Line bundles on perfectoid covers: case of good reduction

Abstract

We study Picard groups and Picard functors of perfectoid spaces which are limits of rigid spaces. For sufficiently large covers that are limits of rigid spaces of good reduction, we show that the Picard functor can be represented by the special fibre. We use our results to answer several open questions about Picard groups of perfectoid spaces from the literature, for example we show that these are not always $p$-divisible. Along the way, we construct a "Hodge--Tate spectral sequence for ${\mathbb{G}}_m$" of independent interest.

Added Thm 5.16; minor edits reflecting new/updated literature

References in corpus (2)

• Perfectoid covers of abelian varieties and the weight-monodromy conjecture
• On Picard Groups of Perfectoid Covers of Toric Varieties

Cited by in corpus (3)

• The $p$-adic Corlette-Simpson correspondence for abeloids
• Pro-étale uniformisation of abelian varieties
• On the $v$-Picard group of Stein spaces