The Base Change Of Fundamental Group Schemes

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 whose objects consist of vector bundles on $X$ and $X_K$ respectively, $\pi(\mathcal{C}_X,x)$ and $\pi(\mathcal{C}_{X_K},x_K)$ the corresponding Tannaka group schemes respectively. We establish a unified criterion determining when the base change homomorphism $\pi(\mathcal{C}_{X_K},x_K)\rightarrow \pi(\mathcal{C}_X,x)_K$ is faithfully flat or an isomorphism. As applications, we recover and generalize base change results for the S, Nori, EN, F, EF, \'et, E\'et, Loc, ELoc, and unipotent-fundamental group schemes under different types of field extensions (e.g., separable, finite Galois, and algebraically closed extensions). Moreover, our approach provides a unified explanation for both positive and negative results, including previously known counterexamples.