A note on fontaine theory using different Lubin-Tate groups

The starting point of Fontaine theory is the possibility of translating the study of a p-adic representation of the absolute Galois group of a finite extension K of Qp into the investigation of a (φ, Γ)-module. This is done by decomposing the Galois group along a totally ramified extension of K, via the theory of the field of norms: the extension used is obtained by means of the cyclotomic tower which, in turn, is associated to the multiplicative Lubin-Tate group. It is known that one can insert different Lubin-Tate groups into the "Fontaine theory" machine to obtain equivalences with new categories of (φ, Γ)-modules (here φ may be iterated). This article uses only (φ, Γ)-theoretical terms to compare the different (φ, Γ) modules arising from various Lubin-Tate groups.