# Arithmetic cohomology

• ## General Introduction

Arithmetic cohomology can mean at least two things. The first is a notion of Geisser, who constructs a cohomology theory with compact supports for separated schemes of finite type over a finite field, which (assuming the Tate conjecture and rat=num) gives an integral model for l-adic cohomology with compact supports when l is different from p. These groups are expected to be finitely generated and related to special values of zeta functions for all schemes as above. This is a variant of Weil-etale cohomology, probably agreeing with it for smooth projective varieties, but being better behaved in general. For l=p his definition using the eh-topology gives a new theory which for smooth and proper schemes agrees with logarithmic de Rham-Witt cohomology. See also Arithmetic homology

Arithmetic cohomology is also sometimes used as a synonym for Absolute cohomology. See also Absolute cohomology, Geometric cohomology

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• ## Online References

Article of Nekovar

Arithmetic cohomology over finite fields and special values of zeta-functions, by Thomas Geisser

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