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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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