Nori's Fundamental Group over a non Algebraically Closed Field

TL;DR

This paper generalizes Nori's fundamental group scheme to arbitrary base points over non algebraically closed fields, linking it to Galois groups and exploring its structure at different levels.

Contribution

It extends Nori's fundamental group scheme definition to arbitrary base points over non algebraically closed fields, connecting it with Galois groups.

Findings

The generalized fundamental group scheme remains non trivial over non algebraically closed fields.

The group scheme relates closely to the absolute Galois group ${ m Gal}(ar{k}/k)$.

Analysis of the group scheme at various levels provides new insights into its structure.

Abstract

In this note we generalize Nori's definition of the fundamental group scheme from a rational point to an arbitrary base point so that when we take $X$ to be a field $k$ and the point to be $k\subseteq \bar{k}$ we still get a non trivial group scheme which is similar to the absolute Galois group ${\rm Gal}(\bar{k}/k)$. Then we try to understand the group scheme at various levels.