{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/277510"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/277510","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"D-cap modules on rigid analytic spaces","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 $X$.","abstract_html":"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&#x27;s Proper Mapping Theorem, considering the &#x27;naive&#x27; pushforward from <span class=\"etd-inline-math\">\\wideparen{\\mathcal{D}}<sub>X</sub></span>-modules to <span class=\"etd-inline-math\">f<sub>*</sub>\\wideparen{\\mathcal{D}}<sub>X</sub></span>-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 <span class=\"etd-inline-math\">\\wideparen{\\mathcal{D}}<sub>X</sub></span>-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 <span class=\"etd-inline-math\">\\wideparen{\\mathcal{D}}<sub>X</sub>(X)</span> form a Frechet--Stein algebra for any smooth affinoid $X$.","abstract_has_math":true,"creators":["Bode, Andreas"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Wadsley, Simon James"],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-07-20","date_published":"2018-07-20","updated_at":"2026-07-22T22:24:10Z","subjects":["D-modules","Rigid analytic geometry","p-adic representation theory"],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/a0d9e6ed-c4c5-43d5-9c7b-b7dd7c9afba4/download","https://creativecommons.org/licenses/by-nc-sa/4.0/"],"identifier_entries":[{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000260435511"],"render_values":[{"text":"0000-0002-6043-5511","href":"https://orcid.org/0000-0002-6043-5511","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.24826","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Wadsley, Simon James"]},{"key":"dc:creator","label":"Author","values":["Bode, Andreas"]},{"key":"dc:creator.authoridentifier","label":"Author Identifier","values":["0000000260435511"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2018-07-20"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/277510"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["D-modules","Rigid analytic geometry","p-adic representation theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/a0d9e6ed-c4c5-43d5-9c7b-b7dd7c9afba4/download","https://creativecommons.org/licenses/by-nc-sa/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.24826"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/e399ed03-f35a-45cc-a7a7-5cedd4911b84/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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 $X$."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["87eda9de84448d1f82354d60eee3eb5f","d59c20f8bf58c1a9afa430aa9c3ffbfb"]},{"key":"dc:title","label":"Title","values":["D-cap modules on rigid analytic spaces"]}]}],"canonical_facts":{"dc:contributor.advisor":["Wadsley, Simon James"],"dc:creator":["Bode, Andreas"],"dc:creator.authoridentifier":["0000000260435511"],"dc:date.issued":["2018-07-20"],"dc:description.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 $X$."],"dc:format.checksum.md5":["87eda9de84448d1f82354d60eee3eb5f","d59c20f8bf58c1a9afa430aa9c3ffbfb"],"dc:identifier.doi":["10.17863/CAM.24826"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/e399ed03-f35a-45cc-a7a7-5cedd4911b84/download"],"dc:language":["en"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/277510"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/a0d9e6ed-c4c5-43d5-9c7b-b7dd7c9afba4/download","https://creativecommons.org/licenses/by-nc-sa/4.0/"],"dc:subject":["D-modules","Rigid analytic geometry","p-adic representation theory"],"dc:title":["D-cap modules on rigid analytic spaces"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:10Z"}