Almost $C_p$-representation

Abstract

Let $\overline{{\mathbb{Q}}_p}$ be an algebraic closure of $\mathbb{Q}$ and $C$ its $p$-adic completion. Let $K$ be a finite extension of $\mathbb{Q}$ contained in $\overline{{\mathbb{Q}}_p}$ and set $G_K=\mathrm{Gal}(\overline{{\mathbb{Q}}_p}/K)$. A ${\mathbb{Q}}_p$-representation (resp. a $C$-representation) of $G_K$ is a finite dimensional $\mathbb{Q}$-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 $\mathbb{Q}$-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-$\mathbb{Q}$-vector spaces $V$ of $X$ and $V^{\prime }$ of $W$ stable under $G_K$ and an isomorphism $X/V\to W/V^{\prime }$. The almost $C$-representations of $G_K$ form an abelian category $\mathcal{C}({\mathcal{G}}_{\mathcal{K}})$. There is a unique additive function $dh:Ob\mathcal{C}({\mathcal{G}}_{\mathcal{K}})\to \mathbb{N}\times \mathbb{Z}$ such that $dh(W)=({\dim }_{C}W,0)$ if $W$ is a $C$-representation and $dh(V)=(0,{\dim }_{{\mathbb{Q}}_p}V)$ if $V$ is a ${\mathbb{Q}}_p$-representation. If $X$ and $Y$ are objects of $\mathcal{C}({\mathcal{G}}_{\mathcal{K}})$, the ${\mathbb{Q}}_p$-vector spaces ${\mathrm{Ext}}_{\mathcal{C}({\mathcal{G}}_{\mathcal{K}})}^i(X,Y)$ are finite dimensional and are zero for $i\notin \{0,1,2\}$. One gets ${\sum }_{i=0}^2(-1{)}^i{\dim }_{\mathbb{Q}}{\mathrm{Ext}}_{\mathcal{C}({\mathcal{G}}_{\mathcal{K}})}^i(X,Y)=-[K:\mathbb{Q}]h(X)h(Y)$. Moreover, there is a natural duality between ${\mathrm{Ext}}_{\mathcal{C}({\mathcal{G}}_{\mathcal{K}})}^i(X,Y)$ and ${\mathrm{Ext}}_{\mathcal{C}({\mathcal{G}}_{\mathcal{K}})}^{2-i}(Y,X(1))$.