Local-global principle for transvection groups

A. Bak, Rabeya Basu, Ravi A. Rao
2010 Proceedings of the American Mathematical Society  
In this article we extend the validity of Suslin's Local-Global Principle for the elementary transvection subgroup of the general linear group GL n (R), the symplectic group Sp 2n (R), and the orthogonal group O 2n (R), where n > 2, to a Local-Global Principle for the elementary transvection subgroup of the automorphism group Aut(P ) of either a projective module P of global rank > 0 and constant local rank > 2, or of a nonsingular symplectic or orthogonal module P of global hyperbolic rank > 0
more » ... hyperbolic rank > 0 and constant local hyperbolic rank > 2. In Suslin's results, the local and global ranks are the same, because he is concerned only with free modules. Our assumption that the global (hyperbolic) rank > 0 is used to define the elementary transvection subgroups. We show further that the elementary transvection subgroup ET(P ) is normal in Aut(P ), that ET(P ) = T(P ), where the latter denotes the full transvection subgroup of Aut(P ), and that the unstable K 1 -group K 1 (Aut(P )) = Aut(P )/ET(P ) = Aut(P )/T(P ) is nilpotent by abelian, provided R has finite stable dimension. The last result extends previous ones of Bak and Hazrat for GL n (R), Sp 2n (R), and O 2n (R). An important application to the results in the current paper can be found in a preprint of Basu and Rao in which the last two named authors studied the decrease in the injective stabilization of classical modules over a nonsingular affine algebra over perfect C 1 -fields. We refer the reader to that article for more details. Suslin's local-global principle. Let R be a commutative ring with identity and let Shortly after his proof of Serre's problem, Suslin-Kopeiko in [14] established an analogue of the Local-Global Principle for the elementary subgroup of the orthogonal group. Around the same time, V.I. Kopeiko proved the analogous result for the elementary subgroup of the symplectic group. In this note we establish an analogous Local-Global Principle for the elementary transvection subgroup of the automorphism group of projective, symplectic and orthogonal modules of global rank at least 1 and local rank at least 3. All previous work on this topic assumed that the global rank is at least 3. By definition the global rank or simply rank of a finitely generated projective R-module (resp. symplectic or orthogonal R-module) is the largest integer r such that r ⊕R (resp. r ⊥ H(R)) is a direct summand (resp. orthogonal summand) of the module. H(R) denotes the hyperbolic plane. Using this principle one can generalize well known facts regarding the group GL n (R) (Sp 2n (R) or O 2n (R)) of automorphisms of the free module n ⊕R of rank n (free hyperbolic module n ⊥ H(R) of rank n) to the automorphism group of finitely generated projective (symplectic or orthogonal) modules of global rank at least 1 and satisfying the local condition mentioned above. Specifically, we shall show that the elementary transvection subgroup is normal and the full automorphism group modulo its elementary transvection subgroup is nilpotent-by-abelian whenever the stable dimension is finite. These generalize results by Suslin and Kopeiko in [9], [13], [14], Taddei in [15], the first author in [1], Vavilov and Hazrat in [8], and others. We treat the above three groups uniformly. Our main results are as follows: Let Q denote a projective, symplectic or orthogonal module of global rank ≥ 1 and satisfying the local conditions stated above. Let G(Q) = the automorphism group of Q, T(Q) = the subgroup generated by transvections, and ET(Q) = the subgroup generated by elementary transvections. Theorem 1. Let R be a commutative ring with identity and let α(X) ∈ G(Q[X]), with α(0) = I n . If α m (X) ∈ ET(Q m [X]), for every maximal ideal m ∈ Max(R), then α(X) ∈ ET(Q[X]). Theorem 2. T(Q) = ET(Q). Hence ET(Q) is a normal subgroup of G(Q). By applying the Local-Global Principle (Theorem 1) we prove Theorem 3. The factor group G(Q) ET(Q) is nilpotent-by-abelian when the stable dimension (i.e. Bass-Serre dimension) is finite. To prove the result we use the ideas of the first author in [1], where he has shown that the group GL n (R)/E n (R) is nilpotent-by-abelian for n ≥ 3, but we avoid the functorial construction of the descending central series. Preliminaries Definition 2.1. Let R be an associative ring with identity. The following condition was introduced by H. Bass: License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use LOCAL-GLOBAL PRINCIPLE FOR TRANSVECTION GROUPS 1193 (R m ) for every (a 1 , . . . , a m+1 ) ∈ Um m+1 (R), there are {x i } (1≤i≤m) ∈ R such that (a 1 + a m+1 x 1 )R + · · · + (a m + a m+1 x m )R = R. The condition (R m ) ⇒ (R m+1 ) for every m > 0. Moreover, for any n ≥ m + 1 the condition (R m ) implies (R n ) with x i = 0 for i ≥ m + 1. By stable range for an associative ring R we mean the least n such that (R n ) holds. The integer n − 1 is called the stable dimension of R and is denoted by sdim(R). Lemma 2.2 (cf. [2] ). If R is a commutative Noetherian ring with identity of Krull dimension d, then sdim(R) ≤ d. A row vector (a 1 , . . . , a n ) ∈ R n is said to be unimodular in R if n i=1 Ra i = R. The set of unimodular vectors of length n in R is denoted by Um n (R). For an ideal I, Um n (R, I) will denote the set of those unimodular vectors which are (1, 0, . . . , 0) modulo I. Definition 2.4. Let M be a finitely generated left module over a ring R. An element m in M is said to be unimodular in M if Rm ∼ = R and Rm is a direct summand of M , i.e. if there exists a finitely generated R-submodule M such that M ∼ = Rm ⊕ M . Definition 2.5. For an element m ∈ M , one can attach an ideal, called the order ideal of m in M , viz. O M (m) = {f (m)|f ∈ M * = Hom(M, R)}. Clearly, m is unimodular if and only if Rm = R and O M (m) = R. Definition 2.6. Following H. Bass ([2], pg. 167) we define a transvection of a finitely generated left R-module as follows: Let M be a finitely generated left Rmodule. Let q ∈ M and ϕ ∈ M * with ϕ(q) = 0 . An automorphism of M of the form 1 + ϕ q (defined by ϕ q (p) = ϕ(p)q, for p ∈ M ), will be called a transvection of M if either q ∈ Um(M ) or ϕ ∈ Um(M * ). We denote by Trans(M ) the subgroup of Aut(M ) generated by transvections of M . Definition 2.7. Let M be a finitely generated left R-module. The automorphisms of the form (p, a) → (p+ax, a) and (p, a) → (p, a+ψ(p)), where x ∈ M and ψ ∈ M * , are called elementary transvections of M ⊕ R. (It is easily verified that these automorphisms are transvections.) The subgroup of Trans(M ⊕ R) generated by the elementary transvections is denoted by ETrans(M ⊕ R). Definition 2.3. Definition 2.8. Let R be an associative ring with identity. To define other classical modules, we need an involutive antihomomorphism (involution, in short) * : R → R (i.e., (x − y) * = x * − y * , (xy) * = y * x * and (x * ) * = x, for any x, y ∈ R). We assume that 1 * = 1. For any left R-module M the involution induces a left module structure to the right R-module M * = Hom(M, R) given by (xf )v = (fv)x * , where v ∈ M , x ∈ R and f ∈ M * . Any right R-module can be viewed as a left Rmodule via the convention ma = a * m for m ∈ M and a ∈ R. Hence if M is a left R-module, then O M (m) has a left R-module structure with scalar multiplication given by λf (m) = f (λm).
doi:10.1090/s0002-9939-09-10198-3 fatcat:ha73jdfmsneb7ffp5tyvgak66u