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A generalization of Sen’s theory. (Une généralisation de la théorie de Sen.) (French) Zbl 1072.11089
Let \(K\) be a local field of characteristic \(0\) with residue field \(k\) of characteristic \(p>0\). Let \({\mathbb C}\) be the completion of an algebraic closure \(\overline{K}\) of \(K\) and let \(G\) be the Galois group \({\text{Gal}}( \overline{K}/K)\). Then \({\mathbb C}\) is algebraically closed and the action of \(G\) extends continuously to \({\mathbb C}\).
The main goal of the paper under review is to generalize the results of S. Sen [Invent. Math. 62, 89–116 (1980; Zbl 0463.12005)] obtained in case \(k\) is perfect. In this work the author does not assume \(k\) to be perfect and assumes that \([k: k^p]= p^h < \infty\), \(h\geq 0\).
Let \({\mathbf {Rep}}_E (J)\) denote the \(E^J\)-linear additive category of finite dimensional \(E\)-representations, where \(E\) is a topological field on which a topological group \(J\) acts. A description of the category of \({\mathbb C}\)-representations is given, generalizing Sen’s theory which corresponds to the case \(h=0\).
Let \(t_1,\dots, t_h\) be elements of the ring of integers of \(K\) such that \(k=k^p[\overline{t}_1, \dots, \overline{t}_h]\). Let \(K_\infty\) be the field generated by the \(p^n\)-th roots of unity and those of \(t_i\) for all \(i=1,\dots, h\) and \(n\) any positive integer and let \(L\) be the closure of \(K_\infty\) in \({\mathbb C}\).
Let \(H\) and \(\Gamma\) denote the Galois groups \({\text{Gal}}(\overline {K}/K_\infty)\) and \({\text{Gal}} (K_\infty/K)\) respectively. For any \(L\)-representation \(X\) of \(\Gamma\), let \(X^{\text{f}}\) denote the union of all finite-dimensional \(K_ \infty\)-subspaces of \(X\) which are stable under the action of \(\Gamma\). The main results of the paper are described by the following two theorems.
(1) We have \({\mathbb C}^H=L\) and for every \({\mathbb C}\)-representation \(W\) of \(H\), the \({\mathbb C}\)-linear map \({\mathbb C}\otimes _L W^H\to W\) induced by the inclusion of \(W^H\) into \(W\) is an isomorphism. The functor \(W\to W^H\) establishes an equivalence between the categories \({\mathbf {Rep}}_{\mathbb C}(H)\) and the category of finite dimensional vector spaces over \(L\) and also induces an equivalence between the categories \({\mathbf {Rep}}_{\mathbb C}(G)\) and \({\mathbf {Rep}}_L(\Gamma)\).
(2) We have \(L^{\text{f}}=K_\infty\) and for any \(L\)-representation \(X\) of \(\Gamma\), the \(L\)-linear application \(L\otimes _{K_\infty} X^{\text{f}}\to X\) induced by the inclusion of \(X^{\text{f}}\) into \(X\) is an isomorphism. The functor \(X\to X^{\text{f}}\) gives a category equivalence between \({\mathbf {Rep}} _L (\Gamma)\) and \({\mathbf {Rep}} _{K_\infty} (\Gamma)\).
In the case of Sen’s paper, \(h=0\), \(\Gamma\) is a one-dimensional \(p\)-adic commutative Lie group. In the general case \(\Gamma\) is a \(p\)-adic Lie group of dimension \(h+1\). This is a Lie group which is an extension of a Lie group of dimension one and a group isomorphic to \({\mathbf Z} _p ^h\), where \({\mathbf Z}_p\) denotes the ring of \(p\)-adic integers.
For the proof of these results, the key technical result is that the inclusion \(K_\infty\hookrightarrow L={\mathbb C}^H\) induces the bijections \[ H^1 (G,{\text{GL}}_d({\mathbb C})) \cong H^1(\Gamma, {\text{GL}}_d(L)) \cong H^1(\Gamma,{\text{GL}}_d(K_\infty)). \]

MSC:
11S25 Galois cohomology
11S15 Ramification and extension theory
11S20 Galois theory
12G05 Galois cohomology
14F30 \(p\)-adic cohomology, crystalline cohomology
22E50 Representations of Lie and linear algebraic groups over local fields
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