Holmstrom Arithmetic cohomology

Arithmetic cohomology

Article of Nekovar

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


Arithmetic cohomology

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Arithmetic cohomology

AAG (Arithmetic algebraic geometry)

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Arithmetic cohomology

Mixed (?), Charp

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Arithmetic cohomology

Arithmetic vs geometric CTs.

Ref: Nekovar, Beilinson’s height pairing article.

Absolute: VS. Geometric: VS with extra structure (“Tannakian”?).

Hochschild-Serre spectral sequence.

It seems geometric theories are related to the def of L-fn and zeta fn, but arithmetic theories capture orders of vanishing and maybe more info about the special values.

Absolute should be related to weight 0 part of geometric.

What is geometric/absolute cohom of Spec Z, its compactification, and the infinite prime?

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Arithmetic cohomology

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

nLab page on Arithmetic cohomology

Created on June 10, 2014 at 21:14:54 by Andreas Holmström