Mixed Weil cohomologies

作者：Cisinski Denis Charles; Deglise Frederic^*

来源：Advances in Mathematics, 2012, 230(1): 55-130.

DOI：10.1016/j.aim.2011.10.021

• Weil cohomology
• Homotopy theory of schemes
• Motivic cohomology
• Kunneth formula
• Poincare duality
• de Rham cohomology
• Rigid cohomology

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 Well 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.

• 出版日期2012-5-1