQ n ( R m [ X ] , Λ m [ X ]) for all m ∈ Max( C ( R )), for each m there exists s ∈ C ( R ) − m such that α s ( X ) ∈ EQ n ( R s [ X ] , Λ s [ X ]). We considera finite cover of C ( R ), say s + · · · + s r = 1. Following Suslin’s trick, let θ ( X, T ) = α s ( X + T ) α s ( T ) − . Then θ ( X, T ) ∈ EQ n (( R s [ T ])[ X ] , (Λ s [ T ])[ X ]) and θ (0 , T ) = I n .Since for l ≫ h s l , . . . , s lr i = R , we chose b , b , . . . , b r ∈ C ( R ), with b i ∈ ( s l ) C ( R ) , l ≫ b + · · · + b r = 1. Then by dilation lemma(Lemma 3.9), applied with base ring R [ T ], there exists β i ( X ) ∈ EQ( M [ X ] , Λ[ X ])(considering M as an R [ T ] module) such that β is i ( X ) = θ ( b i X, T ). Therefore, r Π i =1 β i ( X ) ∈ EQ( M [ X ] , Λ[ X ]). But, α s ··· s r ( X ) = (cid:18) r − Π i =1 θ s ··· ˆ s i ··· s r ( b i X, T ) | T = b i +1 X + ··· + b r X (cid:19) θ s ······ s r − ( b r X, . bserve that as a consequence of the Lemma 3.7 it follows that the mapEQ( R, s k R, s k Λ) → E( R s , Λ s )for k ∈ N , is injective for each s = s i . As α (0) = I n , we conclude α ( X ) ∈ EQ( M [ X ]). ✷ Stabilization of K GQ We recall following result of Bak–Petrov–Tang in [4] for free modules of evensize. We are stating the theorem for commutative ring,s and for this section weshall work for commutative rings with trivial involution. We consider the underline R -module M = V ⊕ H ( R ), where V is a right R -module and H ( R ) is the hyperbolicspace with usual inner product. Theorem 4.1. ( Bak–Petrov–Tang ) Let ( R, Λ) be a commutative form ring withKrull dimention d . If n ≥ max(6 , d +2) , then K , n GQ( R, Λ) is a group, the sta-bilization maps K , n − GQ( R, Λ) −→ K , n GQ( R, Λ) is surjective, and the maps K , n GQ( R, Λ) −→ K , n +2 GQ( R, Λ) are isomorphism of groups. As an application of our local-global principle for the transvection subgroup(Theorem 3.10) we generalize the above stabilization result for the general qua-dratic modules. Let us first recall the following key lemma of Vaserstein ( cf.
Corollary 5.4, [27]). A proof for the absolute case is nicely written in an unpub-lished paper by Maria Saliani ( cf.
Theorem 6.1, [22]). The relative case follows byusing the double ring concept, as it is done in [13] (Theorem 4.1).
Lemma 4.2.
Let ( R, Λ) be an associative ring with identity with Krull dimension d , and I be an ideal of R . Let M = V ⊕ H ( R ) and ( M, h, q ) a general quadraticmodule of rank n ≥ max(6 , d + 2) . Then the group of elementary transvection EQ( M ⊥ H ( R ) , I ) acts transitively on the set Um( M ⊥ H ( R ) , I ) of unimodularelements which are congruent to (0 , . . . , , . . . , modulo I . In other words, GQ((M ⊥ H (R)) , I) = EQ((M ⊥ H (R)) , I)GQ(V , I) . Also, we have the standard fact.
Proposition 4.3.
Let M and V be as above, and ∆ ∈ GQ(M ⊥ H (R)) and n ≥ .If ∆ e n = e n , then ∆ ∈ EQ(M ⊥ H (R))GQ(M) . Proof.
Proof goes as in Lemma 3.6 in [11]. ✷ We deduce the following stabilization result for the general quadratic modules:
Theorem 4.4.
Let ( R, Λ) be an associative ring with identity with Krull dimension d . Let M = V ⊕ H ( R ) and ( M, h, q ) a general quadratic module of rank n ≥ max(6 , d + 2) . Then, the stabilization map K , n GQ( M ) −→ K , n +2 GQ( M ⊥ H ( R )) is isomorphism of groups. roof. It follows from the above result of Bak–Petrov–Tang (Theorem 4.1)that for every localization at maximal ideals of R the map is surjective at thelevel 2 d + 2. Hence the surjectivity follows by local-global principle, and by thesurjectivity result of Bak–Petrov–Tang.In view of their result, it is enough to prove the injectivity for 2 n = 2 d +2. Let n = 2 n + 2. Suppose γ ∈ GQ(M) is such that e γ = γ ⊥ Id lies inEQ( M ⊥ H ( R )). Let φ ( X ) be the isotopy between e γ and identity. Viewing φ ( X ) as a matrix it follows that v ( X ) = φ ( X ) e n is in Um( M ⊥ H ( R ))[ X ]), and v ( X ) = e n modulo ( X − X ). It follows from Lemma 4.2 that over R [ X ] we get σ ( X ) ∈ EQ(( M ⊥ H ( R ))[ X ]) such that σ ( X ) t v ( X ) = e n and σ ( X ) = Id modulo( X − X ). Therefore, σ ( X ) t φ ( X ) e n = e n . Then by Lemma 4.3 over R [ X ] wecan write σ ( X ) t φ ( X ) = ψ ( X ) e φ ( X ) for some ψ ( X ) ∈ EQ(( M ⊥ H ( R ))[ X ]) and e φ ( X ) ∈ GQ(M[X]).Since σ ( X ) = Id modulo ( X − X ), e φ ( X ) is an isotopy between γ and identity.Therefore, after localization at a maximal ideal m , the image e φ m ( X ) is stablyelementary for every maximal ideal m in Max( R ). Hence by the stability theoremof Bak–Petrov–Tang (Theorem 4.1) for the free modules, it follows that e φ m ( X ) ∈ EQ(2 d + 2 , R m [ X ]). Since φ (0) = Id, we get e φ (0) = Id. Hence by the abovelocal-global principle for the transvection subgroups (Theorem 3.10) it followsthat e φ ( X ) ∈ EQ( M [ X ]). Hence γ = e φ (1) ∈ EQ( M ). This proves that the map atthe level 2 d + 4 is injective. Acknowledgment:
My sincere thanks to Sergey Sinchuk for many useful dis-cussions. I am highly grateful to Nikolai Vavilov and Alexei Stepanov for their kindinvitation to visit Chebyshev Laboratory, St. Petersburg, Russia, where I couldfinish my long pending work on this manuscript. I am thankful to St. PetersburgState University and IISER Pune for supporting my visit.
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