D-cap modules on rigid analytic spaces
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Authors
Advisors
Wadsley, Simon James
Date
2018-07-20Awarding Institution
University of Cambridge
Author Affiliation
Pure Mathematics and Mathematical Statistics
Qualification
Doctor of Philosophy (PhD)
Language
English
Type
Thesis
Metadata
Show full item recordCitation
Bode, A. (2018). D-cap modules on rigid analytic spaces (Doctoral thesis). https://doi.org/10.17863/CAM.24826
Abstract
Following the notion of $p$-adic analytic differential operators introduced by Ardakov--Wadsley, we establish a number of properties for coadmissible $\wideparen{\mathcal{D}}$-modules on rigid analytic spaces. Our main result is a $\wideparen{\mathcal{D}}$-module analogue of Kiehl's Proper Mapping Theorem, considering the 'naive' pushforward from $\wideparen{\mathcal{D}}_X$-modules to $f_*\wideparen{\mathcal{D}}_X$-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 $\wideparen{\mathcal{D}}_X$-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 arbitrary twisted $\wideparen{\mathcal{D}}$-modules on analytified partial flag varieties.
Our results rely heavily on the study of completed tensor products for $p$-adic Banach modules, for which we provide several new exactness criteria. We also show that the main results of Ardakov--Wadsley on the algebraic structure of $\wideparen{\mathcal{D}}$ still hold without assuming the existence of a smooth Lie lattice. For instance, we prove that the global sections $\wideparen{\mathcal{D}}_X(X)$ form a Frechet--Stein algebra for any smooth affinoid $X$.
Keywords
D-modules, Rigid analytic geometry, p-adic representation theory
Identifiers
This record's DOI: https://doi.org/10.17863/CAM.24826
Rights
Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
Licence URL: https://creativecommons.org/licenses/by-nc-sa/4.0/
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