Représentations p-adiques et normes universelles, I. Le cas cristallin

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Let V be a crystalline p-adic representation of the absolute Galois group Gₖ of an finite unramified extension K of Qₚ and T a lattice of V stable by Gₖ. We prove the following result: Let Fil¹ V be the maximal sub-representation of V with Hodge-Tate weights strictly positive and Fil¹ T=T ∩ Fil¹ V. Then, the projective limit of the H¹g(K(μₚₙ), T) is equal up to torsion to the projective limit of the H¹(K(μₚₙ), Fil¹ T). So its rank over the Iwasawa algebra is [K:Qₚ] dim Fil¹ V.