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On Witt vector cohomology for singular varieties

Published online by Cambridge University Press:  26 March 2007

Pierre Berthelot
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
Spencer Bloch
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
Hélène Esnault
Mathematik, Universität Duisburg-Essen, FB6, Mathematik, 45117 Essen, Germany
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Over a perfect field $k$ of characteristic $p > 0$, we construct a ‘Witt vector cohomology with compact supports’ for separated $k$-schemes of finite type, extending (after tensorization with ${\mathbb Q}$) the classical theory for proper $k$-schemes. We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope ${<}1$ part of the former. Over a finite field, this allows one to compute congruences for the number of rational points in special examples. In particular, the congruence modulo the cardinality of the finite field of the number of rational points of a theta divisor on an abelian variety does not depend on the choice of the theta divisor. This answers positively a question by J.-P. Serre.

Research Article
Foundation Compositio Mathematica 2007