Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras

TL;DR

This paper develops a new framework using perfectoid big Cohen-Macaulay algebras to analyze singularities in mixed characteristic, extending concepts like rational and log terminal singularities.

Contribution

It introduces a mixed characteristic BCM-variant of key singularity classes and multiplier/test ideals, with new restriction and deformation theorems.

Findings

Restriction theorem for perfectoid BCM multiplier/test ideals

Deformation results for BCM-regular and BCM-rational singularities

Behavioral results of F-regular and F-rational singularities in families

Abstract

We utilize recent results of Andr\'e and Gabber on the existence of weakly functorial integral perfectoid big Cohen-Macaulay (BCM) algebras to study singularities of local rings in mixed characteristic. In particular, we introduce a mixed characteristic BCM-variant of rational/$F$-rational singularities, of log terminal/$F$-regular singularities and of multiplier/test ideals of divisor pairs. We prove a number of results about these objects including a restriction theorem for perfectoid BCM multiplier/test ideals and deformation statements for perfectoid BCM-regular and BCM-rational singularities. As an application, we obtain results on the behavior of $F$-regular and $F$-rational singularities in arithmetic families.