  The classification of the local Galois representations using
$(\varphi,\Gamma)$-modules by Fontaine has been generalized by Kisin and Ren
over 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 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
\'{e}tale $(\varphi_q,\Gamma_{LT})$-module attached to the local Galois
representation. Moreover, we generalize the notion of
$(\varphi_q,\Gamma_{LT})$-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.

Number Theory