UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Cotangent spectra and the determinantal sphere Permalink https://escholarship.org/uc/item/1rx093jf Author Peterson, Eric Christopher Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Cotangent spectra and the determinantal sphere by Eric Christopher Peterson A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Constantine Teleman, Chair Professor Martin C. Olsson Professor Christos H. Papadimitriou Spring 2015 Cotangent spectra and the determinantal sphere Copyright 2015 by Eric Christopher Peterson 1 Abstract Cotangent spectra and the determinantal sphere by Eric Christopher Peterson Doctor of Philosophy in Mathematics University of California, Berkeley Professor Constantine Teleman, Chair We explore the generalization of cellular decomposition in chromatically localized stable cate- gories suggested by Picard–graded homotopy groups. In particular, for K(d) a Morava K–theory, we show that the Eilenberg–Mac Lane space K(Z, d + 1) has a K(d)–local cellular decomposi- tion tightly analogous to the usual decomposition of infinite–dimensional complex projective space (alias K(Z,2)) into affine complex cells. Additionally, we identify these generalized cells in terms of classical invariants — i.e., we show that their associated line bundles over the Lubin– Tate stack are tensor powers of the determinant bundle. (In particular, these methods give the first choice–free construction of the determinantal sphere S[det].) Finally, we investigate the bottom attaching map in this exotic cellular decomposition, and we justify the sense in which it selects a particular K(d)–local homotopy class 1 2 Σ− [det] [det] S ! S generalizing the classical four “Hopf invariant 1” classes h in the cofiber sequences 1 n 2 h 1 2 Σ− ( )^ S −! Pk ! Pk associated to the four normed real division algebras: k RCHO n 1 2 4 8 . h 2 η ν σ We also include a lengthy introduction to the subject of chromatic homotopy theory, outlining all of the tools relevant to the statements of our original results. i Contents Contents i 0 Introduction 1 1 Background on E– and K–theory 4 1.1 Schemes and formal schemes . 5 1.2 Formal Lie groups . 12 1.3 Basic applications to topology . 20 1.4 The Γ –local stable category . 30 2 Ravenel–Wilson and E–theory 40 2.1 Recollections on Hopf rings . 41 2.2 The calculation in E–theory . 45 2.3 Connections to Dieudonné theory . 46 3 The cotangent space construction 53 3.1 Cotensors of coassociative spectra . 54 3.2 The main calculation . 63 3.3 Examples . 65 3.4 Koszul duality and a new Γ –local Hopf map . 69 det 3.5 Comparison of K with Westerland’s Rd .......................... 71 A The Morava K–theory of inverse limits 73 Bibliography 83 ii Acknowledgments Stable homotopy theory is an exceptionally warm and welcoming community, and as a grad- uate student I’ve had the pleasure of traveling to many institutions and interacting with many mathematicians. This has made for a rich scholastic experience—and it now makes for a lengthy acknowledgements page. My first debt of gratitude is to UIUC, where I began studying. Matt Ando showed me an incredible amount of care and attention, and to this day remains the mathematician I have learned the most from. Randy McCarthy also influenced me strongly, first by snatching me out of the jaws of industry and second by being so patient while I found my mathematical sea legs that first year. I would never have begun this endeavor—nevermind finished—if not for the two of them. At Berkeley, poor Constantin has had to tolerate me as a student in a field askew from his own, and it’s to his enormous credit that he’s been unwaiveringly supportive and helpful. My officemate and fellow homotopy theorist, Aaron Mazel-Gee, has been wonderfully challenging at work, a good travel partner, and generally a great friend. Work would have been much lonelier without him. Jose, Kevin, Alex, and Ralph (as well as many others) also made my time here really pleasant. Thanks, guys. Additionally, our friends at Stanford in the south Bay, Cary Malkiewich chief among them, are also great folks and good eggs with their continued attendance of our traveling topology seminar. The good graces of Peter Teichner have allowed me to visit the MPIM in Bonn twice, where the bulk of the work on this document actually took place. I learned a lot there, especially from Justin Noel and from the graduate students Irakli, Lennart, and Karol. Additionally, various saints helped care for Aaron and me while we were there: Dave, Nikki, and Moritz were great company, cheerful Cerolein made our visit as smooth as it could have been, and Nisha kept us well-fed and in good spirits. Nat Stapleton coaxed Haynes Miller into allowing Aaron and me to visit MIT for what was a truly incredible semester, with joie de vivre and joie de maths in equal amounts. Saul, Michael, Michael, Tomer, Rune, Tobi, Vesna, Kyle, Drew, Sebastian, and Nat again all contributed to the magic, and it was a special pleasure to meet and work with Mike Hopkins and Mark Behrens. Several of us also attended the chromatic Talbot workshop that year, and it was a real treat to meet young people so interested in chromatic homotopy—especially Jon, Vitaly, and Sean. I also had valuable mathematical conversations with Kathryn Hess, Tyler Lawson, and Craig Westerland about the mathematics presented here. Additionally, the project is the brainchild of Neil Strickland, and the whole mentality of this document reflects his influence. The intellectual debt owed to him is obvious and profound. I’ve had a great time over the past five years, both at work and at play. For that, I have all my friends to thank, mathematical and otherwise. 1 Chapter 0 Introduction Before proceeding to the body of the thesis, we give an overview of what to expect in more detail than provided in the abstract above. In Chapter 1, we will introduce a sequence of homology theories KΓ and EΓ , the study of which is generally referred to as “chromatic homotopy theory”. Morava first constructed these homology theories by employing a fundamental connection, uncovered by Quillen, between sta- ble homotopy theory and the theory of formal Lie groups. Because of the strength of this con- nection, the Morava K–theories exert a remarkable amount of control over the stable category, and it is often profitable to study a given problem in stable homotopy theory by considering its analogues in the KΓ –local stable category for every KΓ . Our primary goal in this thesis is to compute an involved example of the following maxim: After KΓ –localization, spectra can acquire unusually efficient cellular decompositions. There is an uninteresting sense in which the maxim above is often true: any kind of localiza- tion has the potential to lose information, and hence it may render a complicated object simple if those complications are locally invisible. A more interesting interpretation of the maxim comes from the observation that the notion of “cell” in the KΓ –local stable category is dramatically en- larged from the usual one. In the global stable category, an (n + 1)–cell is attached to a spectrum n n X along a homotopy class S X — i.e., using an element in πnX . The sphere spectra S are ! n n uniquely characterized in the global stable category by the property that S has a mate S− such n n 0 that S S− is the monoidal unit S . In the KΓ –local stable category, however, there are many more such^ objects, and it has been previously observed that it is profitable to grade homotopy groups using the entire collection. If we take these as selecting the loci along which we are permit- ted to attach new cells, then we can build considerably more intricate decompositions. (Precisely how much more intricate is unknown, as there are very few KΓ for which we can exhaustively name the invertible KΓ –local spectra.) However, the KΓ –local category has bad properties which prohibit the easy manufacture of cellular decompositions. First consider, again, the global situation. For an n–connective spec- trum X , the Hurewicz isomorphism H X π X shows that any generating set of H X can be n ∼= n n CHAPTER 0. INTRODUCTION 2 lifted to a sequence of homotopy classes of X which, in turn, can be used to model the bottom cells of X . Continuing inductively shows that the integral homology groups HZ X can be used to describe a (non-functorial) cellular structure on X which is minimal in the sense∗ that it uses the minimum number of cells in each graded dimension. However, in the KΓ –local case, it is a theorem of Hovey and Strickland that the KΓ –local category has no localizing or colocalizing subcategories, and hence no natural notion of cellular or Postnikov decomposition (since a t– structure would, in particular, have such subcategories). Hence, for a KΓ –local spectrum, there is generally no natural way to select a “bottom-most class” on KΓ –homology. This makes extracting a KΓ –local cellular decomposition from KΓ –homology difficult. As a concrete example, we consider the Eilenberg–Mac Lane spaces K(Z, t). Ravenel and Wilson computed the KΓ –cohomology ring KΓ∗(K(Z, t)) to be a finite–dimensional power series ring, and there is a dual description of the KΓ –homology.