Untilting Line Bundles on Perfectoid Spaces

TL;DR

This paper establishes a canonical isomorphism between the Picard groups of a perfectoid space and its tilt, revealing conditions under which they agree, using advanced homological techniques.

Contribution

It constructs a canonical map between Picard groups of perfectoid spaces and their tilts, proving it is an isomorphism under certain conditions, and characterizes Picard group agreement via p-divisibility.

Findings

The map $ heta$ is an isomorphism when $R$ is a perfectoid ring.

Picard groups of $X$ and $X^lat$ agree if and only if $ ext{Pic} X$ is p-divisible.

Higher derived limits of $R^*$ vanish, enabling the main isomorphism.

Abstract

Let $X$ be a perfectoid space with tilt $X^\flat$. We construct a canonical map $\theta:\operatorname{Pic} X^\flat\to\lim\operatorname{Pic} X$ where the (inverse) limit is taken over the $p$-power map, and show that $\theta$ is an isomorphism if $R = \Gamma(X,\mathscr{O}_X)$ is a perfectoid ring. As a consequence we obtain a characterization of when the Picard groups of $X$ and $X^\flat$ agree in terms of the $p$-divisibility of $\operatorname{Pic} X$. The main technical ingredient is the vanishing of higher derived limits of the unit group $R^*$, whence the main result follows from the Grothendieck spectral sequence.