English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

The classification of the local Galois representations using$(\varphi,\Gamma)$-modules by Fontaine has been generalized by Kisin and Renover the Lubin-Tate extensions of local fields using the theory of$(\varphi_q,\Gamma_{LT})$-modules. In this paper, we extend the work of(Fontaine) Herr by introducing a complex which allows us to compute cohomologyover the Lubin-Tate extensions and compare it with the Galois cohomologygroups. We further extend that complex to include certain non-abelianextensions. We then deduce some relations of this cohomology with those arisingfrom $(\psi_q,\Gamma_{LT})$-modules. We also compute the Iwasawa cohomologyover the Lubin-Tate extensions in terms of $\psi_q$-operator acting on the\'{e}tale $(\varphi_q,\Gamma_{LT})$-module attached to the local Galoisrepresentation. Moreover, we generalize the notion of$(\varphi_q,\Gamma_{LT})$-modules over the coefficient ring $R$ and show thatthe equivalence given by Kisin and Ren extends to the Galois representationsover $R$. This equivalence allows us to generalize our results to the case ofcoefficient rings.

Galois Cohomology for Lubin-Tate $(\varphi_q,\Gamma_{LT})$-modules over Coefficient rings