The Fundamental Group Scheme of a non Reduced Scheme

TL;DR

This paper extends the concept of the fundamental group scheme to non reduced schemes over Dedekind schemes and compares it with the scheme of its reduced part.

Contribution

It introduces a new definition of the fundamental group scheme for non reduced schemes and analyzes its relationship with the reduced scheme's fundamental group.

Findings

Extended the fundamental group scheme to non reduced schemes.

Compared fundamental group schemes of affine schemes and their reduced parts.

Provided new insights into the structure of fundamental group schemes in algebraic geometry.

Abstract

We extend the definition of fundamental group scheme to non reduced schemes over any connected Dedekind scheme. Then we compare the fundamental group scheme of an affine scheme with that of its reduced part.