Mixed Weil cohomologies

TL;DR

This paper introduces a framework for mixed Weil cohomologies on smooth schemes over a regular base, establishing their properties, functoriality, and duality, with applications to algebraic de Rham and rigid cohomology.

Contribution

It defines mixed Weil cohomologies via simple axioms and shows they induce symmetric monoidal realizations of motives into derived categories, unifying various cohomology theories.

Findings

Establishes a formalism for mixed Weil cohomologies with key properties.

Proves finiteness and Poincaré duality theorems for these cohomologies.

Provides a comparison framework for different cohomology theories like de Rham and rigid cohomology.

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 K\"unneth 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 Poincar\'e 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.