# Equivariant splitting of the Hodge--de Rham exact sequence

## TL;DR

This paper investigates conditions under which the Hodge-de Rham exact sequence of an algebraic curve with a finite group action splits equivariantly, linking this to the ramification properties of the group action and lifting problems.

## Contribution

It establishes that a G-equivariant splitting of the Hodge-de Rham sequence implies the group action is weakly ramified, extending previous results for hyperelliptic curves.

## Key findings

- G-equivariant splitting implies weak ramification of the group action
- Generalizes K"ock and Tait's result for hyperelliptic curves
- Connects splitting to lifting coverings to Witt vectors

## Abstract

Let $X$ be an algebraic curve with an action of a finite group $G$ over a field $k$. We show that if the Hodge-de Rham short exact sequence of $X$ splits $G$-equivariantly then the action of $G$ on $X$ is weakly ramified. In particular, this generalizes the result of K\"{o}ck and Tait for hyperelliptic curves. We discuss also converse statements and tie this problem to lifting coverings of curves to the ring of Witt vectors of length $2$.