2026-09-23T05:16:17Zhttps://api.repository.cam.ac.uk/server/oai/request

oai:www.repository.cam.ac.uk:1810/2775102024-06-26T13:50:55Zcom_1810_213747com_1810_256064col_1810_213748

D-cap modules on rigid analytic spaces 10.17863/CAM.24826 Bode, Andreas 0000000260435511 Wadsley, Simon James 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 $X$. University of Cambridge 2018-07-20 Thesis Doctoral Doctor of Philosophy (PhD) en https://www.repository.cam.ac.uk/handle/1810/277510 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/a0d9e6ed-c4c5-43d5-9c7b-b7dd7c9afba4/download 87eda9de84448d1f82354d60eee3eb5f https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/e399ed03-f35a-45cc-a7a7-5cedd4911b84/download d59c20f8bf58c1a9afa430aa9c3ffbfb https://creativecommons.org/licenses/by-nc-sa/4.0/ D-modules Rigid analytic geometry p-adic representation theory