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arXiv · 2602.11110

The Base Change Of Fundamental Group Schemes

Lingguang Li·Niantao Tian
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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 whose objects consist of vector bundles on $X$ and $X_K$ respectively, $π(\mathcal{C}_X,x)$ and $π(\mathcal{C}_{X_K},x_K)$ the corresponding Tannaka group schemes respectively. We establish a unified criterion determining when the base change homomorphism $π(\mathcal{C}_{X_K},x_K)\rightarrow π(\mathcal{C}_X,x)_K$ is faithfully flat or an isomorphism. As applications, we recover and generalize base change results for the S, Nori, EN, F, EF, ét, Eét, 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.

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Lingguang Li, Niantao Tian. 2026-06-09. The Base Change Of Fundamental Group Schemes. https://arxiv.org/abs/2602.11110

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