The K\"unneth Formula of Fundamental Group Schemes

TL;DR

This paper establishes the conditions under which the homotopy sequence of fundamental group schemes is exact and derives the K"unneth formula for fundamental group schemes of product schemes, extending to various types over any field.

Contribution

It provides necessary and sufficient conditions for the exactness of the homotopy sequence of fundamental group schemes and proves the K"unneth formula for a broad class of fundamental group schemes.

Findings

Exactness criteria for the homotopy sequence of fundamental group schemes.

K"unneth formula for fundamental group schemes of product schemes.

Extension of the K"unneth formula to various fundamental group schemes over any field.

Abstract

Let $k$ be a field, $f:X\rightarrow S$ a proper morphism between connected schemes proper over $k$, $x\in X(k)$ lying over $s\in S(k)$, $X_s$ the fibre of $f$ over $s$, $\mathcal{C}_X$, $\mathcal{C}_{S}$, $\mathcal{C}_{X_s}$ Tannakian categories over $X,S,X_s$ respectively, $\pi(\mathcal{C}_X,x)$, $\pi(\mathcal{C}_S,s)$, $\pi(\mathcal{C}_{X_s},x)$ the Tannaka group schemes respectively. We give the necessary and sufficient conditions for the exactness of the homotopy sequence $\pi(\mathcal{C}_{X_s},x)\rightarrow \pi(\mathcal{C}_X,x)\rightarrow \pi(\mathcal{C}_S,s)\rightarrow 1$. In particular, we obtain the equivalent conditions for the Kunneth formula of fundamental group schemes for the product $X\times_k Y$ of two connected schemes $X$ and $Y$ proper over $k$. As an application, we obtain the Kunneth formula of certain fundamental group schemes over any field, such as S, Nori, EN, F, Etale, Loc, ELoc and Unipotent fundamental group schemes.