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The Galois Group of a Stable Homotopy Theory

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Advances in Mathematics 291 (2016) 403–541 Contents lists available at ScienceDirect Advances in Mathematics www.elsevier.com/locate/aim The Galois group of a stable homotopy theory Akhil Mathew Department of Mathematics, Harvard University, Cambridge, MA 02138, United States a r t i c l e i n f o a b s t r a c t Article history: To a “stable homotopy theory” (a presentable, symmetric Received 20 July 2014 monoidal stable ∞-category), we naturally associate a catego- Received in revised form 19 ry of finite étale algebra objects and, using Grothendieck’s December 2015 categorical machine, a profinite group that we call the Accepted 30 December 2015 Galois group. We then calculate the Galois groups in several Available online 4 February 2016 Communicated by Mark Behrens examples. For instance, we show that the Galois group of the periodic E∞-algebra of topological modular forms is trivial Keywords: and that the Galois group of K(n)-local stable homotopy Étale fundamental group theory is an extended version of the Morava stabilizer group. Structured ring spectra We also describe the Galois group of the stable module Thick subcategories category of a finite group. A fundamental idea throughout Descent theory is the purely categorical notion of a “descendable” algebra Quillen stratification object and an associated analog of faithfully flat descent in Topological modular forms Axiomatic Galois theory this context. © 2016 Elsevier Inc. All rights reserved. Contents 1. Introduction...................................................... 404 2. Axiomatic stable homotopy theory....................................... 407 2.1. Stable ∞-categories............................................ 407 2.2. Stable homotopy theories and 2-rings................................ 412 2.3. Examples................................................... 415 2.4. Morita theory................................................ 417 E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aim.2015.12.017 0001-8708/© 2016 Elsevier Inc. All rights reserved. 404 A. Mathew / Advances in Mathematics 291 (2016) 403–541 2.5. Étale algebras................................................ 419 3. Descent theory..................................................... 423 3.1. Comonads and descent.......................................... 424 3.2. Pro-objects.................................................. 426 3.3. Descendable algebra objects....................................... 430 3.4. Nilpotence................................................... 433 3.5. Local properties of modules....................................... 435 3.6. First examples................................................ 436 3.7. Application: descent for linear ∞-categories............................ 441 4. Nilpotence and Quillen stratification...................................... 444 4.1. Descent spectral sequences........................................ 444 4.2. Quillen stratification for finite groups................................ 447 4.3. Stratification for Hopf algebras..................................... 449 4.4. Chromatic homotopy theory...................................... 452 5. Axiomatic Galois theory.............................................. 453 5.1. The fundamental group.......................................... 454 5.2. Definitions................................................... 457 5.3. The Galois correspondence....................................... 465 5.4. Profinite groupoids............................................. 470 6. The Galois group and first computations................................... 474 6.1. Two definitions of the Galois group.................................. 475 6.2. Rognes’s Galois theory.......................................... 479 6.3. The connective case............................................ 481 6.4. Galois theory and filtered colimits.................................. 482 6.5. The even periodic and regular case.................................. 488 7. Local systems, cochain algebras, and stacks................................. 493 7.1. Inverse limits and Galois theory.................................... 493 7.2. ∞-categories of local systems...................................... 497 7.3. Stacks and finite groups......................................... 502 8. Invariance properties ................................................ 505 8.1. Surjectivity properties........................................... 506 8.2. Square-zero extensions.......................................... 509 8.3. Stronger invariance results........................................ 510 8.4. Coconnective rational E∞-algebras.................................. 513 9. Stable module ∞-categories............................................ 514 9.1. The case of Z/p ............................................... 515 9.2. Tate spectra for elementary abelian subgroups.......................... 515 9.3. G-Galois extensions for topological groups............................. 519 9.4. The general elementary abelian case................................. 522 9.5. General finite groups............................................ 525 10. Chromatic homotopy theory........................................... 529 10.1. Affineness and TMF............................................ 529 10.2. K(n)-local homotopy theory...................................... 532 10.3. Purity...................................................... 535 Acknowledgments....................................................... 538 References............................................................ 538 1. Introduction Let X be a connected scheme. One of the basic arithmetic invariants that one can extract from X is the étale 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 [30] in terms of the category of finite, étale covers of X. It provides an A. Mathew / Advances in Mathematics 291 (2016) 403–541 405 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 étale cohomology. 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 étale fundamental group, and construct a form of the Galois correspondence, in stable homotopy theory. For example, while the classical theory of [30] 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 E∞-ring spectrum. The idea of a type of Galois theory applicable to structured ring spectra begins with Rognes’s work in [71], where, for a finite group G, the notion of a G-Galois extension of E∞-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. Rognes has also explored 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, as has Hess in [32]. More recently, the PhD thesis of Pauwels [65] has studied Galois theory in tensor-triangulated categories. 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 ∞-category (C, ⊗, 1) where the tensor product commutes with all co- limits. 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 ∞-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 406 A. Mathew / Advances in Mathematics 291 (2016) 403–541 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 E∞-algebra. Given a stable homotopy theory (C, ⊗, 1), the collection of all homotopy classes of maps 1 → 1 is naturally a commutative ring RC. In general, there is
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