The Galois group of a stable homotopy theory Submitted by Akhil Mathew in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Harvard University March 24, 2014 Advisor: Michael J. Hopkins Contents 1. Introduction 3 2. Axiomatic stable homotopy theory 4 3. Descent theory 14 4. Nilpotence and Quillen stratification 27 5. Axiomatic Galois theory 32 6. The Galois group and first computations 46 7. Local systems, cochain algebras, and stacks 59 8. Invariance properties 66 9. Stable module 1-categories 72 10. Chromatic homotopy theory 82 11. Conclusion 88 References 89 Email: [email protected]. 1 1. Introduction Let X be a connected scheme. One of the basic arithmetic invariants that one can extract from X is the ´etale fundamental group π1(X; x) relative to a \basepoint" x ! X (where x is the spectrum of a separably closed field). The fundamental group was defined by Grothendieck [Gro03] in terms of the category of finite, ´etalecovers of X. It provides an analog of the usual fundamental group of a topological space (or rather, its profinite completion), and plays an important role in algebraic geometry and number theory, as a precursor to the theory of ´etalecohomology. From a categorical point of view, it unifies the classical Galois theory of fields and covering space theory via a single framework. In this paper, we will define an analog of the ´etalefundamental group, and construct a form of the Galois correspondence, in stable homotopy theory. For example, while the classical theory of [Gro03] enables one to define the fundamental (or Galois) group of a commutative ring, we will define the fundamental group of the homotopy-theoretic analog: an E1-ring spectrum. The idea of a type of Galois theory applicable to structured ring spectra begins with Rognes's work in [Rog08], where, for a finite group G, the notion of a G-Galois extension of E1-ring spectra A ! B was introduced (and more generally, E-local G-Galois extensions for a spectrum E). Rognes's definition is an analog of the notion of a finite G-torsor of commutative rings in the setting of \brave new" algebra, and it includes many highly non-algebraic examples in stable homotopy theory. For instance, the “complexification” map KO ! KU from real to complex K-theory is a fundamental example of a Z=2-Galois extension. This was taken further by Hess in [Hes09], which discusses the more general theory of Hopf-Galois extensions, intended as a topological version of the idea of a torsor over a group scheme in algebraic geometry. In this paper, we will take the setup of an axiomatic stable homotopy theory. For us, this will mean: Definition 1.1. An axiomatic stable homotopy theory is a presentable, symmetric monoidal stable 1-category (C; ⊗; 1) where the tensor product commutes with all colimits. An axiomatic stable homotopy theory defines, at the level of homotopy categories, a tensor triangulated category. Such axiomatic stable homotopy theories arise not only from stable homotopy theory itself, but also from representation theory and algebra, and we will discuss many examples below. We will associate, to every axiomatic stable homotopy theory C, a profinite group (or, in general, groupoid) which we call the Galois group π1(C). In order to do this, we will give a definition of a finite cover generalizing the notion of a Galois extension, and, using heavily ideas from descent theory, show that these can naturally be arranged into a Galois category in the sense of Grothendieck. We will actually define two flavors of the fundamental group, one of which depends only on the structure of the dualizable objects in C and is appropriate to the study of \small" symmetric monoidal 1-categories. Our thesis is that the Galois group of a stable homotopy theory is a natural invariant that one can attach to it; some of the (better studied) others include the algebraic K-theory (of the compact objects, say), the lattice of thick subcategories, and the Picard group. We will discuss several examples. The classical fundamental group in algebraic geometry can be recovered as the Galois group of the derived category of quasi-coherent sheaves. Rognes's Galois theory (or rather, faithful Galois theory) is the case of C = Mod(R) for R an E1-algebra. Given a stable homotopy theory (C; ⊗; 1), the collection of all homotopy classes of maps 1 ! 1 is naturally a commutative ring RC under composition. In general, there is always a surjection of profinite groups et (1) π1C π1 Specπ0RC: The ´etalefundamental group of SpecRC represents the \algebraic" part of the Galois theory of C. For example, if C = Mod(R) for R an E1-algebra, then the \algebraic" part of the Galois theory of C corresponds to those E1-algebras under R which are finite ´etaleat the level of homotopy groups. It is an insight of Rognes that, in general, the Galois group contains a topological component as well: the map (1) is generally not an isomorphism. The remaining Galois extensions (which behave much differently on the level of homotopy groups) can be quite useful computationally. In the rest of the paper, we will describe several computations of these Galois groups in various settings. Our basic tool is the following result, which is a refinement of (a natural generalization of) the main result of [BR08]. 3 Theorem 1.2. If R is an even periodic E1-ring with π0R regular noetherian, then the Galois group of R is that of the discrete ring π0R: that is, (1) is an isomorphism. Using various techniques of descent theory, and a version of van Kampen's theorem, we are able to compute Galois groups in several other examples of stable homotopy theories \built" from Mod(R) where R is an even periodic E1-ring; these include in particular many arising from both chromatic stable homotopy theory and modular representation theory. In particular, we prove the following three theorems. Theorem 1.3. The Galois group of the 1-category LK(n)Sp of K(n)-local spectra is the extended Morava stabilizer group. Theorem 1.4. The Galois group of the E1-algebra TMF of (periodic) topological modular forms is trivial. Theorem 1.5. There is an effective procedure to describe the Galois group of the stable module 1-category of a finite group. These results suggest a number of other settings in which the computation of Galois groups may be feasible, for example, in stable module 1-categories for finite group schemes. We hope that these results and ideas will, in addition, shed light on some of the other invariants of E1-ring spectra and stable homotopy theories. Acknowledgments. I would like to thank heartily Mike Hopkins for his advice and support over the past few years. His ideas have shaped this project, and I am grateful for the generosity with which he has shared them. In addition, I would like to thank Gijs Heuts, Tyler Lawson, Jacob Lurie, Lennart Meier, Niko Naumann, and Vesna Stojanoska for numerous helpful discussions. Finally, I would like to thank my friends and family for their love and support. This thesis would not have been possible without them. 2. Axiomatic stable homotopy theory As mentioned earlier, the goal of this paper is to extract a Galois group(oid) from a stable homotopy theory. Once again, we restate the definition. Definition 2.1. A stable homotopy theory is a presentable, symmetric monoidal stable 1-category (C; ⊗; 1) where the tensor product commutes with all colimits. In this section, intended mostly as background, we will describe several general features of the setting of stable homotopy theories. We will discuss a number of examples, and then construct a basic class of commmutative algebra objects in any such C (the so-called \´etalealgebras") whose associated corepresentable functors can be described very easily. The homotopy categories of stable homotopy theories, which acquire both a tensor structure and a compatible triangulated structure, have been described at length in the memoir [HPS97]. 2.1. Stable 1-categories. Let C be a stable 1-category in the sense of [Lur12]. Recall that stability is a condition on an 1-category, rather than extra data, in the same manner that, in ordinary category theory, being an abelian category is a property. The homotopy category of a stable 1-category is canonically triangulated, so that stable 1-categories may be viewed as enhancements of triangulated categories; however, as opposed to traditional DG-enhancements, stable 1-categories can be used to model phenomena in stable homotopy theory (such as the 1-category of spectra, or the 1-category of modules over a structured ring spectrum). Here we will describe some general features of stable 1-categories, and in particular the constructions one can perform with them. Most of this is folklore (in the setting of triangulated or DG-categories) or in [Lur12]. Definition 2.2. Let Cat1 be the 1-category of 1-categories. Given 1-categories C; D, the mapping space HomCat1 (C; D) is the maximal 1-groupoid contained in the 1-category Fun(C; D) of functors C!D. st Definition 2.3. We define an 1-category Cat1 of (small) stable 1-categories where: 4 st 1 (1) The objects of Cat1 are the stable 1-categories which are idempotent complete. st (2) Given C; D 2 Cat , the mapping space Hom st (C; D) is a union of connected components in 1 Cat1 HomCat1 (C; D) spanned by those functors which preserve finite limits (or, equivalently, colimits). Such functors are called exact.