On the Weil-étale topos of regular arithmetic schemes

Abstract

We define and study a Weil-étale topos for any regular, proper scheme X over Spec(Z) which has some of the properties suggested by Lichtenbaum for such a topos. In particular, the cohomology with R-coefficients has the expected relation to ζ(X , s) at s = 0 if the Hasse-Weil L-functions L(h i (X Q ), s) have the expected meromorphic continuation and functional equation. If X has characteristic p the cohomology with Z-coefficients also has the expected relation to ζ(X , s) and our cohomology groups recover those previously studied by Lichtenbaum and Geisser.