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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.
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Subspace foliations and collapse of closed flat manifolds
Abstract We study relations between certain totally geodesic foliations of a closed flat manifold and its collapsed Gromov–Hausdorff limits. Our main results explicitly identify such collapsed limits as flat orbifolds, and provide algebraic and geometric criteria to determine whether they are singular.
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- Award ID(s):
- 1904342
- PAR ID:
- 10442962
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Mathematische Nachrichten
- Volume:
- 295
- Issue:
- 12
- ISSN:
- 0025-584X
- Page Range / eLocation ID:
- p. 2338-2356
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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