Arithmetic cohomology over finite fields and special values of zeta-functions

2004/05/10 by Thomas Geisser, Thomas H. Geisser, Geisser, Thomas H.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.math/0405164

28 pages, revised version

openalex publication_date 2004/05/10 · arxiv created 2005/03/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a cohomology theory with compact support Hⁱ_c(X_ar,Z(n)) for separated schemes of finite type over a finite field, which should play a role analog to Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds and rational and numerical equivalence agree up to torsion, then the groups Hⁱ_c(X_ar,Z(n)) are finitely generated, form an integral version of l-adic cohomology with compact support, and admit a formula for the special values of the zeta-function of X.

Citations

• A sheaf-theoretic reformulation of the Tate conjecture

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