Pure subrings of regular local rings, endomorphism rings and Frobenius morphisms

TL;DR

This paper explores the structure of endomorphism rings of pure subrings in regular local rings, their relation to noncommutative crepant resolutions, and the role of Frobenius morphisms in understanding their properties.

Contribution

It proves that certain endomorphism rings are noncommutative crepant resolutions and constructs Morita equivalent resolutions using Frobenius morphisms, linking these to singularity properties.

Findings

Endomorphism rings are noncommutative crepant resolutions when maximal Cohen-Macaulay.

Frobenius morphisms can be used to construct Morita equivalent resolutions.

Wild quotient singularities with unramified smooth covers are not strongly F-regular.

Abstract

The aim of this paper is threefold: first, to prove that the endomorphism ring associated to a pure subring of a regular local ring is a noncommutative crepant resolution if it is maximal Cohen-Macaulay; second, to see that in that situation, a different, but Morita equivalent, noncommutative crepant resolution can be constructed by using Frobenius morphisms; finally, to study the relation between Frobenius morphisms of noncommutative rings and the finiteness of global dimension. As a byproduct, we will obtain a result on wild quotient singularities: If the smooth cover of a wild quotient singularity is unramified in codimension one, then the singularity is not strongly F-regular.