Derived splinters in positive characteristic

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 Kovacs. 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.