On Witt vector cohomology for singular varieties

By P. Berthelot, Spencer Bloch, H. Esnault. Compositio Mathematica, 2005

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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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4c6227fbaa5216157ee304b1625bdf1c3e04d8eb

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Citations in the corpus

Projectivity of the Witt vector affine Grassmannian (2015, methodology): “ ﬁnite global dimension. Finally, in §11.5, we use the discreteness of perfect simplicial commutative rings proven in §11.1 to improve and reprove an h-descent theorem for Witt vector cohomology from [BBE07]. 11.1. Discreteness of perfect simplicial commutative rings. We work in the setting of deriv”
Rational points on log Fano threefolds over a finite field (2015, methodology): “ Xis rationally chain connected. RATIONAL POINTS ON LOG FANO 3-FOLDS 3 Remark 1.6. In Section 5, we shall prove Theorem 1.2 directly from Theorem 1.4 and Theorem 1.5 using the argument in [Esn03] and [BBE07]. Theorem 1.4 can be seen as an analogy of the following fact in characteristic zero which is”