Projective modules over overrings of polynomial rings

TL;DR

This paper investigates properties of projective modules over certain overrings of polynomial rings, establishing conditions for cancellation, existence of unimodular elements, and connections to algebraic K-theory.

Contribution

It extends known results by proving new properties of projective modules over overrings of polynomial rings, including cancellation and K-theoretic isomorphisms.

Findings

L^1(R P) acts transitively on Um(R P) and P is cancellative.

If A is an affine algebra over a field, P has a unimodular element.

The map _r : GL_r(R)/EL^1_r(R) e2 K_1(R) is surjective and an isomorphism when f_i are monic.

Abstract

Let A be a commutative Noetherian ring of dimension d and let P be a projective R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac {1}{f_1\ldots f_m}]-module of rank r\geq max {2,dim A+1, where f_i\in A[Y_i]. Then (i) \EL^1(R\op P) acts transitively on Um(R\oplus P). In particular, P is cancellative. (ii) If A is an affine algebra over a field, then P has a unimodular element. (iii) The natural map \Phi_r : GL_r(R)/EL^1_r(R) \ra K_1(R) is surjective. (iv) Assume f_i is a monic polynomial. Then \Phi_{r+1} is an isomorphism. In the case of Laurent polynomial ring (i.e. f_i=Y_i), (i) is due to Lindel, (ii) is due to Bhatwadekar, Lindel and Rao and (iii, iv) is due to Suslin.