COHOMOLOGY AND OVERCONVERGENCE FOR REPRESENTATIONS OF POWERS OF GALOIS GROUPS

Source abstract

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 $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules. Using Tate duality and a pairing for multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -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 $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.