# The action of the \'etale fundamental group scheme on the connected component of the essentially finite one

## TL;DR

This paper defines an action of the étale fundamental group scheme on the local component of the essentially finite fundamental group scheme, demonstrating faithfulness for curves of genus at least 2.

## Contribution

It introduces a new action of the étale fundamental group scheme on the essentially finite fundamental group scheme's local component, extending previous work by Otabe.

## Key findings

- The representation is faithful for curves of genus ≥ 2.
- The paper generalizes the understanding of fundamental group schemes.
- Provides a new perspective on the structure of fundamental group schemes.

## Abstract

We follow the pattern in a recent paper of Otabe [Ota15] to define an action of the \'etale fundamental group scheme $\pi^\text{et}(X)$ on the local component of the essentially finite fundamental group scheme $\pi^{\mathrm{EF}}(X)$ of Nori. We show that the associated representation is faithful when $X$ is a curve of genus $\geq 2$.