On the Weil-étale topos of regular arithmetic schemes

Abstract

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