Cancellation problem for projective modules over affine algebras

TL;DR

This paper investigates the cancellation property of projective modules over affine algebras, establishing conditions under which such modules are cancellative, and providing an alternative proof of a known result in algebraic geometry.

Contribution

It proves that if all finite extensions of a ring have cancellative free modules, then the projective module over the original ring is also cancellative, offering a new proof of a key algebraic geometry result.

Findings

Proves the cancellation property for projective modules under certain conditions.

Provides an alternative proof of Bhatwadekar's theorem on projective modules.

Explores the cancellative property for modules of rank d-1 over affine algebras.

Abstract

Let A be a ring of dimension d and let P be a projective A-module of rank d. We prove that if for every finite extension R of A, R^d is cancellative, then P is cancellative. This gives an alternate proof of Bhatwadekar's result: every projective module of rank d over an affine algebra of dimension d over a C_1-field of characteristic 0 is cancellative. Let P be a projective module of rank d-1 over an affine agebra of dimension d over an algebraically closed field. Then, it is not known if P is cancellative. We prove some results in this direction also.