Isocrystals and de Rham-Witt connections

TL;DR

This paper introduces a new framework for understanding isocrystals and de Rham-Witt connections through integrable sheaf connections, providing a novel perspective on Frobenius-structured isocrystals in positive characteristic.

Contribution

It develops a new description of convergent and overconvergent isocrystals with Frobenius structure using integrable connections on sheaves of differential graded algebras.

Findings

New characterization of isocrystals with Frobenius structure.

Extension of integrable connections to de Rham-Witt complexes.

Enhanced understanding of p-adic cohomology theories.

Abstract

We introduce the notion of integrable connections for a sheaf of differential graded algebras on a topological space. We then describe them in the finite locally projective setting, when the sheaf is either the de Rham complex of a formal or a weakly formal scheme, or for the convergent or the overconvergent de Rham-Witt complex on a smooth scheme over a perfect field of positive characteristic. This enables us to give a new description of convergent and overconvergent isocrystals with a Frobenius structure.