##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Gabriel Dorfsman-Hopkins

#### UC Berkeley

## Untilting Line Bundles on Perfectoid Spaces

##### Abstract:

Let $X$ be a perfectoid space with tilt $X^\flat$. We build a natural map $\theta:\mathrm{Pic} X^\flat\to\lim\mathrm{Pic} X$ 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 $X^\flat$ 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 $R^*$, whence the main result follows from the Grothendieck spectral sequence.

### November 18, 2021

### 1:00 PM

### APM 6402 and Zoom; see https://www.math.ucsd.edu/$\sim$nts/

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