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The Base Change Of Fundamental Group Schemes

Lingguang Li, Niantao Tian

TL;DR

This work develops a unified framework for base-changing fundamental group schemes arising from Tannakian categories of vector bundles on a connected proper scheme $X/k$. By constructing the saturation $\overline{\mathcal{C}}_X$ and using the universal $\pi(\mathcal{C}_X,x)$-bundle $P\to X$, it derives equivalent criteria for when the base-changed group $π(\mathcal{C}_{X_K},x_K)$ coincides with $π(\mathcal{C}_X,x)_K$, notably via the observable property of the associated base-change functor $η_K^{P_K}$. The paper then applies these principles to a range of fundamental group schemes (S, Nori, EN, F, Loc, ELoc, étale, unipotent), establishing isomorphism or faithful-flatness results under separable, finite Galois, and algebraically closed extensions, and highlighting counterexamples in characteristic $p>0$ that prevent descent in certain cases. It concludes with conjectures on purely inseparable base changes, aiming to extend the reach of base-change phenomena to broader field extensions. Overall, the results provide concrete criteria and broad applicability for understanding how Tannakian fundamental groups behave under field extensions, with implications for arithmetic and geometric aspects of vector bundles in algebraic geometry.

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.

The Base Change Of Fundamental Group Schemes

TL;DR

This work develops a unified framework for base-changing fundamental group schemes arising from Tannakian categories of vector bundles on a connected proper scheme . By constructing the saturation and using the universal -bundle , it derives equivalent criteria for when the base-changed group coincides with , notably via the observable property of the associated base-change functor . The paper then applies these principles to a range of fundamental group schemes (S, Nori, EN, F, Loc, ELoc, étale, unipotent), establishing isomorphism or faithful-flatness results under separable, finite Galois, and algebraically closed extensions, and highlighting counterexamples in characteristic that prevent descent in certain cases. It concludes with conjectures on purely inseparable base changes, aiming to extend the reach of base-change phenomena to broader field extensions. Overall, the results provide concrete criteria and broad applicability for understanding how Tannakian fundamental groups behave under field extensions, with implications for arithmetic and geometric aspects of vector bundles in algebraic geometry.

Abstract

Let be a field, a field extension, a connected scheme proper over , lying over , and the Tannakian categories over and respectively, and the corresponding Tannaka group schemes respectively. We give equivalent conditions to the isomorphisms of fundamental group schemes 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.
Paper Structure (18 sections, 59 theorems, 147 equations)

This paper contains 18 sections, 59 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The saturation of Tannakian categoris
  4. Base change of fundamental group schemes
  5. Base change of fundamental group schemes under arbitrary extension
  6. Base change of fundamental group schemes under finite Galois extension
  7. Base change of fundamental group schemes under algebraically closed extension
  8. Applications
  9. Base change of S-fundamental group scheme
  10. Base change of Nori's fundamental group scheme
  11. Base change of EN-fundamental group scheme
  12. Base change of F-fundamental group scheme
  13. Base change of local fundamental group scheme
  14. Base change of extended local fundamental group scheme
  15. Base change of étale fundamental group scheme
  16. ...and 3 more sections

Key Result

Theorem 1.1

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, $\phi:P\rightarrow X$ the universal principal $\pi(\mathcal{C}_X,x)$-bundle.

Theorems & Definitions (126)

  • Theorem 1.1: Theorem \ref{['fieldgeneral']}
  • Theorem 1.2: Theorem \ref{['fieldfinitegalois']}
  • Theorem 1.3: Theorem \ref{['fieldalgclosed']}
  • Proposition 1.4
  • Proposition 1.5: Proposition \ref{['algclosednotiso']}, Proposition \ref{['containlocnobasechange']} & Corollary \ref{['5.47']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4: DaEs22, Proposition A.6
  • ...and 116 more