Mixed Weil cohomologies

Abstract

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.

update references; hopefully improve the exposition

References in corpus (7)

• Algebras and modules in monoidal model categories
• Triangulated categories of mixed motives
• The homotopy coniveau tower
• Motivic cell structures
• Derived Algebraic Geometry III: Commutative Algebra
• Sur le théorème de l'indice des équations différentielles p-adiques. III
• Algebraic K-theory, A^1-homotopy and Riemann-Roch theorems

Cited by in corpus (22)

• Triangulated categories of mixed motives
• Decomposition de motifs abeliens
• Orientation theory in arithmetic geometry
• Arakelov motivic cohomology I
• The universal six-functor formalism
• The Monsky-Washnitzer and the overconvergent realizations
• The rigid syntomic ring spectrum
• On Modules Over Motivic Ring Spectra
• Bivariant theories in motivic stable homotopy
• Perverse motives and graded derived category $\mathcal{O}$
• Tannaka duality and stable infinity-categories
• Motivic complexes and special values of zeta functions
• The p-cohomology of algebraic varieties and special values of zeta functions
• Overconvergent Chern Classes and Higher Cycle Classes
• Mixed motives and quotient stacks: Abelian varieties
• Motivic Birational Covers and Finite Filtrations on Chow Groups
• Comparison between Rigid and Overconvergent Cohomology with Coefficients
• On the Riemann-Roch formula without projective hypothesis
• Modules over the de Rham cohomology spectrum
• Mixed Motives
• Embedding of category of twisted Chow-Witt correspondences into geometric stable $\A^1$-derived category over a field
• On the higher Riemann-Roch without denominators