Let
$K$ be a finite
extension of
${\mathbb{Q}}_{p}$ 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
$\mathcal{\mathcal{M}}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)$ of coherent
${\mathcal{O}}_{X}$modules
equipped with a continuous and semilinear action
of ${G}_{\phantom{\rule{0.3em}{0ex}}K}$.
An
almost ${\u2102}_{p}$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
$d\in \mathbb{N}$,
two
${G}_{\phantom{\rule{0.3em}{0ex}}K}$stable finite
dimensional sub${\mathbb{Q}}_{p}$vector
spaces
${U}_{+}$ of
$V$,
${U}_{}$ of
${\u2102}_{p}^{d}$, and a
${G}_{\phantom{\rule{0.3em}{0ex}}K}$equivariant
isomorphism
$$V\u2215{U}_{+}\to {\u2102}_{p}^{d}\u2215{U}_{}.$$
These representations form an abelian category
$\mathcal{C}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)$. The main purpose of this
paper is to prove that
$\mathcal{C}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)$
can be recovered from
$\mathcal{\mathcal{M}}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)$
by a simple construction (and viceversa) inducing, in particular, an equivalence of
triangulated categories
$${D}^{b}\left(\mathcal{\mathcal{M}}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)\right)\to {D}^{b}\left(\mathcal{C}\left({G}_{\phantom{\rule{0.3em}{0ex}}K}\right)\right).$$
