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Arithmetic cohomology over finite fields and special values of \(\zeta\)-functions. (English) Zbl 1104.14011

In the present paper, the author, using ideas of Voevodsky, constructs a modified version \(H^i_c(X_{\text{ar}},\mathbb Z(n))\) of Weil-étale cohomology, which he calls arithmetic cohomology (with compact support). Arithmetic cohomology groups are expected to be finitely generated and related to special values of \(\zeta\)-functions for every separated scheme of finite type over a finite field and generalize Lichtenbaum’s Weil-étale cohomology groups defined for smooth and projective schemes.

MSC:

14F20 Étale and other Grothendieck topologies and (co)homologies
14F42 Motivic cohomology; motivic homotopy theory
11G25 Varieties over finite and local fields
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