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 $f^{\ast }{\mathcal{C}}_X\subseteq {\mathcal{C}}_Y$, we give the necessary and sufficient conditions for the exactness of the natural sequence of $k$-group schemes $1\to \pi ({\mathcal{C}}_Y,y)\to \pi ({\mathcal{C}}_X,x)\to G\to 1.$ 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.

arXiv comment10 pages, comments are welcome!