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