# Almost $C_{p}$-representation

### Jean-Marc Fontaine

## Abstract

Let $Q_{p} $ be an algebraic closure of $Q$ and $C$ its $p$-adic completion. Let $K$ be a finite extension of $Q$ contained in $Q_{p} $ and set $G_{K}=Gal(Q_{p} /K)$. A $Q_{p}$-representation (resp. a $C$-representation) of $G_{K}$ is a finite dimensional $Q$-vector space (resp. $C$-vector space) equipped with a linear (resp. semi-linear) continuous action of $G_{K}$. A Banach representation of $G_{K}$ is a topological $Q$-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 $G_{K}$. An almost $C$-representation of $G_{K}$ 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-$Q$-vector spaces $V$ of $X$ and $V_{′}$ of $W$ stable under $G_{K}$ and an isomorphism $X/V→W/V_{′}$. The almost $C$-representations of $G_{K}$ form an abelian category $C(G_{K})$. There is a unique additive function $dh:ObC(G_{K})→N×Z$ such that $dh(W)=(dim_{C}W,0)$ if $W$ is a $C$-representation and $dh(V)=(0,dim_{Q_{p}}V)$ if $V$ is a $Q_{p}$-representation. If $X$ and $Y$ are objects of $C(G_{K})$, the $Q_{p}$-vector spaces $Ext_{C(G_{K})}(X,Y)$ are finite dimensional and are zero for $i∈{0,1,2}$. One gets $∑_{i=0}(−1)_{i}dim_{Q}Ext_{C(G_{K})}(X,Y)=−[K:Q]h(X)h(Y)$. Moreover, there is a natural duality between $Ext_{C(G_{K})}(X,Y)$ and $Ext_{C(G_{K})}(Y,X(1))$.