We give a relation between the dimension of the tangent space of the deformation functor of curves with automorphisms and the Galois module structure of the space of 2-holomorphic differentials. We prove a homological version of the local-global principle similar to the one of J.Bertin, A. Mezard. Let G be a cyclic subgroup of the group of automorphisms of a curve X, so that the order of G is equal to the characteristic. By using the results of S. Nakajima on the Galois module structure of the space of 2-holomorphic differentials, we compute the dimension of the tangent space of the deformation functor
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