A Proper Mapping Theorem for coadmissible D-cap-modules

TL;DR

This paper proves a proper mapping theorem for coadmissible D-cap-modules on rigid analytic varieties, extending classical results and ensuring the preservation of coadmissibility under pushforward for proper morphisms.

Contribution

It establishes a D-cap-module analogue of Kiehl's Proper Mapping Theorem, including higher direct images and applications to the Beilinson--Bernstein correspondence.

Findings

Proved coadmissibility of higher direct images under proper pushforward.

Extended results to twisted D-cap-modules on partial flag varieties.

Provided geometric justification for coadmissibility preservation in global sections.

Abstract

We study the behaviour of D-cap-modules on rigid analytic varieties under pushforward along a proper morphism. We prove a D-cap-module analogue of Kiehl's Proper Mapping Theorem, considering the derived sheaf-theoretic pushforward from $\mathcal{D}_X$-cap-modules to $f_*\mathcal{D}_X$-cap-modules for proper morphisms $f: X\to Y$. Under assumptions which can be naturally interpreted as a certain properness condition on the cotangent bundle, we show that any coadmissible $\mathcal{D}_X$-cap-module has coadmissible higher direct images. This implies among other things a purely geometric justification of the fact that the global sections functor in the rigid analytic Beilinson--Bernstein correspondence preserves coadmissibility, and we are able to extend this result to twisted D-cap-modules on analytified partial flag varieties.