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

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 $\geq 2$.