Almost $C_p$-representation

Abstract

Let $\overline{\Bbb Q_p}$ be an algebraic closure of $\Bbb Q$ and $C$ its $p$-adic completion. Let $K$ be a finite extension of $\Bbb Q$ contained in $\overline{\Bbb Q_p}$ and set $G_K=\mathrm{Gal}(\overline{\Bbb Q_p}/K)$. A $\Bbb Q_p$-representation (resp. a $C$-representation) of $G_K$ is a finite dimensional $\Bbb Q$-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 $\Bbb Q$-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-$\Bbb Q$-vector spaces $V$ of $X$ and $V'$ of $W$ stable under $G_K$ and an isomorphism $X/V\to W/V'$. The almost $C$-representations of $G_K$ form an abelian category $\cal C(G_K)$. There is a unique additive function $dh: {Ob}\cal C(G_K)\to \Bbb N\times\Bbb Z$ such that $dh(W)=(\dim_{C}W,0)$ if $W$ is a $C$-representation and $dh(V)=(0,\dim_{\Bbb Q_p}V)$ if $V$ is a $\Bbb Q_p$-representation. If $X$ and $Y$ are objects of $\cal C(G_K)$, the $\Bbb Q_p$-vector spaces $\mathrm{Ext}^{i}_{\cal C(G_K)}(X,Y)$ are finite dimensional and are zero for $i\not\in\{0,1,2\}$. One gets $\sum_{i=0}^{2}(-1)^{i}\dim_{\Bbb Q}\mathrm{Ext}^{i}_{\cal C(G_K)}(X,Y)=-[K:\Bbb Q]h(X)h(Y)$. Moreover, there is a natural duality between $\mathrm{Ext}^i_{\cal C(G_K)}(X,Y)$ and $\mathrm{Ext}^{2-i}_{\cal C(G_K)}(Y,X(1))$.