A Kunz-type characterization of regular rings via alterations

TL;DR

This paper characterizes regular local rings over a field using alterations, showing that regularity is equivalent to finite projective dimension of the pushforward of the structure sheaf under all regular alterations.

Contribution

It provides a Kunz-type criterion for regularity based on the behavior of alterations, extending classical characterizations to a broader context.

Findings

Regular local rings are characterized by finite projective dimension under all regular alterations.

The criterion applies to rings essentially of finite type over a field.

In characteristic zero, the projective dimension is zero, simplifying the criterion.

Abstract

We prove that a local domain $R$, essentially of finite type over a field, is regular if and only if for every regular alteration $\pi : X \to Spec R$, we have that $R \pi_* \mathcal{O}_X$ has finite (equivalently zero in characteristic zero) projective dimension.