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Symmetry and parity in Frobenius action on cohomology

Part of: (Co)homology theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  08 December 2011

Junecue Suh
Junecue Suh*
Affiliation:
Department of Mathematics, FAS Harvard University, One Oxford Street, Cambridge, MA 02138, USA (email: [email protected])
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Abstract

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We prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincaré duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz.

Keywords

zeta functions over finite fieldsétale cohomologycrystalline cohomologyNewton polygon of Frobenius

MSC classification

Secondary: 14F20: Étale and other Grothendieck topologies and (co)homologies 11G25: Varieties over finite and local fields 14F30: $p$-adic cohomology, crystalline cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 1 , January 2012 , pp. 295 - 303
Copyright
Copyright © Foundation Compositio Mathematica 2011

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