On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic p

Details On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic p On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic p

1992-02-13

arxiv.org/abs/alg-geom/9202009v1

Inventiones Mathematicae 108 (1992), 11-13

Mathematics

Algebraic Geometry

4 pages, LaTeX

Scientific paper

10.1007/BF02100595

This paper is about sheaf cohomology for varieties (schemes) in characteristic $p>0$. We assume the presence of a Frobenius splitting. (See V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Annals of Math. 122 (1985), 27--40). The main result is that a non-zero higher direct image under a proper map of the ideal sheaf of a compatibly Frobenius split subvariety can not have a support whose inverse image is contained in that subvariety. Earlier vanishing theorems for Frobenius split varieties were based on direct limits and Serre's vanishing theorem, but our theorem is based on inverse limits and Grothendieck's theorem on formal functions. The result implies a Grauert--Riemenschneider type theorem.

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