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

Abstract

We construct cohomology groups with compact support $H_c^i(X_{\mathrm{ar}},\mathbb{Z}(n))$ for separated schemes of finite type over a finite field which generalize Lichtenbaum's Weil-étale cohomology groups for smooth and projective schemes (see [22]). In particular, if Tate's conjecture holds, and rational and numerical equivalence agree up to torsion, then the groups $H_c^i(X_{\mathrm{ar}},\mathbb{Z}(n))$ are finitely generated, form an integral model of $l$-adic cohomology with compact support, and admit a formula for the special values of the $\zeta$-function of $X$