Raja Sridharan

The Euler Class Group of a Noetherian Ring

For a commutative Noetherian ring A of finite Krull dimension containing the field of rational numbers, an Abelian group called the Euler class group is defined. An element of this group is attached to a projective A-module of rank = dimA and it is shown that the vanishing of this element is necessary and sufficient for P to split off a free summand of rank 1. As one of the applications of this result, it is shown that for any n-dimensional real affine domain, a projective module of rank n (with trivial determinant), all of whose generic sections have n generated vanishing ideals, necessarily splits off a free direct summand of rank 1.

A note on the fundamental theorem of projective geometry

Keywords: foundations of geometry Reference CMA-ARTICLE-1969-001doi:10.1007/BF02564531 Record created on 2008-12-16, modified on 2016-08-08

Ketu and the second invariant of a quadratic space

Keywords: Chern classes ; quadratic forms ; Clifford invariant ; Giffen invariant Reference CMA-ARTICLE-1993-001doi:10.1007/BF00961215 Record created on 2008-12-16, modified on 2016-08-08

The Euler class groups of polynomial rings and unimodular elements in projective modules

Abstract Let A be a Noetherian ring of Krull dimension n containing the field of rationals. Let P be a projective A [ T ]-module of rank n with trivial determinant such that the A -module P / TP has a free summand of rank one. It is proved that if n is even, then P has a free summand of rank one if it maps onto an ideal I of A [ T ] of height n which is generated by n elements.

Indecomposable quadratic bundles of rank 4n over the real affine plane

Keywords: real affine plane ; quadratic space extended from subring ; hermitian bundle ; non extended quadratic spaces ; indecomposable quadratic bundles ; indecomposable quadratic spaces Reference CMA-ARTICLE-1983-001doi:10.1007/BF02095998 Record created on 2008-12-16, modified on 2016-08-08

A local global principle for quadratic forms over polynomial rings

If, further, K has no real completion (i.e., K is either a totally imaginary number field or a function field), then every quadratic space over K of rank > 5 is isotropic and in view of a theorem of Ojanguren [5] it follows that any quadratic space of rank > 5 over K[X, ,..., X,] is extended from K so that Theorem 1, in this case, holds for all quadratic spaces, without restriction on the rank. Our proof of Theorem 1 uses the following result which seems to be of some independent interest.

Maslov Index and a Central Extension of the Symplectic Group

In this paper we show that over any field K of characteristic different from 2, the Maslov index gives rise to a 2-cocycle on the stable symplectic group with values in the Witt group. We also show that this cocycle admits a natural reduction to I 2 .K/ and that the induced natural homomorph- ism fromK2Sp.K/! I 2 .K/ is indeed the homomorphism given by the symplectic symbolfx;yg

Projective generation of curves in polynomial extensions of an affine domain (II)

Let Abe an affine domain of dimension nover an algebraically closed field kof characteristic 0. Let I⊂ A[T]be a local complete intersection ideal of height nsuch that I/I2 is generated by n elements. It is proved that there exists a projective A[T]module Pof rank nsuch that Iis a surjective image of P.

Graded Witt Ring and Unramified Cohomology

It is proved that for a smooth alfine curve X over a local ring or global field, the graded Witt ring of X is isomorphic to the graded unramified cohomology ring of X. If X is projective and has a rational point, the same result holds if and only if every quadratic space defined on the complement of a rational point extends to X. Such an extension is possible, for instance, if the canonical line bundle on X is a square in Pic X.

A Note on Purely Inseparable Extensions.

Let K be a field of characteristic p > 0 and let L be a subfield of K such that K/L is a finite, purely inseparable extension of exponent 1. A Galois theory for such extensions using the concept of restricted Lie ring was initiated by Jacobson [4]. This theory gives a bijective correspondence between restricted Lie subrings of DerLK (the Lie ring o f all derivations of K vanishing on L) and subextensions of K/L. M. Gerstenhaber [1] showed that there is a bijection already between restricted subspaces of Der L K and subextensions of K/L and as a consequence deduced that every restricted subspace of Der L K is actually a Lie subring. In a subsequent paper [2], he generalized this Galois theory to the infinite case and showed that with the natural Krull-topology on DerK, there is a bijective correspondence between closed restricted subspaces o f DerK and subfields o f K containing K p. His proof, however, seems to be incomplete since Lemma 4 of [2] which is used to prove the theorem is incorrect. 1) It is probable that this trouble may be circumvented and a suitably modified proof still holds. However, the aim of this note is to give another p roof (Theorem 2) of the Galois correspondence, which, even in the finite dimensional case is different from the existing proofs in the literature (for example [3]). We also show (Theorem 1) that any restricted subspace of D e r K (whether closed or not) is a Lie subring, by giving more or less an explicit formula for the commuta tor o f two derivations. In this paper, we use the notat ion and terminology of [2]. The authors have pleasure in thanking Prof. B. Eckmann for his keen interest in this work.