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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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Complements on adic spaces

Complements on adic spaces

Chapter:
(p.35) Lecture 5 Complements on adic spaces
Source:
Berkeley Lectures on p-adic Geometry
Author(s):

Peter Scholze

Jared Weinstein

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691202082.003.0005

This chapter analyzes a collection of complements in the theory of adic spaces. These complements include adic morphisms, analytic adic spaces, and Cartier divisors. It turns out that there is a very general criterion for sheafyness. In general, uniformity does not guarantee sheafyness, but a strengthening of the uniformity condition does. Moreover, sheafyness, without any extra assumptions, implies other good properties. Ultimately, it is not immediately clear how to get a good theory of coherent sheaves on adic spaces. The chapter then considers Cartier divisors on adic spaces. The term closed Cartier divisor is meant to evoke a closed immersion of adic spaces.

Keywords:   adic spaces, adic morphisms, analytic adic spaces, Cartier divisors, sheafyness, uniformity, closed Cartier divisor

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