Test ideals in mixed characteristic: a unified theory up to perturbation

TL;DR

This paper introduces a unified test ideal in mixed characteristic that aligns with multiplier ideals after inverting p, is computable via alterations, and satisfies key properties, using the p-adic Riemann-Hilbert functor.

Contribution

It defines a comprehensive test ideal in mixed characteristic that unifies previous notions and satisfies expected properties, advancing the understanding of singularities in mixed characteristic.

Findings

The test ideal agrees with multiplier ideals after inverting p.

It can be computed from a sufficiently large alteration.

It satisfies the full suite of properties of multiplier or test ideals.

Abstract

Let $X$ be an integral scheme of finite type over a complete DVR of mixed characteristic. We provide a definition of a test ideal which agrees with the multiplier ideal after inverting $p$, is computed from a sufficiently large alteration, agrees with previous mixed characteristic BCM test ideals after completing at any point of residue characteristic $p$ (up to small perturbation), and which satisfies the full suite of expected properties of a multiplier or test ideal. This object is obtained via the $p$-adic Riemann-Hilbert functor.