Let K/Q p be a finite extension with ring of integers o, let G be a connected reductive split Q p -group of Borel subgroup P = T N and let α be a simple root of T in N . We associate to a finitely generated module D over the Fontaine ring over o endowed with a semilinearétale action of the monoid T + (acting on the Fontaine ring via α), a G(Q p )equivariant sheaf of o-modules on the compact space G(Q p )/P (Q p ). Our construction generalizes the representation D ⊠ P 1 of GL(2, Q p ) associated by Colmez to a (ϕ, Γ)module D endowed with a character of Q * p .

We show that the category of continuous representations of the dth direct power of the absolute Galois group of Q p on finite dimensional F p -vector spaces (resp. finitely generated Z p -modules, resp. finite dimensional Q p -vector spaces) is equivalent to the category of étale (ϕ, Γ)-modules over a d-variable Laurent-series ring over F p (resp. over Z p , resp. over Q p ).Moreover, for each element α ∈ ∆ we have the partial Frobenius ϕ α , and group Γ α ∼ = Gal(Q p (µ p ∞ )/Q p ) acting on the variable X α in the usual way and commuting with the other *

We show that the Galois cohomology groups of p-adic representations of a direct power of Gal(Q p /Q p ) can be computed via the generalization of Herr's complex to multivariable (ϕ, Γ)-modules. Using Tate duality and a pairing for multivariable (ϕ, Γ)modules we extend this to analogues of the Iwasawa cohomology. We show that all padic representations of a direct power of Gal(Q p /Q p ) are overconvergent and, moreover, passing to overconvergent multivariable (ϕ, Γ)-modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.