Full faithfulness for overconvergent F-isocrystals

TL;DR

This paper proves that the functor from overconvergent F-isocrystals to convergent F-isocrystals is fully faithful on smooth varieties over characteristic p, using quasi-unipotence and projective module theorems.

Contribution

It establishes the full faithfulness of the forgetful functor for overconvergent F-isocrystals, combining recent quasi-unipotence results with a new projective module theorem.

Findings

The forgetful functor is fully faithful.

Every finite projective module over rings of overconvergent power series is free.

The proof integrates quasi-unipotence and algebraic module results.

Abstract

Let X be a smooth variety over a field of characteristic p>0. We prove that the forgetful functor from the category of overconvergent F-isocrystals on X to the category of convergent F-isocrystals is fully faithful. The argument uses the quasi-unipotence theorem for overconvergent F-isocrystals (recently proved independently by Andre, Mebkhout, and the author; see math.AG/0110124), plus arguments of de Jong. In the process, we establish a theorem of Quillen-Suslin type (i.e., every finite projective module is free) over rings of overconvergent power series.