On the behavior of test ideals under finite morphisms

TL;DR

This paper establishes how test ideals and $F$-singularities transform under finite morphisms between normal varieties in positive characteristic, linking Frobenius-related homomorphisms and multiplier ideal transformations.

Contribution

It provides new transformation rules for test ideals under finite morphisms and relates Frobenius splittings to multiplier ideals, extending understanding in positive characteristic algebraic geometry.

Findings

Derived transformation rules for test ideals under finite morphisms.

Connected Frobenius splittings to multiplier ideal transformations.

Identified conditions for surjectivity of the trace map.

Abstract

We derive transformation rules for test ideals and $F$-singularities under an arbitrary finite surjective morphism $\pi : Y \to X$ of normal varieties in prime characteristic $p > 0$. The main technique is to relate homomorphisms $F_{*} O_{X} \to O_{X}$, such as Frobenius splittings, to homomorphisms $F_{*} O_{Y} \to O_{Y}$. In the simplest cases, these rules mirror transformation rules for multiplier ideals in characteristic zero. As a corollary, we deduce sufficient conditions which imply that trace is surjective, i.e. $Tr_{Y/X}(\pi_{*}O_{Y}) = O_{X}$.