  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 $D^b(\mathcal{M}(G_K))\to D^b(\mathcal{C}(G_K))$.
