Flach, Matthias; Morin, Baptiste

On the Weil-étale topos of regular arithmetic schemes

Doc. Math., J. DMV 17, 313-399 (2012)


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

Mathematics Subject Classification

14F20, 11S40, 11G40, 18F10