Equivariant splitting of the Hodge-de Rham exact sequence. (English) Zbl 1479.14030
Summary: Let \(X\) be an algebraic curve with a faithful 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 Köck 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.
MSC:
| 14F40 | de Rham cohomology and algebraic geometry |
| 14G17 | Positive characteristic ground fields in algebraic geometry |
| 14H37 | Automorphisms of curves |
| 14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |
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