Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



On the behavior of test ideals under finite morphisms

Authors: Karl Schwede and Kevin Tucker
Journal: J. Algebraic Geom. 23 (2014), 399-443
Published electronically: September 9, 2013
MathSciNet review: 3205587
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Abstract: We derive precise transformation rules for test ideals under an arbitrary finite surjective morphism $ \pi \colon Y \to X$ of normal varieties in prime characteristic $ p > 0$. Specifically, given a $ \mathbb{Q}$-divisor $ \Delta _{X}$ on $ X$ and any $ \mathcal {O}_{X}$-linear map $ \mathfrak{T} \colon K(Y) \to K(X)$, we associate a $ \mathbb{Q}$-divisor $ \Delta _{Y}$ on $ Y$ such that $ \mathfrak{T} ( \pi _{*}\tau (Y;\Delta _{Y})) = \tau (X;\Delta _{X})$. When $ \pi $ is separable and $ \mathfrak{T} = \operatorname {Tr}_{Y/X}$ is the field trace, we have $ \Delta _{Y} = \pi ^{*} \Delta _{X} - \operatorname {Ram}_{\pi }$, where $ \operatorname {Ram}_{\pi }$ is the ramification divisor. If, in addition, $ \operatorname {Tr}_{Y/X}(\pi _{*}\mathcal {O}_{Y}) = \mathcal {O}_{X}$, we conclude that $ \pi _{*}\tau (Y;\Delta _{Y}) \cap K(X) = \tau (X;\Delta _{X})$ and thereby recover the analogous transformation rule to multiplier ideals in characteristic zero. Our main technique is a careful study of when an $ \mathcal {O}_{X}$-linear map $ F_{*} \mathcal {O}_{X} \to \mathcal {O}_{X}$ lifts to an $ \mathcal {O}_{Y}$-linear map $ F_{*} \mathcal {O}_{Y} \to \mathcal {O}_{Y}$, and the results obtained about these liftings are of independent interest as they relate to the theory of Frobenius splittings. In particular, again assuming $ \operatorname {Tr}_{Y/X}(\pi _{*}\mathcal {O}_{Y}) = \mathcal {O}_{X}$, we obtain transformation results for $ F$-pure singularities under finite maps which mirror those for log canonical singularities in characteristic zero. Finally, we explore new conditions on the singularities of the ramification locus, which imply that, for a finite extension of normal domains $ R \subseteq S$ in characteristic $ p > 0$, the trace map $ \mathfrak{T} : \operatorname {Frac} S \to \operatorname {Frac} R$ sends $ S$ onto $ R$.

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Additional Information

Karl Schwede
Affiliation: Department of Mathematics, The Pennsylvania State University, 318C McAlister Building, University Park, Pennsylvania 16802

Kevin Tucker
Affiliation: Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607

Received by editor(s): March 20, 2011
Received by editor(s) in revised form: August 25, 2011
Published electronically: September 9, 2013
Additional Notes: The first author was partially supported by a National Science Foundation postdoctoral fellowship, RTG grant number 0502170 and NSF DMS 1064485/0969145. The second author was partially supported by RTG grant number 0502170 and a National Science Foundation postdoctoral fellowship DMS 1004344
Article copyright: © Copyright 2013 University Press, Inc.
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