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

Let K be a finite extension of p and GK the absolute Galois group. Then GK acts on the fundamental curve X of p-adic Hodge theory and we may consider the abelian category (GK) of coherent OX-modules equipped with a continuous and semilinear action of GK.

An almost p-representation of GK is a p-adic Banach space V equipped with a linear and continuous action of GK such that there exists d , two GK-stable finite dimensional sub-p-vector spaces U+ of V , U of pd, and a GK-equivariant isomorphism

VU+ pdU .

These representations form an abelian category C(GK). The main purpose of this paper is to prove that C(GK) can be recovered from (GK) by a simple construction (and vice-versa) inducing, in particular, an equivalence of triangulated categories

Db((G K)) Db(C(G K)).

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$p$-adic Hodge theory, vector bundle
Mathematical Subject Classification 2010
Primary: 11S20, 14H60
Received: 9 February 2019
Revised: 4 August 2019
Accepted: 4 August 2019
Published: 14 October 2019
Jean-Marc Fontaine
Université de Paris-Sud
91405 Orsay