DOCUMENTA MATHEMATICA, Vol. 17 (2012), 313-399

M. Flach and B. Morin

On the Weil-Étale Topos of Regular Arithmetic Schemes

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

2010 Mathematics Subject Classification: Primary: 14F20, 11S40, Secondary: 11G40, 18F10

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