Period Rings with Big Coefficients and Application II

TL;DR

This paper advances the noncommutative deformation theory of p-adic Hodge structures, focusing on descent of modules with large coefficients and exploring geometric aspects relevant to noncommutative Iwasawa theory and analytic geometry.

Contribution

It introduces the study of descent of pseudocoherent modules with large noncommutative coefficients and systematically explores noncommutative geometric aspects of deformed Hodge structures.

Findings

Improved noncommutative Hodge-Iwasawa theory with applications to Tamagawa number conjectures.

Generalized Kedlaya-Liu glueing of pseudocoherent Banach modules to large noncommutative coefficients.

Initiated systematic study of noncommutative geometric aspects of p-adic Hodge structures.

Abstract

We continue our study on the corresponding noncommutative deformation of the relative $p$-adic Hodge structures of Kedlaya-Liu along our previous work. In this paper, we are going to initiate the study of the corresponding descent of pseudocoherent modules carrying large noncommutative coefficients. And also we are going to more systematically study the corresponding noncommutative geometric aspects of noncommutative deformation of Hodge structures, which will definitely also provide the insights not only for noncommutative Iwasawa theory but also for noncommutative analytic geometry. The noncommutative Hodge-Iwasawa theory is now improved along some very well-defined direction (we will expect many well-targeted applications to noncommutative Tamagawa number conjectures from the modern perspectives of Burns-Flach-Fukaya-Kato), while the corresponding Kedlaya-Liu glueing of pseudocoherent Banach modules with certain stability is also generalized to the large noncommutative coefficient case.