The Künneth Formula Of Fundamental Group Schemes

Abstract

Let $k$ be a field, $f:X\to S$ 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 a unified criterion for the exactness of the homotopy sequence of Tannakian fundamental group schemes $\pi ({\mathcal{C}}_{X_s},x)\to \pi ({\mathcal{C}}_X,x)\to \pi ({\mathcal{C}}_S,s)\to 1$. In particular, we obtain the equivalent conditions for the Künneth 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 Künneth formula of certain fundamental group schemes over any field, such as S, N, EN, F, EF, ét, Eét, Loc, ELoc and uni-fundamental group schemes.

21 pages, Comments welcome!