Regular rings over valuation rings

TL;DR

This paper explores the concept of regularity in local rings over valuation rings, extending classical results and applying them to derive a version of Kodaira's vanishing theorem.

Contribution

It introduces new properties and results for regularity of local rings over valuation rings, including openness, Cohen--Macaulay algebras, and cotangent complexes, with applications to vanishing theorems.

Findings

Parallel results to Noetherian regularity established

Openness of regular loci demonstrated

A version of Kodaira's vanishing theorem proved in positive characteristic

Abstract

Bertin (1972) defined regularity for coherent local rings, and Knaf (2004) studied the property for a local ring $A$ essentially finitely presented over a valuation ring $V$. We discuss several properties of this notion of regularity for such $A$, obtaining results parallel to results for regularity of Noetherian local rings. We include classical and modern topics: openness of loci, perfectoid big Cohen--Macaulay algebras, and cotangent complexes. We also give an application to Noetherian rings, showing a version of Kodaira's vanishing theorem in large enough residue characteristics.