A Krull-Schmdit type theorem for coherent sheaves

Let X be projective variety over an algebraically closed field k and G be a finite group with g.c.d.(char(k), |G|) = 1. We prove that any representations of G on a coherent sheaf, ρ : G → End(ℰ), has a natural decomposition ℰ ≃ ⊕ V ⊗_k ℱ_V, where G acts trivially on ℱ_V and the sum run over all irreducible representations of G over k.