Ravi A. Rao

Local-global principle for transvection groups

In this article we extend the validity of Suslins 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 and constant local hyperbolic rank > 2. In Suslins 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.

On Stably Free Modules over Affine Algebras

If X is a smooth affine variety of dimension d over an algebraically closed field k, and if (d−1)!∈k× then any stably trivial vector bundle of rank (d−1) over X is trivial. The hypothesis that X is smooth can be weakened to X is normal if d≥4.

A structure theorem for the elementary unimodular vector group

Given a pair of vectors v,w ∈ R r+1 with = v · w T = 1, A. Suslin constructed a matrix S r (v,w) ∈ Sl 2 r(R). We study the subgroup SUm r (R) generated by these matrices, and its (elementary) subgroup EUm r (R) generated by the matrices S r (e 1 e, e 1 e T-1 ), for e ∈ E r+1 (R). The basic calculus for EUm r (R) is developed via a key lemma, and a fundamental property of Suslin matrices is derived.

A stably elementary homotopy

If R is an affine algebra of dimension d over a perfect C 1 field and σ ∈ SL d+1 (R) is a stably elementary matrix, we show that there is a stably elementary matrix σ(X) ∈ SL d+1 (R[X]) with σ(1) = σ and σ(0) = I d+1 .

On completing unimodular polynomial vectors of length three

It is shown that if R is a local ring of dimension three, with ? € R , then a polynomial three vector (vQ(X), vx(X), v2(X)) over R[X] can be completed to an invertible matrix if and only if it is unimodular. In particular, if 1/3! € J! , then every stably free projective R[XX, ... , .Y„]-module is free.

SOME REMARKS ON SYMPLECTIC INJECTIVE STABILITY

It is shown that if A is an affine algebra of odd dimension d over an infinite field of cohomological dimension at most one, with (d + 1)!A = A, and with 4|(d - 1), then Um d+1 (A) = e 1 Sp d+1 (A). As a consequence it is shown that if A is a non-singular affine algebra of dimension d over an infinite field of cohomological dimension at most one, and d!A = A, and 4|d, then Sp d (A) ∩ ESp d+2 (A) = ESp d (A). This result is a partial analogue for even-dimensional algebras of the one obtained by Basu and Rao for odd-dimensional algebras earlier.

The orbit set and the orthogonal quotient

We study the injectivity of the map (of pointed sets) Umr+1(R)/Er+1(R) → SO2(r+1)(R)/EO2(r+1)(R), which we had constructed in [12]. We obtain a necessary and sufficient condition for the kernel to be non-trivial under some additional assumptions on r.

Stability results for projective modules over Rees algebras

We provide a class of commutative Noetherian domains R of dimension d such that every finitely generated projective R -module P of rank d splits off a free summand of rank one. On this class, we also show that P is cancellative. At the end we give some applications to the number of generators of a module over the Rees algebras.

A Study of Suslin Matrices: Their Properties and Uses

We describe recent developments in the study of unimodular rows over a commutative ring by studying the associated group \(SUm_r(R)\), generated by Suslin matrices associated to a pair of rows v, w with \(\langle v, w \rangle = 1\). We also sketch some futuristic developments which we expect on how this association will help to solve a long standing conjecture of Bass–Suslin (initially in the metastable range, and later the entire expectation) regarding the completion of unimodular polynomial rows over a local ring, as well as how this study will lead to understanding the geometry and physics of the orbit space of unimodular rows under the action of the elementary subgroup.