$F$-divided sheaves trivialized by dominant maps are essentially finite

TL;DR

This paper extends the understanding of when vector bundles are essentially finite, showing that on certain algebraic stacks over fields of positive characteristic, trivialization by proper maps implies essential finiteness, generalizing prior results.

Contribution

It generalizes the Biswas-Dos Santos result to non-proper, non-smooth algebraic stacks over arbitrary fields of characteristic p>0.

Findings

Vector bundles trivialized by proper and flat maps on pseudo-proper stacks are essentially finite.

Extension of essential finiteness results to algebraic stacks in positive characteristic.

Partial generalization of previous theorems to broader classes of stacks.

Abstract

By a result of Biswas and Dos Santos, on a smooth and projective variety over an algebraically closed field, a vector bundle trivialized by a proper and surjective map is essentially finite, that is it corresponds to a representation of the Nori fundamental group scheme. In this paper we obtain similar results for non-proper non-smooth algebraic stacks over arbitrary fields of characteristic $p>0$. As by-product we have the following partial generalization of the Biswas-Dos Santos' result in positive characteristic: on a pseudo-proper and inflexible stack of finite type over $k$ a vector bundle which is trivialized by a proper and flat map is essentially finite.