Presque C P -représentations

Let Qp be an algebraic closure of Qp and C its p-adic completion. Let K be a finite extension of Qp contained in Qp and set GK = Gal(Qp/K). A Qp-representation (resp. a C-representation) of GK is a finite dimensional Qp-vector space (resp. C-vector space) equipped with a linear (resp. semi-linear) continuous action of GK . A banach representation of GK is a topological Qp-vector space, whose topology may be defined by a norm with respect to which it is complete, equipped with a linear and continuous action of GK . An almost C-representation of GK is a banach representation X which is almost isomorphic to a C-representation, i.e. such that there exists a C-representation W , finite dimensional sub-Qpvector spaces V of X and V ′ of W stable under GK and an isomorphism X/V → W/V . The almost C-representations of GK form an abelian category C(GK). There is a unique additive function dh : ObC(GK) → N × Z such that dh(W ) = (dimC W, 0) if W is a C-representation and dh(V ) = (0,dimQp V ) if V is a Qp-representation. If X and Y are objects of C(GK), the Qp-vector spaces ExtiC(GK)(X,Y ) are finite dimensional and are zero for i 6∈ {0, 1, 2}. One gets ∑2 i=0(−1) dimQpExt i C(GK)(X,Y ) = −[K : Qp]h(X)h(Y ). Moreover, there is a natural duality between ExtiC(GK)(X,Y ) and Ext C(GK)(Y,X(1)). 2000 Mathematics Subject Classification: Primary: 11F80, 11S20 11S25. Secondary:11G25, 11S31, 14F30, 14G22. Mots clés: Corps p-adiques, représentations galoisiennes p-adiques, cohomologie galoisienne, espaces de Banach p-adiques.