The Monsky-Washnitzer and the overconvergent realizations

TL;DR

This paper develops new realization functors connecting algebraic and analytic motives over non-archimedean fields, providing a motivic perspective on overconvergent cohomology theories and their finite dimensionality.

Contribution

It introduces the dagger and Monsky-Washnitzer realization functors using motivic language, offering a new direct definition of overconvergent cohomology theories.

Findings

Provides a motivic framework for overconvergent de Rham and rigid cohomology.

Shows finite dimensionality of these cohomologies follows from Betti cohomology.

Establishes new functors linking algebraic and analytic motives.

Abstract

We construct the dagger realization functor for analytic motives over non-archimedean fields of mixed characteristic, as well as the Monsky-Washnitzer realization functor for algebraic motives over a discrete field of positive characteristic. In particular, the motivic language on the classic \'etale site provides a new direct definition of the overconvergent de Rham cohomology and rigid cohomology and shows that their finite dimensionality follows formally from the one of Betti cohomology for smooth projective complex varieties.