Diamantine Picard functors of rigid spaces

TL;DR

This paper explores the relationship between different Picard functors of smooth proper rigid spaces over perfectoid fields, establishing their equivalences and implications for the $p$-adic Simpson correspondence.

Contribution

It demonstrates that the étale Picard functor is the diamondification of the rigid analytic Picard functor and relates the $v$-Picard functor to the rigid one via the Hodge–Tate sequence, advancing the moduli perspective.

Findings

Étale Picard functor is the diamondification of the rigid analytic Picard functor.

The $v$-Picard functor relates to the rigid one through the Hodge–Tate sequence.

Results facilitate a new moduli perspective for the $p$-adic Simpson correspondence.

Abstract

For a connected smooth proper rigid space $X$ over a perfectoid field extension of $\mathbb Q_p$, we show that the \'etale 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 Hodge--Tate 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.