Almost $C_p$ Galois representations and vector bundles

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $G_K$ the absolute Galois group. Then $G_K$ acts on the fundamental curve $X$ of $p$-adic Hodge theory and we may consider the abelian category $\mathcal{M}(G_K)$ of coherent $\mathcal{O}_X$-modules equipped with a continuous and semi-linear action of $G_K$. An almost $C_p$-representation of $G_K$ is a $p$-adic Banach space $V$ equipped with a linear and continuous action of $G_K$ such that there exists $d\in\mathbb{N}$, two $G_K$-stable finite dimensional sub-$\mathbb{Q}_p$-vector spaces $U_+$ of $V$, $U_-$ of $C_p^d$, and a $G_K$-equivariant isomorphism $V/U_+\to C_p^d/U_-$. These representations form an abelian category $\mathcal{C}(G_K)$. The main purpose of this paper is to prove that $\mathcal{C}(G_K)$ can be recovered from $\mathcal{M}(G_K)$ by a simple construction (and conversely) inducing, in particular, an equivalence of triangulated categories $D^b(\mathcal{M}(G_K))\to D^b(\mathcal{C}(G_K))$.