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Berkeley Lectures on p-adic Geometry(AMS-207)$
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Peter Scholze and Jared Weinstein

Print publication date: 2020

Print ISBN-13: 9780691202082

Published to Princeton Scholarship Online: January 2021

DOI: 10.23943/princeton/9780691202082.001.0001

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The v-topology

The v-topology

(p.149) Lecture 17 The v-topology
Berkeley Lectures on p-adic Geometry

Peter Scholze

Jared Weinstein

Princeton University Press

This chapter describes the v-topology. It develops a powerful technique for proving results about diamonds. There is a topology even finer than the pro-étale topology, the v-topology, which is reminiscent of the fpqc topology on schemes but which is more “topological” in nature. The class of v-covers is extremely general, which will reduce many proofs to very simple base cases. The chapter provides a sample application of this philosophy by establishing a general classification of p-divisible groups over integral perfectoid rings in terms of Breuil-Kisin-Fargues modules. Another use of the v-topology is to prove that certain pro-étale sheaves on Perf are diamonds without finding an explicit pro-étale cover.

Keywords:   v-topology, diamonds, v-covers, p-divisible groups, perfectoid rings, Breuil-Kisin-Fargues modules, pro-étale sheaves

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