MULTIVARIABLE (phi,Gamma)-MODULES AND REPRESENTATIONS OF PRODUCTS OF GALOIS GROUPS: THE CASE OF THE IMPERFECT RESIDUE FIELD

Let K be a complete discretely valued field with mixed characteristic (0,p) and imperfect residue field k(alpha). Let Delta be a finite set. We construct an equivalence of categories between finite dimensional F-p-representations of the product of Delta copies of the absolute Galois group of K and multivariable etale (phi,Gamma)-modules over a multivariable Laurent series ring over k(alpha).