#### DOCUMENTA MATHEMATICA, Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 285-385

Jean-Marc Fontaine

Presque $C_p$-Représentations

Let $\qpb$ be an algebraic closure of $\qp$ and $C$ its $p$-adic completion. Let $K$ be a finite extension of $\qp$ contained in $\qpb$ and set $G_K={ Gal}(\qpb/K)$. A $\qp$-{repre\-sen\-ta\-tion} (resp. a $C$-{repre\-sen\-ta\-tion}) {of} $G_K$ is a finite dimensional $\qp$-vector space (resp. $C$-vector space) equipped with a linear (resp. semi-linear) continuous action of $G_K$. A {banach repre\-sen\-ta\-tion of} $G_K$ 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 $G_K$. \hbox{ An almost $C$-repre\-sen\-ta\-tion of $G_K$} is a banach repre\-sen\-ta\-tion $X$ which is {almost isomorphic} to a $C$-repre\-sen\-ta\-tion, i.e. such that there exists a $C$-repre\-sen\-ta\-tion $W$, finite dimensional sub-$\qp$-vector spaces $V$ of $X$ and $V'$ of $W$ stable under $G_K$ and an isomorphism $X/V\f W/V'$. The almost $C$-repre\-sen\-ta\-tions of $G_K$ form an abelian category $\cg$. There is a unique additive function $dh: {Ob}\cg\f \n\times\z$ such that $dh(W)=(\dim_{C}W,0)$ if $W$ is a $C$-repre\-sen\-ta\-tion and $dh(V)=(0,\dim_{\qp}V)$ if $V$ is a $\qp$-repre\-sen\-ta\-tion. If $X$ and $Y$ are objects of $\cg$, the $\qp$-vector spaces $\ext^{i}_{\cg}(X,Y)$ are finite dimensional and are zero for $i\not\in\{0,1,2\}$. One gets $\sum_{i=0}^{2}(-1)^{i}{ dim}_{\qp}{Ext}^{i}_{\cg}(X,Y)=-[K:\qp]h(X)h(Y)$. Moreover, there is a natural duality between $\ext^{i}_{\cg}(X,Y)$ and $\ext^{2-i}_{\cg}(Y,X(1))$.

2000 Mathematics Subject Classification: Primary: 11F80, 11S20 11S25. Secondary:11G25, 11S31, 14F30, 14G22.

Keywords and Phrases: $p$-adic fields, $p$-adic Galois representations, Galois cohomology, $p$-adic Banach spaces.

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