  In the Lubin-Tate setting we compare different categories of
$(\varphi_L,\Gamma_L)$-modules over various perfect or imperfect coefficient
rings. Moreover, we study their associated Herr-complexes. Finally, we show
that a Lubin Tate extension gives rise to a weakly decompleting, but not
decompleting tower in the sense of Kedlaya and Liu.
