Cohomology and overconvergence for representations of powers of Galois groups

Abstract

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

58 pages, final version, to appear in Journal of the Institute of Mathematics of Jussieu