Mixed Weil cohomologies

By Denis-Charles Cisinski, Frédéric Déglise. 2007

56 citations recorded by the source corpus.

Abstract We define, for a regular scheme S and a given field of characteristic zero K, the notion of K-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 G m behaves correctly), and Kunneth 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 K. This implies a finiteness theorem and a Poincare 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/86a8b7a9591c6e84c54eec499c7c37ab574d5157