# Cohomology and overconvergence for representations of powers of Galois groups

## TL;DR

This paper extends the theory of Galois cohomology for p-adic representations to multivariable settings, demonstrating overconvergence and equivalence of categories for multivariable $(,g)$-modules, and confirming the Herr complex computes cohomology.

## Contribution

It generalizes Herr's complex to multivariable $(,g)$-modules, proves overconvergence for all p-adic representations of power Galois groups, and establishes the Herr complex as a cohomology computational tool.

## Key findings

- Galois cohomology groups can be computed via multivariable Herr's complex.
- All p-adic representations of power Galois groups are overconvergent.
- Overconvergent Herr complex computes Galois cohomology.

## 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.