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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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Adic spaces II

Adic spaces II

(p.17) Lecture 3 Adic spaces II
Berkeley Lectures on p-adic Geometry

Peter Scholze

Jared Weinstein

Princeton University Press

This chapter defines adic spaces. A scheme is a ringed space which locally looks like the spectrum of a ring. An adic space will be something similar. The chapter then identifies the adic version of “locally ringed space.” Briefly, it is a topologically ringed topological space equipped with valuations. The chapter also reflects on the role of A+ in the definition of adic spaces. The subring A+ in a Huber pair may seem unnecessary at first: why not just consider all continuous valuations on A? Specifying A+ keeps track of which inequalities have been enforced among the continuous valuations. Finally, the chapter differentiates between sheafy and non-sheafy Huber pairs.

Keywords:   adic spaces, locally ringed spaces, topological space, valuations, Huber pairs, continuous valuations, sheafy Huber pairs, non-sheafy Huber pairs

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