Category and Cohomology of Hodge-Iwasawa Modules

TL;DR

This paper explores the categories and cohomologies of Hodge-Iwasawa modules, emphasizing their role in Iwasawa theory, deformation of local systems, and applications in arithmetic and analytic geometry.

Contribution

It introduces new cohomological frameworks for Hodge-Iwasawa modules, connecting them to Iwasawa theory and geometric applications.

Findings

Cohomologies are essential for Iwasawa theoretic developments.

Applications include local systems over analytic spaces and arithmetic Riemann-Hilbert correspondence.

Provides insights into equivariant Iwasawa theory and geometrization.

Abstract

In this paper we study the corresponding categories and the corresponding cohomologies of the Hodge-Iwasawa modules we developed in our series papers on Hodge-Iwasawa theory. The corresponding cohomologies will be essential in the corresponding development of the contact with the corresponding Iwasawa theoretic consideration, while they are as well very crucial in the corresponding study of the corresponding deformations of local systems over general analytic spaces. We contact with some applications in analytic geometry and arithmetic geometry which all have their own interests and deserve further study for us in the future, including local systems over general analytic spaces after Kedlaya-Liu, arithmetic Riemann-Hilbert correspondence in families after Liu-Zhu, and equivariant Iwasawa theory and geometrization of equivariant Iwasawa theory after Berger-Fourquaux and Nakamura.