Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Künneth Formula of Fundamental Group Schemes

Lingguang Li, Niantao Tian

Abstract

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, $π(\mathcal{C}_X,x)$, $π(\mathcal{C}_S,s)$, $π(\mathcal{C}_{X_s},x)$ the Tannaka group schemes respectively. We give the necessary and sufficient conditions for the exactness of the homotopy sequence $π(\mathcal{C}_{X_s},x)\rightarrow π(\mathcal{C}_X,x)\rightarrow π(\mathcal{C}_S,s)\rightarrow 1$. In particular, we obtain the equivalent conditions for the Kunneth 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 Kunneth formula of certain fundamental group schemes over any field, such as S, Nori, EN, F, Etale, Loc, ELoc and Unipotent fundamental group schemes.

The Künneth Formula of Fundamental Group Schemes

Abstract

Let be a field, a proper morphism between connected schemes proper over , lying over , the fibre of over , , , Tannakian categories over respectively, , , the Tannaka group schemes respectively. We give the necessary and sufficient conditions for the exactness of the homotopy sequence . In particular, we obtain the equivalent conditions for the Kunneth formula of fundamental group schemes for the product of two connected schemes and proper over . As an application, we obtain the Kunneth formula of certain fundamental group schemes over any field, such as S, Nori, EN, F, Etale, Loc, ELoc and Unipotent fundamental group schemes.
Paper Structure (5 sections, 29 theorems, 122 equations)

This paper contains 5 sections, 29 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Homotopy Sequence of Fundamental Group Schemes
  4. The Künneth Formula of Fundamental Group Schemes
  5. Application

Key Result

Theorem 1.1

Let $k$ be a field, $f:X\rightarrow S$ a morphism of connected schemes proper over a field $k$ with $f_*\mathcal{O}_X\cong\mathcal{O}_S$, $x\in X(k)$ lying over $s\in S(k)$, $\mathcal{C}_X,\mathcal{C}_S,\mathcal{C}_{X_s}$ the Tannakian categories over $X,S,X_s$ respectively. If pullback induces func The above equivalent conditions imply the following exact homotopy sequence Moreover, if we suppos

Theorems & Definitions (53)

  • Theorem 1.1: Theorem \ref{['Main']}
  • Theorem 1.2: Theorem \ref{['kunnethMain']}
  • Proposition 1.3: Proposition \ref{['kunnethdescent']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: Mil12, Chapter VII, Proposition 2.3 & 7.2
  • Lemma 2.6: Mil12, Chapter VIII, Proposition 10.2
  • ...and 43 more