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 a unified criterion for the exactness of the homotopy sequence of Tannakian fundamental group schemes $\pi(\mathcal{C}_{X_s},x)\rightarrow \pi(\mathcal{C}_X,x)\rightarrow \pi(\mathcal{C}_S,s)\rightarrow 1$. In particular, we obtain the equivalent conditions for the K\"unneth 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 K\"unneth formula of certain fundamental group schemes over any field, such as S, N, EN, F, EF, \'et, E\'et, Loc, ELoc and uni-fundamental group schemes.