| Summary: | We study the behavior of D-modules on rigid analytic varieties under pushforward along a proper morphism. We prove a D-module analogue of Kiehl’s proper mapping theorem, considering the derived sheaf-theoretic pushforward from DX-modules to f∗Dx-modules for proper morphisms f : X → Y . Under assumptions which can be naturally interpreted as a certain properness condition on the cotangent bundle, we show that any coadmissible Dx-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 DÛ-modules on analytified partial flag varieties
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