zbMATH — the first resource for mathematics

\(\Phi\)-\(\Gamma\)-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.

11S23 Integral representations
11S25 Galois cohomology
11S20 Galois theory
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.