Holonomic \'etale sheaves are constructible

TL;DR

This paper proves that holonomic étale sheaves over a perfect field are constructible, establishing an étale analogue of Kashiwara's theorem and extending Beilinson's work on constructible sheaves.

Contribution

It introduces the notion of holonomicity for étale sheaves and proves their constructibility over perfect fields, providing a new link between holonomicity and constructibility.

Findings

Holonomic étale sheaves are constructible over perfect fields.

Establishes the étale analogue of Kashiwara's theorem.

Extends Beilinson's result on constructible sheaves.

Abstract

Building on Beilinson's work, ``constructible sheaves are holonomic,'' we introduce the notion of holonomicity for \'etale sheaves, without assuming a priori constructibility. Over a perfect base field, we establish the converse of Beilinson's result, showing that holonomic sheaves are indeed constructible. This can be seen as an \'etale analogue of Kashiwara's theorem on holonomic ${\mathcal D}_X$-modules.