On the bumpy fundamental group scheme

TL;DR

This paper explores the construction of a new 'bumpy' fundamental group scheme over schemes, extending known cases and introducing a novel category to handle cases where traditional constructions are unknown, especially over higher-dimensional bases.

Contribution

It introduces a new category of torsors with a novel topology to define a fundamental group scheme in cases where classical methods are insufficient.

Findings

The 'bumpy' fundamental group scheme exists over any noetherian regular scheme.

It coincides with the classical fundamental group scheme over Dedekind schemes.

The new category of torsors is cofiltered, enabling the construction of the bumpy fundamental group scheme.

Abstract

In this short paper we first recall the definition and the construction of the fundamental group scheme of a scheme $X$ in the known cases: when it is defined over a field and when it is defined over a Dedekind scheme. It classifies all the finite (or quasi-finite) fpqc torsors over $X$. When $X$ is defined over a noetherian regular scheme $S$ of any dimension we do not know if such an object can be constructed. This is why we introduce a new category, containing the fpqc torsors, whose objects are torsors for a new topology. We prove that this new category is cofiltered thus generating a fundamental group scheme over $S$, said \textit{bumpy} as it may not be flat in general. We prove that it is flat when $S$ is a Dedekind scheme, thus coinciding with the \textit{classical} one.