发表机构
Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education（教育部智能计算与应用重点实验室（同济大学））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决Tannakian基本群概型的实现问题，结合正合序列、Godeaux--Serre构造及Lefschetz型定理，实现了两类群概型对应的光滑射影簇基本群概型。
AI 中文摘要
我们研究Tannakian基本群概型的实现问题：给定仿射k-群概型G，何时存在光滑射影连通带点k-簇(X,x)，其基本群概型同构于G？我们建立与主丛相关的基本群概型正合序列，并将其与Godeaux--Serre构造及Lefschetz型定理结合。作为应用，我们证明每个有限平展k-群概型都可作为光滑射影簇的S-、Nori及推广的Nori基本群概型实现，而每个有限常群概型可作为F-及平展变体实现。
英文摘要
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.
Comments10 pages, comments are welcome!
