To a "stable homotopy theory" (a presentable, symmetric monoidal stable ∞-category), we naturally associate a category of finiteétale algebra objects and, using Grothendieck's categorical machine, a profinite group that we call the Galois group. We then calculate the Galois groups in several examples. For instance, we show that the Galois group of the periodic E∞-algebra of topological modular forms is trivial and that the Galois group of K(n)-local stable homotopy theory is an extended version of the Morava stabilizer group. We also describe the Galois group of the stable module category of a finite group. A fundamental idea throughout is the purely categorical notion of a "descendable" algebra object and an associated analog of faithfully flat descent in this context.

Abstract. This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of E∞-ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus obtained. As a major application, we calculate the Picard groups of the periodic spectrum of topological modular forms T M F and the nonperiodic and non-connective T mf . We find that Pic(T M F ) is cyclic of order 576, generated by the suspension ΣT M F (a result originally due to Hopkins), while Pic(T mf ) = Z ⊕ Z/24. In particular, we show that there exists an invertible T mf -module which is not equivalent to a suspension of T mf .

Abstract. Let G be a finite group and let F be a family of subgroups of G. We introduce a class of G-equivariant spectra that we call F -nilpotent. This definition fits into the general theory of torsion, complete, and nilpotent objects in a symmetric monoidal stable ∞-category, with which we begin. We then develop some of the basic properties of F -nilpotent G-spectra, which are explored further in the sequel to this paper.In the rest of the paper, we prove several general structure theorems for ∞-categories of module spectra over objects such as equivariant real and complex K-theory and Borel-equivariant M U . Using these structure theorems and a technique with the flag variety dating back to Quillen, we then show that large classes of equivariant cohomology theories for which a type of complex-orientability holds are nilpotent for the family of abelian subgroups. In particular, we prove that equivariant real and complex K-theory, as well as the Borel-equivariant versions of complex-oriented theories, have this property.

We compute the mod 2 homology of the spectrum tmf of topological modular forms by proving a 2-local equivalence tmf ∧ DA(1) ≃ tmf 1 (3) ≃ BP 2 , where DA(1) is an eight cell complex whose cohomology doubles the subalgebra A(1) of the Steenrod algebra generated by Sq 1 and Sq 2 . To do so, we give, using the language of stacks, a modular description of the elliptic homology of DA(1) via level three structures. We briefly discuss analogs at odd primes and recover the stack-theoretic description of the Adams-Novikov spectral sequence for tmf.Date: December 21, 2015.

Let G be a finite group. To any family F of subgroups of G, we associate a thick ⊗-ideal F Nil of the category of G-spectra with the property that every G-spectrum in F Nil (which we call F-nilpotent) can be reconstructed from its underlying H-spectra as H varies over F. A similar result holds for calculating G-equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition E ∈ F Nil implies strong collapse results for this spectral sequence as well as its dual homotopy colimit spectral sequence. As applications, we obtain Artin and Brauer type induction theorems for G-equivariant E-homology and cohomology, and generalizations of Quillen's Fp-isomorphism theorem when E is a homotopy commutative G-ring spectrum.We show that the subcategory F Nil contains many G-spectra of interest for relatively small families F. These include G-equivariant real and complex K-theory as well as the Borel-equivariant cohomology theories associated to complex oriented ring spectra, the Lnlocal sphere, the classical bordism theories, connective real K-theory, and any of the standard variants of topological modular forms. In each of these cases we identify the minimal family such that these results hold.