Let X be a perfectoid space with tilt X ♭ . We build a natural map θ : Pic X ♭ → lim Pic X where the (inverse) limit is taken over the p-power map, and show that θ is an isomorphism if R = Γ(X, O X ) is a perfectoid ring. As a consequence we obtain a characterization of when the Picard groups of X and X ♭ 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.

Let X be a proper smooth toric variety over a perfectoid field of prime residue characteristic p. We study the perfectoid space X perf which covers X constructed by Scholze, showing that Pic(X perf ) is canonically isomorphic to Pic(X)[p −1 ]. We also compute the cohomology of line bundles on X perf and establish analogs of Demazure and Batyrev-Borisov vanishing. This generalizes the first author's analogous results for projectivoid space.

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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.