Line bundles on perfectoid covers: case of good reduction

TL;DR

This paper investigates the Picard groups of perfectoid spaces, demonstrating their relation to special fibers in certain covers, and introduces a new Hodge--Tate spectral sequence for .

Contribution

It establishes that Picard functors of certain perfectoid spaces are represented by their special fibers and constructs a novel Hodge--Tate spectral sequence for .

Findings

Picard groups of perfectoid spaces are not always p-divisible.

Picard functors can be represented by special fibers in certain covers.

Introduces a Hodge--Tate spectral sequence for .

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.