Global generation of test ideals in mixed characteristic and applications

TL;DR

This paper introduces a new notion of test ideals in mixed characteristic schemes and proves global generation results, extending classical results from characteristic zero and positive characteristic to mixed characteristic settings.

Contribution

It defines a $+$-test ideal for schemes over mixed characteristic rings and establishes global generation theorems, broadening the scope of multiplier and test ideal techniques.

Findings

Global generation results for $+$-test ideals in mixed characteristic.

Applications to vanishing orders and base loci in mixed characteristic.

Extension of classical characteristic zero and positive characteristic results.

Abstract

Suppose that $X$ is an integral scheme (quasi-)projective over a complete local ring of mixed characteristic. Using ideas of Takamatsu-Yoshikawa and Bhatt-Ma-et. al, we define a notion of a $+$-test ideal on $X$, including for divisors and linear series. We obtain global generation results in this setting that generalize the well known global generation results obtained via multiplier ideal sheaf techniques in characteristic $0$ and via test ideals in characteristic $p>0$. We also obtain applications to the order of vanishing of linear series and to the diminished base locus in mixed characteristic similar to results of Ein-Lazarsfeld-Mustata-Nakamaye-Popa, Nakayama, and Mustata in the equal characteristic case.