Repr\'esentations p-adiques et normes universelles, I. Le cas cristallin

Let V be a crystalline p-adic representation of the absolute Galois group G_K of an finite unramified extension K of Q_p and T a lattice of V stable by G_K. We prove the following result: Let Fil^1 V be the maximal sub-representation of V with Hodge-Tate weights strictly positive and Fil^1 T=T \cap Fil^1 V. Then, the projective limit of the H^1_g(K(\mu_{p^n}), T) is equal up to torsion to the projective limit of the H^1(K(\mu_{p^n}), Fil^1 T). So its rank over the Iwasawa algebra is [K:Q_p] dim Fil^1 V.