The Base Change Of Fundamental Group Schemes

TL;DR

This paper investigates how fundamental group schemes behave under base change, providing conditions for their isomorphism and extending results to various types of fundamental group schemes over field extensions.

Contribution

It establishes equivalent conditions for the isomorphism of fundamental group schemes after base change and generalizes known base change results for multiple fundamental group schemes.

Findings

Provides criteria for fundamental group scheme isomorphisms under base change.

Generalizes base change properties for Nori, étale, and unipotent fundamental group schemes.

Extends results to separable extensions and algebraically closed field extensions.

Abstract

Let $k$ be a field, $K/k$ a field extension, $X$ a connected scheme proper over $k$, $x_K\in X_K(K)$ lying over $x\in X(k)$, $\mathcal{C}_X$ and $\mathcal{C}_{X_K}$ the Tannakian categories over $X$ and $X_K$ respectively, $\pi(\mathcal{C}_X,x)$ and $\pi(\mathcal{C}_{X_K},x_K)$ the corresponding Tannaka group schemes respectively. We give equivalent conditions to the isomorphisms of fundamental group schemes $$\pi(\mathcal{C}_{X_K},x_K)\xrightarrow{\cong} \pi(\mathcal{C}_X,x)_K.$$ As application, we generalize the base change of certain fundamental group schemes under separable extension and extension of algebraically closed fields, such as S, Nori, EN, F, \'Etale, Loc, ELoc and Unipotent fundamental group schemes.