Almost -representation

  • Jean-Marc Fontaine


Let be an algebraic closure of and its -adic completion. Let be a finite extension of contained in and set . A -representation (resp. a -representation) of is a finite dimensional -vector space (resp. -vector space) equipped with a linear (resp. semi-linear) continuous action of . A Banach representation of is a topological -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 . An almost -representation of is a Banach representation which is almost isomorphic to a -representation, i.e. such that there exists a -representation , finite dimensional sub--vector spaces of and of stable under and an isomorphism . The almost -representations of form an abelian category . There is a unique additive function such that if is a -representation and if is a -representation. If and are objects of , the -vector spaces are finite dimensional and are zero for . One gets . Moreover, there is a natural duality between and .