Exact sequences of fundamental group schemes for $G$-torsors

Abstract

Let $k$ be a field, $G$ an affine $k$-group scheme, $X$ a connected scheme proper over $k$, $f:(Y,y)\to(X,x)$ a pointed $G$-torsor, $\mathcal{C}_X$ and $\mathcal{C}_Y $ the Tannakian categories of vector bundles on $X$ and $Y$, such that $f^*\mathcal{C}_X\subseteq \mathcal{C}_Y$, we give the necessary and sufficient conditions for the exactness of the natural sequence of $k$-group schemes $1\rightarrow π(\mathcal{C}_Y,y) \rightarrow π(\mathcal{C}_X,x) \rightarrow G \rightarrow 1.$ This provides a common framework of exact sequences for the $S$-, Nori, $F$-, étale fundamental group schemes and their extended variants. Finally, an isogeny of an elliptic curve shows that the relevant preservation property fails for the local, extended local, and unipotent categories.