Untilting Line Bundles on Perfectoid Spaces

Abstract

Abstract Let $X$ be a perfectoid space with tilt $X^\flat $. We build a natural map $\theta :\Pic X^\flat \to \lim \Pic X$ where the (inverse) limit is taken over the $p$-power map and show that $\theta $ is an isomorphism if $R = \Gamma (X,\sO _X)$ 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 $\Pic X$. 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.