On a conjecture of Vorst

TL;DR

This paper proves that under certain conditions involving perfect fields, resolution of singularities, and K_{d+1}-regularity, a localization of a finite type algebra is guaranteed to be a regular ring.

Contribution

It establishes a new criterion linking K_{d+1}-regularity and regularity of rings over perfect fields with resolution of singularities.

Findings

R is regular if it is K_{d+1}-regular over the specified conditions.

The result depends on the assumption of strong resolution of singularities.

The proof applies to localizations of finite type algebras over an infinite perfect field.

Abstract

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type and suppose that R is K_{d+1}-regular. Then R is a regular ring.