Bousfield Classes and Ohkawa's Theorem: Nagoya, Japan, August 28-30, 2015 (Springer Proceedings in Mathematics & Statistics, 309)

Foreword
Contents
Memories on Ohkawa\'s Mathematical Life in Hiroshima
	1 Master Thesis
	2 MathSciNet
	3 RIMS Kokyuroku
	4 Some Comments
Depth and Simplicity of Ohkawa\'s Argument
	1 Introduction
	2 Homology Theories
	3 Spectra and Representability
	4 Bousfield Equivalence Classes of Spectra
	5 Okhawa\'s Argument
	6 Other Proofs and Extensions of Ohkawa\'s Theorem
	7 Nonrepresentable Homology Theories
	References
From Ohkawa to Strong Generation via Approximable Triangulated Categories—A Variation on the Theme of Amnon Neeman\'s Nagoya Lecture Series
	1 Introduction
	2 Ohkawa\'s Theorem on Bousfield Classes Forming a Set, and Its Shadows in Algebraic Geometry
		2.1 Bousfield Localizations
		2.2 Bousfield Classes and Ohkawa\'s Theorem
		2.3 Casacuberta–Gutiérrez-Rosický Theorem, Motivic Analogue of Ohkawa\'s Theorem
		2.4 Localizing Tensor Ideals of Derived Categories and the Fundamental Theorem of Hopkins, Neeman, Thomason and Others
	3 Hopkins–Smith Theorem and Its Motivic Analogue
	4 `3́9`42`\"̇613A``45`47`\"603ADbcoh(X) and `3́9`42`\"̇613A``45`47`\"603ADperf(X)
		4.1 `3́9`42`\"̇613A``45`47`\"603ADbcoh(X)
		4.2 `3́9`42`\"̇613A``45`47`\"603ADperf(X)
		4.3 `3́9`42`\"̇613A``45`47`\"603ADbcoh(X) and `3́9`42`\"̇613A``45`47`\"603ADperf(X) Determine Each Other
	5 Strong Generation in Derived Categories of Schemes
		5.1 Strong Generation of `3́9`42`\"̇613A``45`47`\"603ADperf(X)
		5.2 Strong Generation of `3́9`42`\"̇613A``45`47`\"603ADbcoh(X)
	References
Combinatorial Homotopy Categories
	1 Introduction
	2 Combinatorial Model Categories
	3 Restricted Yoneda Embedding
	4 Ohkawa\'s Theorem
	5 Generalized Brown Representability
	References
Notes on an Algebraic Stable Homotopy Category
	1 Introduction
	2 Ohkawa Theorem
	3 Bousfield Classes and Supports on mathcalG-Finite Objects
	References
Thick Ideals in Equivariant and Motivic Stable Homotopy Categories
	1 Introduction
	2 Thick Ideals in Classical Stable Homotopy Theory
	3 Thick Ideals in Equivariant Stable Homotopy Theory
		3.1 Equivariant Stable Homotopy Theory
		3.2 Equivariant Morava K-Theories
		3.3 Nilpotence and Lattices of Thick Ideals
		3.4 Thick Ideals and Equivariant Morava K-Theories
		3.5 Thick Ideals in mathcalSH(mathbbZ/2)f
	4 Comparison Functors
		4.1 Symmetric mathbbCP1-Spectra
		4.2 mathbbZ/2-Equivariant Symmetric Spectra
		4.3 Complex and Real Topological Realisation Functors
		4.4 Realisation Functors for Other Fields
		4.5 Constant Presheaf Functors
	5 Thick Ideals Discovered by Comparison Functors
		5.1 Consequences of the Properties of Rk, R\'k, ck and c\'k
		5.2 Finite Motivic Spectra
		5.3 Motivic Thick Ideals
	6 Thick Ideals Associated with Cohomology Theories
		6.1 Equivalence of Homology and Cohomology Theories
		6.2 Thick Ideals
		6.3 Construction and Properties of AK(n)
		6.4 Thick Ideals and Morava K-Theories
	7 mathcalSH(k)f Has More Thick Ideals than mathcalSHfin
		7.1 The Motivic Hopf Map
		7.2 Prime Ideals
		7.3 Prime Ideals in the Topological Categories mathcalSHfin and mathcalSH(mathbbZ/2)f
		7.4 Prime Ideals in the Motivic Category mathcalSH(k)f
	8 Motivic Type-n Spectra
		8.1 Universal Coefficient and Künneth Theorems
		8.2 The Motivic Steenrod Algebra
		8.3 The Motivic Adams Spectral Sequence
		8.4 Vanishing Criterion for Motivic Morava K-Theory
		8.5 Construction of Motivic Type-n Spectra
		8.6 The Constant Type-n Spectrum
	9 Bousfield Classes
		9.1 vn-Torsion
		9.2 Properties of Bousfield Classes
		9.3 The Action of vi on AP(n)
		9.4 Bousfield Classes of AK(n) and AB(n)
		9.5 Decomposition of langleAE(n)rangle
		9.6 AK(n) and AK(n+1)
	References
Some Observations About Motivic Tensor Triangulated Geometry over a Finite Field
	1 Introduction
	2 Tensor Triangulated Geometry
	3 Motivic Categories
		3.1 Grothendieck Motives
		3.2 Voevodsky Motives
		3.3 Morel–Voevodsky\'s Stable Homotopy Category
	4 Observations
		4.1 Rational Coefficients
		4.2 The Structural Morphism
		4.3 Equivariant Stable Homotopy Theory
		4.4 Final Observations
	References
Operations on Integral Lifts of K(n)
	1 Introduction
	2 Notation and Recollections
	3 Some Koszul Constructions
	4 Some Trivial Spectral Sequences
	References
A Short Introduction to the Telescope and Chromatic Splitting Conjectures
	1 Motivation: Freyd\'s Generating Hypothesis
	2 Recollections on Bousfield Localization
	3 The Telescope Conjecture
	4 Classification of Smashing Bousfield Localizations
	5 The Chromatic Splitting Conjecture
	6 An Algebraic Analogue
	References
Spectral Algebra Models of Unstable vn-Periodic Homotopy Theory
	1 Introduction
	2 Models of ``Unstable Homotopy Theory\'\'
	3 Koszul Duality
	4 Models of Rational and p-Adic Homotopy Theory
	5 vn-Periodic Homotopy Theory
	6 The Comparison Map
	7 Outline of the Proof of the Main Theorem
	8 Consequences
	9 The Arone-Ching Approach
	10 The Heuts Approach
	References
On Quasi-Categories of Comodules  and Landweber Exactness
	1 Introduction
	2 Notation
	3 Review of Quasi-Categories
	4 Opposite Monoidal Quasi-Categories and Opposite Tensored Quasi-Categories over Monoidal Quasi-Categories
		4.1 Opposite CoCartesian Fibrations
		4.2 Opposite Monoidal Quasi-Categories
		4.3 Opposites of Tensored Quasi-Categories Over Monoidal Quasi-Categories
	5 Quasi-Categories of Comodules
		5.1 Monoidal Structure on ABModA(mathcalC)op
		5.2 Comparison Maps
		5.3 Cotensor Products for Comodules in Quasi-Categories
		5.4 Equivalence of Quasi-Categories of Comodules
	6 Comodules in the Quasi-Category of Spectra
		6.1 Cotensor Product and Its Derived Functor in Algebraic Setting
		6.2 Bousfield–Kan Spectral Sequences
		6.3 Complex Oriented Spectra
		6.4 The E(n)-Local Category
		6.5 Connective Cases
		6.6 A Model of the K(n)-Local Category
	7 Proof of Proposition 1
		7.1 Examples of Inner Anodyne Maps
		7.2 Opposite Marked Anodyne Maps
		7.3 The Marked Simplicial Set widetildemathcalO(Δn)+
		7.4 Proof of Proposition 1
	References
Koszul Duality for En-Algebras in a Filtered Category
	1 Introduction
		1.1 Overview
		1.2 Basic Constructions
		1.3 Specific Results
		1.4 Further Consequences
		1.5 Outline
	2 Filtration of a Stable Category
		2.1 Complementary Localizations of a Stable Category
		2.2 Filtration
	3 Completion
	4 The Completion as a Complete Category
	5 Totalization in a Filtered Category
	6 Monoidal Structure on a Filtered Category
		6.1 Monoidal Filtered Category
		6.2 Completion of a Monoidal Structure
	7 Applications to the Koszul Duality
		7.1 Notation
		7.2 Fundamental Results
		7.3 Positivity of the Koszul Dual
		7.4 Koszul Duality
		7.5 Constructions of Positive Algebras
	References
Some Technical Aspects of Factorization Algebras on Manifolds
	1 Introduction
	2 Prefactorization Algebras
	3 Assumption on the Target Category
	4 Factorization Algebras
	5 Topological Chiral Homology
	6 Descent Properties of Factorization Algebras
	7 Product Formulae on Factorization Algebras
	References
A Role of the L2 Method in the Study  of Analytic Families
	1 L2 Method of Solving the bar Equation
	2 L2 Extension Theorems and Suita Conjecture
	3 Bergman Kernel in Analytic Families
	4 Rigidity Theorems by the L2 Technique
	5 A Splitting Theorem
	References