DRINFELD'S LEMMA FOR PERFECTOID SPACES AND OVERCONVERGENCE OF MULTIVARIATE (phi, Gamma)-MODULES

Let p be a prime, let K be a finite extension of Q(p), and let n be a positive integer. We construct equivalences of categories between continuous p-adic representations of the n-fold product of the absolute Galois group G(K) and (phi, Gamma)-modules over one of several rings of n-variable power series. The case n = 1 recovers the original construction of Fontaine and the subsequent refinement by Cherbonnier-Colmez; for general n, the case K = Q(p) had been previously treated by the third author. To handle general K uniformly, we use a form of Drinfeld's lemma on the profinite fundamental groups of products of spaces in characteristic p, but for perfectoid spaces instead of schemes. We also construct the multivariate analogue of the Herr complex to compute Galois cohomology; the case K = Q(p) had been previously treated by Pal and the third author, and we reduce to this case using a form of Shapiro's lemma.