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Derived splinters in positive characteristic

Part of: Local theory (Co)homology theory

Published online by Cambridge University Press:  10 July 2012

Bhargav Bhatt
Bhargav Bhatt*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA (email: [email protected])
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Abstract

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This paper introduces the notion of a derived splinter. Roughly speaking, a scheme is a derived splinter if it splits off from the coherent cohomology of any proper cover. Over a field of characteristic 0, this condition characterises rational singularities, as suggested by the work of Kovács. Our main theorem asserts that over a field of characteristic p, derived splinters are the same as (underived) splinters, i.e. schemes that split off from any finite cover. Using this result, we answer some questions of Karen Smith concerning the extension of Serre/Kodaira-type vanishing results beyond the class of ample line bundles in positive characteristic; these are purely projective geometric statements independent of singularity considerations. In fact, we can prove ‘up to finite cover’ analogues in characteristic p of many vanishing theorems known in characteristic 0. All these results fit naturally in the study of F-singularities, and are motivated by a desire to understand the direct summand conjecture.

Keywords

splintersrational singularitiesvanishing theoremsFrobeniusderived categoriesF-regular rings

MSC classification

Secondary: 14F17: Vanishing theorems 14B15: Local cohomology 14B05: Singularities
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1757 - 1786
Copyright
Copyright © Foundation Compositio Mathematica 2012

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