M. Flach and B. Morin

摘要

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 $zeta(X,s)$ 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 $zeta(X,s)$ and our cohomology groups recover those previously studied by Lichtenbaum and Geisser.