#### Vol. 2, No. 3, 2020

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

### Jean-Marc Fontaine

Vol. 2 (2020), No. 3, 667–732
##### Abstract

Let $K$ be a finite extension of ${ℚ}_{p}$ and ${G}_{\phantom{\rule{0.3em}{0ex}}K}$ the absolute Galois group. Then ${G}_{\phantom{\rule{0.3em}{0ex}}K}$ acts on the fundamental curve $X$ of $p$-adic Hodge theory and we may consider the abelian category $\mathsc{ℳ}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)$ of coherent ${\mathsc{O}}_{X}$-modules equipped with a continuous and semilinear action of ${G}_{\phantom{\rule{0.3em}{0ex}}K}$.

An almost ${ℂ}_{p}$-representation of ${G}_{\phantom{\rule{0.3em}{0ex}}K}$ is a $p$-adic Banach space $V$ equipped with a linear and continuous action of ${G}_{\phantom{\rule{0.3em}{0ex}}K}$ such that there exists $d\in ℕ$, two ${G}_{\phantom{\rule{0.3em}{0ex}}K}$-stable finite dimensional sub-${ℚ}_{p}$-vector spaces ${U}_{+}$ of $V$, ${U}_{-}$ of ${ℂ}_{p}^{d}$, and a ${G}_{\phantom{\rule{0.3em}{0ex}}K}$-equivariant isomorphism

$V∕{U}_{+}\to {ℂ}_{p}^{d}∕{U}_{-}.$

These representations form an abelian category $\mathsc{C}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)$. The main purpose of this paper is to prove that $\mathsc{C}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)$ can be recovered from $\mathsc{ℳ}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)$ by a simple construction (and vice-versa) inducing, in particular, an equivalence of triangulated categories

${D}^{b}\left(\mathsc{ℳ}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)\right)\to {D}^{b}\left(\mathsc{C}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)\right).$

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$p$-adic Hodge theory, vector bundle