On the behavior of test ideals under finite morphisms

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. $Tr_{Y/X}(\pi_{*}O_{Y}) = O_{X}$.Comment: 33 pages. The appendix has been removed (it will appear in a different work). Minor changes and typos corrected throughout. To appear in the Journal of Algebraic Geometr

An algorithm for computing compatibly Frobenius split subvarieties

Let $R$ be a ring of prime characteristic $p$, and let $F^e_* R$ denote $R$ viewed as an $R$-module via the $e$th iterated Frobenius map. Given a surjective map $\phi : F^e_* R \to R$ (for example a Frobenius splitting), we exhibit an algorithm which produces all the $\phi$-compatible ideals. We also explore a variant of this algorithm under the hypothesis that $\phi$ is not necessarily a Frobenius splitting (or even surjective). This algorithm, and the original, have been implemented in Macaulay2.Comment: 15 pages, many statements clarified and numerous other substantial improvements to the exposition (thanks to the referees). To appear in the Journal of Symbolic Computatio