an:01567057
Zbl 0984.11062
Dee, Jonathan
\(\Phi\)-\(\Gamma\)-modules for families of Galois representations
EN
J. Algebra 235, No. 2, 636-664 (2001).
00073101
2001
j
11S23 11S25 11S20
local fields; \(p\)-adic representations; complete noetherian rings
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.
Cornelius Greither (Neubiberg)