Almost ${\mathbb C}_p$ Galois representations and vector bundles

Abstract

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 $\mathcal{ℳ}\left(G_K\right)$ of coherent ${\mathcal{O}}_X$-modules equipped with a continuous and semilinear 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 $\mathcal{C}\left(G_K\right)$. The main purpose of this paper is to prove that $\mathcal{C}\left(G_K\right)$ can be recovered from $\mathcal{ℳ}\left(G_K\right)$ by a simple construction (and vice-versa) inducing, in particular, an equivalence of triangulated categories

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