Compairing categories of Lubin-Tate $({\phi }_L,{\mathrm{\Gamma }}_L)$-modules

In the Lubin-Tate setting we compare different categories of $({\phi }_L,{\mathrm{\Gamma }}_L)$-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.