# On the Weil-étale topos of regular arithmetic schemes

### Matthias Flach

Department of Mathematics Department of Mathematics Caltech Caltech Pasadena CA 91125 Pasadena CA 91125 USA USA### Baptiste Morin

Department of Mathematics Department of Mathematics Caltech Caltech Pasadena CA 91125 Pasadena CA 91125 USA USA

## 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.

## Cite this article

Matthias Flach, Baptiste Morin, On the Weil-étale topos of regular arithmetic schemes. Doc. Math. 17 (2012), pp. 313–399

DOI 10.4171/DM/369