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Berkeley Lectures on p-adic Geometry(AMS-207)$
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Berkeley Lectures on p-adic Geometry: (AMS-207)

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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Diamonds

Diamonds

Chapter:
(p.56) Lecture 8 Diamonds
Source:
Berkeley Lectures on p-adic Geometry
Author(s):

Peter Scholze

Jared Weinstein

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691202082.003.0008

This chapter investigates the notion of a diamond. The idea is that there should be a functor which “forgets the structure morphism to Zp.” The desired quotient in the example provided in the chapter exists in a category of sheaves on the site of perfectoid spaces with pro-étale covers. The chapter then defines pro-étale morphisms between perfectoid spaces. A morphism of perfectoid spaces is pro-étale if it is locally (on the source and target) affinoid pro-étale. The intuitive definition of diamonds involved the tilting functor in case of perfectoid spaces of characteristic 0. For this reason, diamonds are defined as certain pro-étale sheaves on the category of perfectoid spaces of characteristic p.

Keywords:   diamond, perfectoid spaces, pro-étale morphisms, pro-étale covers, affinoid pro-étale, tilting functor, pro-étale sheaves

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