Derived splinters in positive characteristic

TL;DR

This paper introduces derived splinters in positive characteristic, proves their equivalence to splinters, and extends several vanishing theorems known in characteristic zero to positive characteristic, advancing the understanding of F-singularities.

Contribution

It establishes the equivalence of derived splinters and splinters in characteristic p and extends classical vanishing theorems to this setting.

Findings

Derived splinters coincide with splinters in characteristic p.

Extended vanishing theorems to positive characteristic.

Connected results to F-singularities and the direct summand conjecture.

Abstract

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 by a result of Kov\'acs. Our main theorem asserts that over a field of characteristic p, derived splinters are the same as (underived) splinters, i.e., as schemes that split off from any finite cover. Using this result, we answer some questions of Karen Smith concerning extending 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 one knows 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