Cancellation of projective modules in polynomial rings of prime characteristic

TL;DR

This paper proves a criterion for the cancellation property of projective modules over polynomial rings in prime characteristic and applies it to confirm the Bass-Quillen conjecture in certain three-dimensional cases.

Contribution

It establishes a new equivalence for the cancellation property of projective modules over polynomial rings in prime characteristic, extending the understanding of module cancellation.

Findings

Cancellation of projective modules is characterized by their reductions modulo the ideal generated by variables.

The Bass-Quillen conjecture is affirmed in dimension three when 2 is invertible in the base ring.

The results apply specifically to commutative Noetherian rings of prime characteristic with finite dimension.

Abstract

Let $A$ be a commutative Noetherian ring of characteristic $p>0$, such that $\dim(A)=d$. Let $P$ be a projective $A[T_1,...,T_n]$-module of rank $d$. We show that $P$ is cancellative if and only if $P/<T_1,...,T_n>P$ is cancellative. We deduce some applications. In one of the interesting consequences, we show that the Bass-Quillen conjecture has an affirmative answer in dimension three, when $2$ is invertible.