# A note on effective descent for overconvergent isocrystals

## TL;DR

This paper proves that certain morphisms are effective descent maps for overconvergent isocrystals and demonstrates that Frobenius pull-back induces an equivalence on overconvergent categories over perfect fields.

## Contribution

It clarifies the proof of effective descent for overconvergent isocrystals and establishes Frobenius pull-back as an equivalence in this setting.

## Key findings

- Proper surjective and faithfully flat maps are effective descent morphisms.
- Frobenius pull-back functor is an equivalence on overconvergent isocrystals over perfect fields.

## Abstract

In this short note we explain the proof that proper surjective and faithfully flat maps are morphisms of effective descent for overconvergent isocrystals. We then show how to deduce the folklore theorem that for an arbitrary variety over a perfect field of characteristic $p$, the Frobenius pull-back functor is an equivalence on the overconvergent category.