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KU-Local Stable Homotopy Theory

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KU-Local Stable Homotopy Theory A Stack-theoretic perspective on KU-local stable homotopy theory by Jeroen W. van der Meer A master thesis supervised by Lennart Meier co-supervised by Bas Edixhoven To be defended on July 05, 2019 Mathematisch Instituut Universiteit Leiden Contents Preface 5 Goal...............................................5 Outline and dependencies...................................6 Conventions...........................................7 Acknowledgements.......................................8 A final word...........................................8 1 Spectra 9 1.1 The stable homotopy category...............................9 1.2 First examples of spectra................................. 16 1.3 Complex orientations................................... 19 1.4 Bousfield localizations................................... 24 2 Hopf algebroids and algebraic stacks 29 2.1 Groupoid objects and Hopf algebroids.......................... 29 2.2 Stacks over sites...................................... 32 2.3 Algebraic stacks...................................... 38 2.4 Quasi-coherent sheaves and substacks......................... 40 2.5 Flat Hopf algebroids and Adams stacks......................... 43 3 The moduli stack of formal groups 46 3.1 Constructions of the stack of formal groups....................... 46 3.2 The height filtration.................................... 49 3.3 The Landweber exact functor theorem.......................... 53 3.4 Lubin–Tate deformation theory.............................. 61 4 The KU(p)-local stable homotopy category 71 4.1 Overview of Bousfield’s results.............................. 71 4.2 A fracture square..................................... 74 4.3 The category of KU(p)-comodules revisited........................ 84 4.4 Conclusion and scope for future work.......................... 95 A Formal group laws 97 A.1 First definitions...................................... 97 A.2 Lazard’s theorem...................................... 101 A.3 Heights........................................... 103 A.4 Endomorphisms of formal group laws.......................... 108 Bibliography 114 Index 116 Preface Goal Stable homotopy theory is the branch of mathematics that studies phenomena in homotopy theory that arise after repeated application of the suspension operation. Classical examples of such phenomena include the Freudenthal suspension theorem (homotopy groups of spheres sta- bilize after applying the suspension functor sufficiently many times), the suspension isomorphism (reduced cohomology of a space and of its suspension coincide), and Bott periodicity (K-theory is periodic under the suspension operation). In the same way that algebraic topologists do much of their work in the category of spaces, ho- motopy theorists studying stable phenomena often work in the stable homotopy category. Roughly speaking, this category is what we obtain when we take the homotopy category of topological spaces, and force the suspension functor to be an equivalence of categories. We are left with only those phenomena that survive repeated applications of the suspension functor, which should be precisely the stable phenomena that the homotopy theorists are after. The stable homotopy category is rather complicated, and in order to study it, it is helpful to zoom in a bit further and restrict attention to more digestible pieces of this category. This approach has been quite successful, and its success can be seen, for instance, in the chromatic approach to stable homotopy theory. This approach asserts that structures in the stable homotopy category should loosely correspond to structures in the category of sheaves over a particular space that we call the stack of formal groups. This stack of formal groups admits a filtration, and the chromatic point of view dictates that this ought to translate to a filtration of pieces of the stable homotopy category, each piece more complicated than the previous one, but all of them simpler than the entire thing. For the most part, this thesis will confine itself to only one digestible piece: we will be interested in the aspects of the stable homotopy category that can be detected by K-homology, or the K-local stable homotopy category for short. Building on earlier work due to Adams, this K-local category was first systematically studied several decades ago by Alridge Bousfield who, in 1979, published a paper, [2], in which he gave an algebraic classification of the objects in the K-local stable homotopy 5 category (or, of the K-local spectra for short, in the same way that objects of the stable homotopy category are called spectra), at least when further localized at an odd prime. The paper of Bousfield is almost entirely algebraic. The fact that we can reduce questions in homotopy theory to algebra is essentially thanks to a tool called the Adams spectral sequence. In the special case of the K-local category, this tool roughly tells us the following: given two spaces X and Y , if one understands enough of the (purely algebraic!) structure of Hom K∗(X), K∗(Y ) , where K∗ means K-homology, then one can deduce what the homotopy classes of maps from X to Y should be in the K-local stable homotopy category. Bousfield’s paper predates the chromatic approach that we mentioned earlier. Recall that this approach studies a certain filtration of increasingly complicated pieces of the stable homotopy category. As it turns out, the very first piece of this filtration is strongly related to the K-local category that Bousfield was interested in, and when suitably defined, they are in fact the same thing. This brings us to the goal of this thesis: with the modern, chromatic language at our fingertips, we will shed a new light on the techniques used in the original paper, only to find that, at least in hindsight, the complicated algebraic structures that were initially considered are geometric in origin. In fact, they are but a minor reflection of the deep and mysterious geometric structures governing stable phenomena — structures which, for the most part, remain to be understood to this very day. Outline and dependencies In AppendixA we study formal group laws over commutative rings. We begin by introducing the basic definitions, and proceed to study heights of formal group laws, ending with a mostly independent section on endomorphisms of formal group laws. Most of the results are well-known, though we occasionally use non-standard terminology when discussing heights. Chapter1 serves as an introduction to the basic language of stable homotopy theory that we will need in the rest of this thesis. None of the results in this chapter are new, and the chapter is mostly included for the sake of bookkeeping, not to mention because of the lack of a suitable reference to which to refer. In Chapter2 we introduce Hopf algebroids and algebraic stacks. The purpose of this chapter is two-fold. First, it seems that Hopf algebroids are less familiar than algebraic stacks to most algebraic geometers, and vice versa for topologists, so that it is reasonable to give a combined discussion on both notions. Second, we use this chapter to fix definitions that may otherwise be ambiguous due to inconsistent conventions in the literature. With the language of spectra and algebraic stacks at hand, we are ready to give a detailed study of the geometric structure of the stack of formal groups in Chapter3. We explain how 6 heights of formal group laws give rise to a filtration of the stack into substacks, and proceed to look into the geometry of each of the layers of this stack. We also discuss Landweber’s exact functor theorem, allowing us to construct spectra in the stable homotopy category using maps into the stack of formal groups. We use this theorem to give detailed constructions of various spectra that carry the name ‘Morava E-theory’. It is my hope that the statements and proofs in this chapter fill up some gaps in the mathematical literature. Finally, in Chapter4 we use the machinery developed in the previous chapter to study the K-local stable homotopy category. We start out by outlining the key constructions and result in Bousfield’s paper. After that, we investigate the category of sheaves over certain substacks of the stack of formal groups; by that point we will know that this category is intimately related to the K-local category that we are ultimately interested in. Finally, we use the conclusions of our investigation to revisit Bousfield’s results, and indicate how our results may shed a conceptually simplifying light on the original ideas. In case the reader is interested only in some specific parts of this thesis, the following depen- dency diagram may be of help. A A.1 A.4 A.2 A.3 1 4 1.4 1.2 1.3 1.1 all 2 3 all Conventions Whenever we work with spaces, we assume that we are working inside some convenient category of topological spaces, such as CW complexes or compactly generated weak Hausdorff spaces. At no point will the precise choice of category of spaces be relevant. Likewise, whenever we work with spectra, we will be working in the usual stable homotopy category. Apart from a few constructions in Section 1.2, a particular choice of model for the stable homotopy category will never be relevant. This means that the reader unfamiliar with stable homotopy theory can safely assume the existence of the category as a black box without much loss of continuity. 7 In order for our story to fit within the framework of classical algebraic geometry, all our rings are assumed to be commutative and with unity. In practice, stable homotopy groups of commutative ring
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