Diamantine Picard functors of rigid spaces
Abstract
For a connected smooth proper rigid space $X$ over a perfectoid field extension of $\mathbb Q_p$, we show that the étale Picard functor of $X$ defined on perfectoid test objects is the diamondification of the rigid analytic Picard functor. In particular, it is represented by a rigid analytic group variety if and only if the rigid analytic Picard functor is. Second, we study the $v$Picard functor that parametrises line bundles in the finer $v$topology on the diamond associated to $X$ and relate this to the rigid analytic Picard functor by a geometrisation of the multiplicative HodgeTate sequence. The motivation is an application to the $p$adic Simpson correspondence, namely our results pave the way towards the first instance of a new moduli theoretic perspective.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 DOI:
 10.48550/arXiv.2103.16557
 arXiv:
 arXiv:2103.16557
 Bibcode:
 2021arXiv210316557H
 Keywords:

 Mathematics  Algebraic Geometry;
 14F20;
 14G22;
 14G45
 EPrint:
 Edited some technical results for future applications. Sections 4,5 of v1 now part of arxiv:2207.13657