# Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers

## TL;DR

This paper introduces a new mixed characteristic analog of multiplier and test ideals using perfectoid algebras, and applies it to establish uniform bounds on symbolic powers of radical ideals in regular rings.

## Contribution

It develops a novel mixed characteristic framework for multiplier and test ideals and extends uniform symbolic power bounds to all excellent regular rings.

## Key findings

- Established subadditivity of the new ideal.
- Derived uniform bounds on symbolic powers.
- Extended results from equal characteristic to mixed characteristic.

## Abstract

Using perfectoid algebras, we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively $p > 0$, in the case of a regular ambient ring. We prove several properties about this ideal such as subadditivity. We then use these techniques to derive a uniform bound on the growth of symbolic powers of radical ideals in all excellent regular rings. The analogous result was shown in equal characteristic by Ein-Lazarsfeld-Smith and Hochster-Huneke.