On D-cap-Modules of Finite Length on Rigid Analytic Spaces

TL;DR

This paper demonstrates that certain D-modules on quasi-compact smooth rigid analytic spaces are of finite length, introducing Hilbert polynomials for modules over completed Weyl algebras as a key tool.

Contribution

It establishes finite length properties of D-cap-modules and introduces Hilbert polynomials for modules over completed Weyl algebras in the rigid analytic setting.

Findings

Extension functor preserves finite length of D-modules on rigid spaces.

Meromorphic connections and local cohomology groups are of finite length as weakly holonomic D-cap-modules.

Hilbert polynomials are developed for modules over completed Weyl algebras.

Abstract

We show that for quasi-compact smooth rigid analytic spaces, the extension functor sends holonomic D-modules to coadmissible D-cap-modules which are of finite length as weakly holonomic D-cap-modules. Using this, we show that the meromorphic connections considered by Bode--Bitoun and the local cohomology groups considered by Ardakov--Bode--Wadsley are of finite length as weakly holonomic D-cap-modules for quasi-compact smooth rigid analytic spaces. As a central tool, we introduce and study Hilbert polynomials for finitely generated modules over completed Weyl algebras.