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Almost $$C_p$$-representation. (Presque $$C_p$$-représentations.) (French) Zbl 1130.11321
Summary: 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 $$V'$$ 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 $$\mathcal C(G_K)$$. There is a unique additive function $$dh: \text{Ob}\mathcal C(G_K)\to \mathbb N\times\mathbb Z$$ 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 $$\mathcal C(G_K)$$, the $$\mathbb Q_p$$-vector spaces $$\mathrm{Ext}^{i}_{\mathcal C(G_K)}(X,Y)$$ are finite dimensional and are zero for $$i\not\in\{0,1,2\}$$. One gets $$\sum_{i=0}^{2}(-1)^{i}\dim_{\mathbb Q}\mathrm{Ext}^{i}_{\mathcal C(G_K)}(X,Y)=-[K:\mathbb Q]h(X)h(Y)$$. Moreover, there is a natural duality between $$\mathrm{Ext}^i_{\mathcal C(G_K)}(X,Y)$$ and $$\mathrm{Ext}^{2-i}_{\mathcal C(G_K)}(Y,X(1))$$.

##### MSC:
 11F80 Galois representations 11S20 Galois theory 11S25 Galois cohomology 11G25 Varieties over finite and local fields 11S31 Class field theory; $$p$$-adic formal groups 14F30 $$p$$-adic cohomology, crystalline cohomology 14G22 Rigid analytic geometry
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