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THE p-COHOMOLOGY OF ALGEBRAIC VARIETIES AND SPECIAL VALUES OF ZETA FUNCTIONS

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  02 June 2014

James S. Milne  and
Niranjan Ramachandran
James S. Milne
Affiliation:
Mathematics Department, University of Michigan, Ann Arbor, MI 48109, USA (jmilne@umich.edu) URL: www.jmilne.org/math/
Niranjan Ramachandran
Affiliation:
Mathematics Department, University of Maryland, College Park, MD 20742, USA (atma@math.umd.edu) URL: www2.math.umd.edu/∼atma/
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Abstract

The $p$-cohomology of an algebraic variety in characteristic $p$ lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl, Illusie, Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc., and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for $p$-cohomology in characteristic $p$.

Keywords

zeta functionscrystalline cohomology

MSC classification

Primary: 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture) 11G20: Curves over finite and local fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 4 , October 2015 , pp. 801 - 835
Copyright
© Cambridge University Press 2014 

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