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$$\Phi$$-$$\Gamma$$-modules for families of Galois representations. (English) Zbl 0984.11062
This paper shows that Fontaine’s “Linearisation Approach” for $$Z_p$$-adic representations of an absolute local Galois group $$G_K$$ carries over to a setting where the base ring $$Z_p$$ is replaced by a general coefficient ring $$R$$, that is, $$R$$ is noetherian complete with finite residue field of characteristic $$p$$. More specifically the author constructs categories of $$\Phi$$-modules, and of $$\Phi$$-$$\Gamma$$-modules depending on $$R$$, which give back the categories Fontaine worked with, on setting $$R=Z_p$$. He then proceeds to carry over the equivalences constructed by Fontaine (or a slight variant thereof) to the new setting: the category of $$R$$-modules of finite type with a continuous $$R$$-linear action of $$G_K$$ is equivalent to the abovementioned category of $$\Phi$$-$$\Gamma$$-modules. The author begins with the equal characteristic case where one just deals with $$\Phi$$-modules, and then achieves the transition to the unequal characteristic case by standard constructions.

##### MSC:
 11S23 Integral representations 11S25 Galois cohomology 11S20 Galois theory
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##### References:
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