Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Li, Lingguang, Wang, Hao
Institution: Tongji University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202170529