On the Weil-´ topos of regular arithmetic schemes

We define and study a Weil-´etale topos for any regular, proper scheme X over Spec(Z) which has some of the properties sug- gested by Lichtenbaum for such a topos. In particular, the cohomol- ogy with ˜ R-coefficients has the expected relation to�(X;s) at s = 0 if the Hasse-Weil L-functions L(h i (XQ);s) have the expected meromor- phic continuation and functional equation. If X has characteristic p the cohomology with Z-coefficients also has the expected relation to �(X;s) and our cohomology groups recover those previously studied by Lichtenbaum and Geisser.