The K\"unneth Formula of Fundamental Group Schemes

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.