D-cap modules on rigid analytic spaces

Change log

Following the notion of p-adic analytic differential operators introduced by Ardakov--Wadsley, we establish a number of properties for coadmissible \wideparenD-modules on rigid analytic spaces. Our main result is a \wideparenD-module analogue of Kiehl's Proper Mapping Theorem, considering the 'naive' pushforward from \wideparenDX-modules to f∗\wideparenDX-modules for proper morphisms f:XY. Under assumptions which can be naturally interpreted as a certain properness condition on the cotangent bundle, we show that any coadmissible \wideparenDX-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 \wideparenD-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 \wideparenD still hold without assuming the existence of a smooth Lie lattice. For instance, we prove that the global sections \wideparenDX(X) form a Frechet--Stein algebra for any smooth affinoid X.

Wadsley, Simon James
D-modules, Rigid analytic geometry, p-adic representation theory
Doctor of Philosophy (PhD)
Awarding Institution
University of Cambridge