The Base Change Of Fundamental Group Schemes

Abstract: Let $k$ be a field, $K/k$ a field extension, $X$ a connected scheme proper over $k$, $x_K\in X_K(K)$ lying over $x\in X(k)$, $\mathcal{C}X$ and $\mathcal{C}{X_K}$ the Tannakian categories over $X$ and $X_K$ respectively, $π(\mathcal{C}X,x)$ and $π(\mathcal{C}{X_K},x_K)$ the corresponding Tannaka group schemes respectively. We give equivalent conditions to the isomorphisms of fundamental group schemes $$π(\mathcal{C}_{X_K},x_K)\xrightarrow{\cong} π(\mathcal{C}_X,x)_K.$$ As application, we generalize the base change of certain fundamental group schemes under separable extension and extension of algebraically closed fields, such as S, Nori, EN, F, Étale, Loc, ELoc and Unipotent fundamental group schemes.