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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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[1] Cherbonnier, F.; Colmez, P., Théorie d’Iwasawa des représentations p-adiques d’un corps local, J. amer. math. soc., 12, 241-268, (1999) · Zbl 0933.11056
[2] Grothendieck, A.; Dieudonné, J., Eléments de géométrie algébrique 0_IV, Publ. math. inst. hautes etudes sci., 20, (1964)
[3] Fontaine, J.-M., Représentations p-adiques des corps locaux, I, The Grothendieck festschrift, (1990), Birkhäuser Basel, p. 249-310 · Zbl 0743.11066
[4] Fontaine, J.-M., Groupes p-divisible sur LES corps locaux, Astérisque, 47-48, (1977)
[5] Fröhlich, A.; Taylor, M.J., Algebraic number theory, (1991), Cambridge Univ. Press Cambridge · Zbl 0744.11001
[6] Herr, L., Sur la cohomologie galoisienne des corps p-adique, Bull. soc. math. France, 126, 563-600, (1998) · Zbl 0967.11050
[7] Matsumura, H., Commutative ring theory, (1989), Cambridge Univ. Press Cambridge
[8] Mazur, B., Deformation theory of Galois representations, Modular forms and Fermat’s last theorem, (1997), Springer New York, p. 243-313 · Zbl 0901.11015
[9] Tate, J., Relations between K2 and Galois cohomology, Invent. math., 36, 257-274, (1976) · Zbl 0359.12011
[10] Weibel, C., An introduction to homological algebra, (1994), Cambridge Univ. Press Cambridge · Zbl 0797.18001
[11] Wintenberger, J.-P., Le corps des normes de certaines extensions infinies des corps locaux; applications, Ann. sci. ecole norm. super., 16, 59-89, (1983) · Zbl 0516.12015
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