Untilting Line Bundles on Perfectoid Spaces
Gabriel Dorfsman-Hopkins (UC Berkeley)
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.
number theory
Audience: researchers in the topic
( paper )
Comments: pre-talk at 1:20pm
Series comments: Most talks are preceded by a pre-talk for graduate students and postdocs. The pre-talks start 40 minutes prior to the posted time (usually at 1:20pm Pacific) and last about 30 minutes.
Organizers: | Kiran Kedlaya*, Alina Bucur, Aaron Pollack, Cristian Popescu, Claus Sorensen |
*contact for this listing |