 Title Pages
 Foreword

Lecture 1 Introduction 
Lecture 2 Adic spaces 
Lecture 3 Adic spaces II 
Lecture 4 Examples of adic spaces 
Lecture 5 Complements on adic spaces 
Lecture 6 Perfectoid rings 
Lecture 7 Perfectoid spaces 
Lecture 8 Diamonds 
Lecture 9 Diamonds II 
Lecture 10 Diamonds associated with adic spaces 
Lecture 11 Mixedcharacteristic shtukas 
Lecture 12 Shtukas with one leg 
Lecture 13 Shtukas with one leg II 
Lecture 14 Shtukas with one leg III 
Lecture 15 Examples of diamonds 
Lecture 16 Drinfeld’s lemma for diamonds 
Lecture 17 The vtopology 
Lecture 18 vsheaves associated with perfect and formal schemes 
Lecture 19 The ${B}_{\text{dR}}^{+}$affine Grassmannian 
Lecture 20 Families of affine Grassmannians 
Lecture 21 Affine flag varieties 
Lecture 22 Vector bundles and Gtorsors on the relative FarguesFontaine curve 
Lecture 23 Moduli spaces of shtukas 
Lecture 24 Local Shimura varieties 
Lecture 25 Integral models of local Shimura varieties  Bibliography
 Index
Adic spaces II
Adic spaces II
 Chapter:
 (p.17) Lecture 3 Adic spaces II
 Source:
 Berkeley Lectures on padic Geometry
 Author(s):
Peter Scholze
Jared Weinstein
 Publisher:
 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 nonsheafy Huber pairs.
Keywords: adic spaces, locally ringed spaces, topological space, valuations, Huber pairs, continuous valuations, sheafy Huber pairs, nonsheafy Huber pairs
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 Title Pages
 Foreword

Lecture 1 Introduction 
Lecture 2 Adic spaces 
Lecture 3 Adic spaces II 
Lecture 4 Examples of adic spaces 
Lecture 5 Complements on adic spaces 
Lecture 6 Perfectoid rings 
Lecture 7 Perfectoid spaces 
Lecture 8 Diamonds 
Lecture 9 Diamonds II 
Lecture 10 Diamonds associated with adic spaces 
Lecture 11 Mixedcharacteristic shtukas 
Lecture 12 Shtukas with one leg 
Lecture 13 Shtukas with one leg II 
Lecture 14 Shtukas with one leg III 
Lecture 15 Examples of diamonds 
Lecture 16 Drinfeld’s lemma for diamonds 
Lecture 17 The vtopology 
Lecture 18 vsheaves associated with perfect and formal schemes 
Lecture 19 The ${B}_{\text{dR}}^{+}$affine Grassmannian 
Lecture 20 Families of affine Grassmannians 
Lecture 21 Affine flag varieties 
Lecture 22 Vector bundles and Gtorsors on the relative FarguesFontaine curve 
Lecture 23 Moduli spaces of shtukas 
Lecture 24 Local Shimura varieties 
Lecture 25 Integral models of local Shimura varieties  Bibliography
 Index