Représentations $p$-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 $\mathbb{Q}_p$, and let $T$ be a lattice of $V$ stable by $G_K$. We prove the following result: Let $\mathrm{Fil}^1V$ be the maximal sub-representation of $V$ with Hodge-Tate weights strictly positive and $\mathrm{Fil}^1T=T\cap \mathrm{Fil}^1V$. Then, the projective limit of $H^1_g(K(\mu_{p^n}), T)$ is equal up to torsion to the projective limit of $H^1(K(\mu_{p^n}), \mathrm{Fil} ^1T)$. So its rank over the Iwasawa algebra is $[K:\mathbb{Q}_p]\operatorname{dim}\mathrm{Fil}^1 V$.