Explicitly Extending Frobenius Splittings over Finite Maps

Schwede, Karl; Tucker, Kevin

doi:10.1080/00927872.2014.888559

Cited by 3 publications

(2 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Although more restrictive, we have found this approach to be highly instructive. This description will appear separately in [ST11].…”

Section: Extending Finite Maps Over Finite Separable Extensionsmentioning

confidence: 99%

See 1 more Smart Citation

On the behavior of test ideals under finite morphisms

Schwede¹,

Tucker²

2013

J. Algebraic Geom.

Self Cite

View full text Add to dashboard Cite

Abstract. We derive precise transformation rules for test ideals under an arbitrary finite surjective morphism π : Y → X of normal varieties in prime characteristic p > 0. Specifically, given a Q-divisor ∆X on X and any OX -linear map T : K(Y ) → K(X), we associate a Q-divisor ∆Y on Y such that T(π * τ (Y ; ∆Y )) = τ (X; ∆X ). When π is separable and T = Tr Y /X is the field trace, we have ∆Y = π * ∆X − Ramπ where Ramπ is the ramification divisor. If in addition Tr Y /X (π * OY ) = OX , we conclude π * τ (Y ; ∆Y ) ∩ K(X) = τ (X; ∆X ) and thereby recover the analogous transformation rule to multiplier ideals in characteristic zero. Our main technique is a careful study of when an OX-linear map F * OX → OX lifts to an OY -linear map F * OY → OY , and the results obtained about these liftings are of independent interest as they relate to the theory of Frobenius splittings. In particular, again assuming Tr Y /X (π * OY ) = OX , 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 ⊆ S in characteristic p > 0, the trace map Tr : Frac S → Frac R sends S onto R.

show abstract

“…Although more restrictive, we have found this approach to be highly instructive. This description will appear separately in [ST11].…”

Section: Extending Finite Maps Over Finite Separable Extensionsmentioning

confidence: 99%

“…This material has been incorporated into a separate paper [BST11], joint with Manuel Blickle. The other appendix, on explicitly lifting finite maps under tame ramification, will become a separate work [ST11]. Both appendixes can be viewed in older versions of the arXiv version of this paper.…”

Section: Introductionmentioning

confidence: 99%

On the behavior of test ideals under finite morphisms

Schwede¹,

Tucker²

2013

J. Algebraic Geom.

Self Cite

View full text Add to dashboard Cite

show abstract

On the behavior of F-signatures, splitting primes, and test modules under finite covers

Carvajal-Rojas

Stäbler

2023

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

On tame ramification and centers of F$F$‐purity

Carvajal‐Rojas,

Fayolle

2024

Journal of London Math Soc

View full text Add to dashboard Cite

We introduce a notion of tame ramification for general finite covers. When specialized to the separable case, it extends to higher dimensions the classical notion of tame ramification for Dedekind domains and curves and sits nicely in between other notions of tame ramification in arithmetic geometry. However, when applied to the Frobenius map, it naturally yields the notion of center of ‐purity (aka compatibly ‐split subvariety). As an application, we describe the behavior of centers of ‐purity under finite covers — it all comes down to a transitivity property for tame ramification in towers.

show abstract

Explicitly Extending Frobenius Splittings over Finite Maps

Cited by 3 publications

References 6 publications

On the behavior of test ideals under finite morphisms

On the behavior of test ideals under finite morphisms

On the behavior of F-signatures, splitting primes, and test modules under finite covers

On tame ramification and centers of F$F$‐purity

Contact Info

Product

Resources

About