On the Weil-étale topos of regular arithmetic schemes

Details On the Weil-étale topos of regular arithmetic schemes On the Weil-étale topos of regular arithmetic schemes

2010-10-19

arxiv.org/abs/1010.3833v1

Mathematics

Number Theory

84 pages. Submitted

Scientific paper

We define and study a Weil-\'etale 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.

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