Galois Cohomology for Lubin-Tate $({\phi }_q,{\Gamma }_{LT})$-modules over Coefficient rings

Abstract

The classification of the local Galois representations using $(\phi ,\Gamma )$-modules by Fontaine has been generalized by Kisin and Ren over the Lubin-Tate extensions of local fields using the theory of $({\phi }_q,{\Gamma }_{LT})$-modules. In this paper, we extend the work of (Fontaine) Herr by introducing a complex which allows us to compute cohomology over the Lubin-Tate extensions and compare it with the Galois cohomology groups. We further extend that complex to include certain non-abelian extensions. We then deduce some relations of this cohomology with those arising from $({\psi }_q,{\Gamma }_{LT})$-modules. We also compute the Iwasawa cohomology over the Lubin-Tate extensions in terms of ${\psi }_q$-operator acting on the étale $({\phi }_q,{\Gamma }_{LT})$-module attached to the local Galois representation. Moreover, we generalize the notion of $({\phi }_q,{\Gamma }_{LT})$-modules over the coefficient ring $R$ and show that the equivalence given by Kisin and Ren extends to the Galois representations over $R$. This equivalence allows us to generalize our results to the case of coefficient rings.

Accepted in Research in Number Theory