Vector bundles trivialized by proper morphisms and the fundamental group scheme, II

TL;DR

This paper investigates when vector bundles become trivial after proper morphisms, showing that under separability conditions, the associated Tannakian category corresponds to finite étale group schemes, with counterexamples provided otherwise.

Contribution

It establishes a link between trivialized vector bundles and finite étale group schemes under separability, extending understanding of vector bundle trivialization.

Findings

Tannakian category of certain vector bundles is equivalent to representations of a finite étale group scheme.

Counterexample shows the necessity of separability for the main result.

Provides conditions under which vector bundles trivialized by proper morphisms relate to fundamental group schemes.

Abstract

Let $X$ be a projective and smooth variety over an algebraically closed field $k$. Let $f:Y\rightarrow X$ be a proper and surjective morphism of $k$-varieties. Assuming that $f$ is separable, we prove that the Tannakian category associated to the vector bundles $E$ on $X$ such that $f^*E$ is trivial is equivalent to the category of representations of a finite and etale group scheme. We give a counterexample to this conclusion in the absence of separability.