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

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.