Two small remarks on Nori fundamental group scheme

TL;DR

This paper investigates the structure of Nori's fundamental group scheme over certain algebraic schemes in characteristic p, focusing on its decomposition and categorical properties.

Contribution

It provides new insights into the decomposition of Nori's fundamental group scheme and describes the functor related to the maximal reduced subgroup in categorical terms.

Findings

Decomposition of Nori's fundamental group scheme into local and étale parts.

Relation between the scheme's geometry and the splitting of the group scheme.

Categorical description of the functor for the maximal reduced subgroup.

Abstract

For $X$ a complete, reduced, geometrically connected scheme over a perfect field of characteristic $p>0$, we analyze the decomposition of Nori's fundamental group scheme into its local and \'etale parts and raise the question of the relation between the geometry and the splitting of the group scheme. We also describe in categorial terms the functor which corresponds to the inclusion of the maximal reduced subgroup scheme.