Formal Laurent series rings and the Hermite ring conjecture

TL;DR

This paper investigates projective modules over formal Laurent series rings, linking the problem to major conjectures in algebra, and provides a reduction step that simplifies the proof of the Hermite ring conjecture for all commutative local rings.

Contribution

It establishes a reduction step for the Hermite ring conjecture, connecting it to the case of complete intersection UFDs, advancing understanding in algebraic K-theory.

Findings

Reduction step for the Hermite ring conjecture

Equivalence of the conjecture for all local rings and for complete intersection UFDs

Relation between projective modules over Laurent series rings and major algebraic conjectures

Abstract

We study the question if projective modules over formal Laurent series rings are extended. We relate this question to the Bass-Quillen conjecture for commutative regular local rings and to the Hermite ring conjecture for all commutative local rings. Using our result about projective modules over formal Laurent series rings, we prove a reduction step for the Hermite ring conjecture. We show that the Hermite ring conjecture holds for all commutative local rings if and only if it holds for complete intersection rings which are also unique factorization domains.