D\'eformations \'equivariantes des courbes stables I : \'Etude cohomologique

Let $k$ be a field, $C\to \text{Spec}k$ be a stable curve and let $G$ be a finite group acting faithfully on the curve $C\to \text{Spec}k$. In this article, we compute the vector space ${\text{Ext}}_G^1({\mathrm{\Omega }}_{C\mathrm{/}k},{\text{O}}_C)$, the sheaf ${\mathrm{\Omega }}_{C\mathrm{/}k}$ being the sheaf of relative differentials. This vector space is naturally isomorphic to the set of first order $G$-equivariant deformations of $C$. The computation we do here will be used in a future article.