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Almost \(C_p\)-representation. (Presque \(C_p\)-repr√©sentations.) (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 \(\mathrm{Ext}^{2-i}_{\mathcal C(G_K)}(Y,X(1))\).

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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