Untilting Line Bundles on Perfectoid Spaces

Abstract

Let $X$ be a perfectoid space with tilt $X^{♭}$. We construct a canonical map $\theta :\mathrm{Pic}{X}^{♭}\to \lim \mathrm{Pic}X$ where the (inverse) limit is taken over the $p$-power map, and show that $\theta$ is an isomorphism if $R=\Gamma (X,{\mathcal{O}}_X)$ 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 $\mathrm{Pic}X$. The main technical ingredient is the vanishing of higher derived limits of the unit group $R^{\ast }$, whence the main result follows from the Grothendieck spectral sequence.

16 pages. Final version. v2: Updated to correct and simplify the proof of the $p$-divisibility of $R^{\ast }/R^{\circ \ast }$ (Proposition 2.9). v3: Proposition 2.9 strengthened and section 3 greatly simplified. To appear in International Mathematics Research Notices