# Characteristic-free test ideals

## TL;DR

This paper generalizes the concept of test ideals beyond tight closure, applying it to arbitrary closure operations, and demonstrates their utility in classifying singularities across different characteristics, including mixed characteristic.

## Contribution

It introduces a characteristic-free notion of test ideals applicable to various closure operations, extending their use in singularity classification.

## Key findings

- Test ideals can be defined for arbitrary closure operations.
- These test ideals retain key properties of tight closure test ideals.
- Applications include classification of singularities in mixed characteristic.

## Abstract

Tight closure test ideals have been central to the classification of singularities in rings of characteristic $p>0$, and via reduction to characteristic $p$, in equal characteristic zero as well. A summary of their properties and applications can be found in "A survey of test ideals" by Karl Schwede and Kevin Tucker. In this paper, we extend the notion of a test ideal to arbitrary closure operations, particularly those coming from big Cohen-Macaulay modules and algebras, and prove that it shares key properties of tight closure test ideals. Our main results show how these test ideals can be used to give a characteristic-free classification of singularities, including a few specific results on the mixed characteristic case. We also compute examples of these test ideals.