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

This volume originated in the workshop held at Nagoya University, August 28–30, 2015, focusing on the surprising and mysterious Ohkawa's theorem: the Bousfield classes in the stable homotopy category SH form a set. An inspiring, extensive mathematical story can be narrated starting with Ohkawa's theorem, evolving naturally with a chain of motivational questions: Ohkawa's theorem states that the Bousfield classes of the stable homotopy category SH surprisingly forms a set, which is still very mysterious. Are there any toy models where analogous Bousfield classes form a set with a clear meaning? The fundamental theorem of Hopkins, Neeman, Thomason, and others states that the analogue of the Bousfield classes in the derived category of quasi-coherent sheaves D qc ( X ) form a set with a clear algebro-geometric description. However, Hopkins was actually motivated not by Ohkawa's theorem but by his own theorem with Smith in the triangulated subcategory SH c , consisting of compact objects in SH . Now the following questions naturally occur: (1) Having theorems of Ohkawa and Hopkins-Smith in SH , are there analogues for the Morel-Voevodsky A 1 -stable homotopy category SH ( k ), which subsumes SH when k is a subfield of C ?, (2) Was it not natural for Hopkins to have considered D qc ( X ) c instead of D qc ( X )? However, whereas there is a conceptually simple algebro-geometrical interpretation D qc ( X ) c = D perf ( X ), it is its close relative D b coh ( X ) that traditionally, ever since Oka and Cartan, has been intensively studied because of its rich geometric and physical information. This book contains developments for the rest of the story and much more, including the chromatics homotopy theory, which the Hopkins–Smith theorem is based upon, and applications of Lurie's higher algebra, all by distinguished contributors. Foreword 7 Contents 8 Memories on Ohkawa's Mathematical Life in Hiroshima 10 1 Master Thesis 10 2 MathSciNet 10 3 RIMS Kokyuroku 11 4 Some Comments 11 Depth and Simplicity of Ohkawa's Argument 12 1 Introduction 12 2 Homology Theories 13 3 Spectra and Representability 15 4 Bousfield Equivalence Classes of Spectra 18 5 Okhawa's Argument 19 6 Other Proofs and Extensions of Ohkawa's Theorem 21 7 Nonrepresentable Homology Theories 22 References 23 From Ohkawa to Strong Generation via Approximable Triangulated Categories—A Variation on the Theme of Amnon Neeman's Nagoya Lecture Series 25 1 Introduction 25 2 Ohkawa's Theorem on Bousfield Classes Forming a Set, and Its Shadows in Algebraic Geometry 30 2.1 Bousfield Localizations 31 2.2 Bousfield Classes and Ohkawa's Theorem 40 2.3 Casacuberta–Gutiérrez-Rosický Theorem, Motivic Analogue of Ohkawa's Theorem 41 2.4 Localizing Tensor Ideals of Derived Categories and the Fundamental Theorem of Hopkins, Neeman, Thomason and Others 42 3 Hopkins–Smith Theorem and Its Motivic Analogue 49 4 `3́9`42`"̇613A``45`47`"603ADbcoh(X) and `3́9`42`"̇613A``45`47`"603ADperf(X) 57 4.1 `3́9`42`"̇613A``45`47`"603ADbcoh(X) 57 4.2 `3́9`42`"̇613A``45`47`"603ADperf(X) 64 4.3 `3́9`42`"̇613A``45`47`"603ADbcoh(X) and `3́9`42`"̇613A``45`47`"603ADperf(X) Determine Each Other 73 5 Strong Generation in Derived Categories of Schemes 80 5.1 Strong Generation of `3́9`42`"̇613A``45`47`"603ADperf(X) 83 5.2 Strong Generation of `3́9`42`"̇613A``45`47`"603ADbcoh(X) 84 References 91 Combinatorial Homotopy Categories 97 1 Introduction 97 2 Combinatorial Model Categories 98 3 Restricted Yoneda Embedding 99 4 Ohkawa's Theorem 103 5 Generalized Brown Representability 105 References 108 Notes on an Algebraic Stable Homotopy Category 110 1 Introduction 110 2 Ohkawa Theorem 111 3 Bousfield Classes and Supports on mathcalG-Finite Objects 113 References 115 Thick Ideals in Equivariant and Motivic Stable Homotopy Categories 116 1 Introduction 116 2 Thick Ideals in Classical Stable Homotopy Theory 121 3 Thick Ideals in Equivariant Stable Homotopy Theory 123 3.1 Equivariant Stable Homotopy Theory 124 3.2 Equivariant Morava K-Theories 126 3.3 Nilpotence and Lattices of Thick Ideals 128 3.4 Thick Ideals and Equivariant Morava K-Theories 133 3.5 Thick Ideals in mathcalSH(mathbbZ/2)f 137 4 Comparison Functors 138 4.1 Symmetric mathbbCP1-Spectra 138 4.2 mathbbZ/2-Equivariant Symmetric Spectra 139 4.3 Complex and Real Topological Realisation Functors 140 4.4 Realisation Functors for Other Fields 144 4.5 Constant Presheaf Functors 145 5 Thick Ideals Discovered by Comparison Functors 149 5.1 Consequences of the Properties of Rk, R'k, ck and c'k 149 5.2 Finite Motivic Spectra 150 5.3 Motivic Thick Ideals 154 6 Thick Ideals Associated with Cohomology Theories 158 6.1 Equivalence of Homology and Cohomology Theories 158 6.2 Thick Ideals 160 6.3 Construction and Properties of AK(n) 161 6.4 Thick Ideals and Morava K-Theories 163 7 mathcalSH(k)f Has More Thick Ideals than mathcalSHfin 164 7.1 The Motivic Hopf Map 164 7.2 Prime Ideals 169 7.3 Prime Ideals in the Topological Categories mathcalSHfin and mathcalSH(mathbbZ/2)f 169 7.4 Prime Ideals in the Motivic Category mathcalSH(k)f 172 8 Motivic Type-n Spectra 174 8.1 Universal Coefficient and Künneth Theorems 175 8.2 The Motivic Steenrod Algebra 176 8.3 The Motivic Adams Spectral Sequence 177 8.4 Vanishing Criterion for Motivic Morava K-Theory 179 8.5 Construction of Motivic Type-n Spectra 180 8.6 The Constant Type-n Spectrum 184 9 Bousfield Classes 186 9.1 vn-Torsion 187 9.2 Properties of Bousfield Classes 192 9.3 The Action of vi on AP(n) 193 9.4 Bousfield Classes of AK(n) and AB(n) 204 9.5 Decomposition of langleAE(n)rangle 218 9.6 AK(n) and AK(n+1) 219 References 223 Some Observations About Motivic Tensor Triangulated Geometry over a Finite Field 227 1 Introduction 227 2 Tensor Triangulated Geometry 228 3 Motivic Categories 233 3.1 Grothendieck Motives 233 3.2 Voevodsky Motives 235 3.3 Morel–Voevodsky's Stable Homotopy Category 238 4 Observations 242 4.1 Rational Coefficients 243 4.2 The Structural Morphism 244 4.3 Equivariant Stable Homotopy Theory 246 4.4 Final Observations 246 References 247 Operations on Integral Lifts of K(n) 250 1 Introduction 250 2 Notation and Recollections 252 3 Some Koszul Constructions 256 4 Some Trivial Spectral Sequences 262 References 263 A Short Introduction to the Telescope and Chromatic Splitting Conjectures 266 1 Motivation: Freyd's Generating Hypothesis 267 2 Recollections on Bousfield Localization 268 3 The Telescope Conjecture 269 4 Classification of Smashing Bousfield Localizations 271 5 The Chromatic Splitting Conjecture 273 6 An Algebraic Analogue 275 References 276 Spectral Algebra Models of Unstable vn-Periodic Homotopy Theory 279 1 Introduction 279 2 Models of ``Unstable Homotopy Theory'' 282 3 Koszul Duality 284 4 Models of Rational and p-Adic Homotopy Theory 290 5 vn-Periodic Homotopy Theory 293 6 The Comparison Map 299 7 Outline of the Proof of the Main Theorem 300 8 Consequences 307 9 The Arone-Ching Approach 311 10 The Heuts Approach 316 References 324 On Quasi-Categories of Comodules and Landweber Exactness 328 1 Introduction 328 2 Notation 331 3 Review of Quasi-Categories 331 4 Opposite Monoidal Quasi-Categories and Opposite Tensored Quasi-Categories over Monoidal Quasi-Categories 335 4.1 Opposite CoCartesian Fibrations 335 4.2 Opposite Monoidal Quasi-Categories 340 4.3 Opposites of Tensored Quasi-Categories Over Monoidal Quasi-Categories 342 5 Quasi-Categories of Comodules 343 5.1 Monoidal Structure on ABModA(mathcalC)op 343 5.2 Comparison Maps 345 5.3 Cotensor Products for Comodules in Quasi-Categories 347 5.4 Equivalence of Quasi-Categories of Comodules 351 6 Comodules in the Quasi-Category of Spectra 353 6.1 Cotensor Product and Its Derived Functor in Algebraic Setting 353 6.2 Bousfield–Kan Spectral Sequences 356 6.3 Complex Oriented Spectra 360 6.4 The E(n)-Local Category 364 6.5 Connective Cases 366 6.6 A Model of the K(n)-Local Category 370 7 Proof of Proposition 1 373 7.1 Examples of Inner Anodyne Maps 373 7.2 Opposite Marked Anodyne Maps 374 7.3 The Marked Simplicial Set widetildemathcalO(Δn)+ 376 7.4 Proof of Proposition 1 379 References 382 Koszul Duality for En-Algebras in a Filtered Category 384 1 Introduction 384 1.1 Overview 384 1.2 Basic Constructions 385 1.3 Specific Results 386 1.4 Further Consequences 388 1.5 Outline 390 2 Filtration of a Stable Category 390 2.1 Complementary Localizations of a Stable Category 390 2.2 Filtration 392 3 Completion 395 4 The Completion as a Complete Category 397 5 Totalization in a Filtered Category 399 6 Monoidal Structure on a Filtered Category 401 6.1 Monoidal Filtered Category 401 6.2 Completion of a Monoidal Structure 402 7 Applications to the Koszul Duality 403 7.1 Notation 403 7.2 Fundamental Results 403 7.3 Positivity of the Koszul Dual 404 7.4 Koszul Duality 406 7.5 Constructions of Positive Algebras 407 References 409 Some Technical Aspects of Factorization Algebras on Manifolds 410 1 Introduction 410 2 Prefactorization Algebras 411 3 Assumption on the Target Category 411 4 Factorization Algebras 412 5 Topological Chiral Homology 413 6 Descent Properties of Factorization Algebras 417 7 Product Formulae on Factorization Algebras 420 References 424 A Role of the L2 Method in the Study of Analytic Families 426 1 L2 Method of Solving the bar Equation 427 2 L2 Extension Theorems and Suita Conjecture 429 3 Bergman Kernel in Analytic Families 430 4 Rigidity Theorems by the L2 Technique 433 5 A Splitting Theorem 435 References 436