On the behavior of test ideals under finite morphisms

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 $F_{\ast }{O}_X\to O_X$, such as Frobenius splittings, to homomorphisms $F_{\ast }{O}_Y\to O_Y$. 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. $T{r}_{Y/X}({\pi }_{\ast }{O}_Y)=O_X$.

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 Geometry

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• Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras
• Finite torsors over strongly $F$-regular singularities
• Test ideals in rings with finitely generated anti-canonical algebras
• On the three dimensional minimal model program in positive characteristic
• Frobenius Splitting in Commutative Algebra
• The weak ordinarity conjecture and $F$-singularities
• Étale fundamental groups of strongly $F$-regular schemes
• On the behavior of $F$-signatures, splitting primes, and test modules under finite covers
• $F$-singularities: applications of characteristic $p$ methods to singularity theory
• A dual to tight closure theory
• Tame fundamental groups of pure pairs and Abhyankar's lemma
• Global $F$-splitting of surfaces admitting an int-amplified endomorphim
• Inversion of adjunction for $F$-signature
• F-singularities via alterations
• On tame ramification and centers of $F$-purity
• On proper splinters in positive characteristic
• Centers of perfectoid purity
• Compatible ideals in Gorenstein rings
• On the superadditivity of anticanonical Iitaka dimension
• Uniform approximation of Abhyankar valuation ideals in function field of prime characteristic
• Explicitly Extending Frobenius Splittings over Finite Maps