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Article Dans Une Revue Advances in Mathematics Année : 2012

Mixed Weil cohomologies

Résumé

We define, for a regular scheme $S$ and a given field of characteristic zero $\KK$, the notion of $\KK$-linear mixed Weil cohomology on smooth $S$-schemes by a simple set of properties, mainly: Nisnevich descent, homotopy invariance, stability (which means that the cohomology of $\GG_{m}$ behaves correctly), and Künneth formula. We prove that any mixed Weil cohomology defined on smooth $S$-schemes induces a symmetric monoidal realization of some suitable triangulated category of motives over $S$ to the derived category of the field $\KK$. This implies a finiteness theorem and a Poincaré duality theorem for such a cohomology with respect to smooth and projective $S$-schemes (which can be extended to smooth $S$-schemes when $S$ is the spectrum of a perfect field). This formalism also provides a convenient tool to understand the comparison of such cohomology theories. Our main examples are algebraic de Rham cohomology and rigid cohomology, and the Berthelot-Ogus isomorphism relating them.

Dates et versions

hal-00268007 , version 1 (31-03-2008)

Identifiants

Citer

Denis-Charles Cisinski, Frédéric Déglise. Mixed Weil cohomologies. Advances in Mathematics, 2012, 230 (1), pp.55-130. ⟨10.1016/j.aim.2011.10.021⟩. ⟨hal-00268007⟩
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