We introduce the notion of a Galois extension of commutative S-algebras (E ∞ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg–MacLane spectra of commutative rings, real and complex topological K-theory, Lubin–Tate spectra and cochain S-algebras. We establish the main theorem of Galois theory in this generality. Its proof involves the notions of separable (and étale) extensions of commutative S-algebras, and the Goerss–Hopkins–Miller theory for E ∞ mapping spaces. We show that the global sphere spectrum S is separably closed (using Minkowski’s discriminant theorem), and we estimate the separable closure of its localization with respect to each of the Morava K-theories. We also define Hopf–Galois extensions of commutative S-algebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin–Tate Galois extensions.

In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞-ring spectra. In that paper, we discussed the the Hopkins-Miller theorem on the Lubin-Tate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞-ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is non-empty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.

Abstract. Let K(n) be the n th Morava K–theory at a prime p, and let T(n) be the telescope of a vn–self map of a finite complex of type n. In this paper we study the K(n)∗–homology of Ω ∞ X, the 0 th space of a spectrum X, and many related matters. We give a sampling of our results. Let PX be the free commutative S–algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map sn(X) : LT(n)P(X) → LT(n)Σ ∞ (Ω ∞ X)+ of commutative algebras over the localized sphere spectrum LT(n)S. The induced map of commutative, cocommutative K(n)∗–Hopf algebras sn(X) ∗ : K(n)∗(PX) → K(n)∗(Ω ∞ X), satistfies the following properties. It is always monic. It is an isomorphism if X is n–connected, πn+1(X) is torsion, and T(i)∗(X) = 0 for 1 ≤ i ≤ n−1. It is an isomorphism only if K(i)∗(X) = 0 for 1 ≤ i ≤ n − 1. It is universal: the domain of sn(X) ∗ preserves K(n)∗–isomorphisms, and if F is any functor preserving K(n)∗–isomorphisms, then any natural transformation F(X) → K(n)∗(Ω ∞ X) factors uniquely through sn(X)∗. The construction of our natural transformation uses the telescopic functors constructed and studied previously by Bousfield and the author, and thus depends heavily on the Nilpotence Theorem of Devanitz, Hopkins, and Smith. Our proof that sn(X) ∗ is always monic uses Topological André–Quillen Homology and Goodwillie Calculus in nonconnective settings.

Abstract. This paper is based on talks I gave in Nagoya and Kinosaki in August of 2003. I survey, from my own perspective, Goodwillie’s work on towers associated to continuous functors between topological model categories, and then include a discussion of applications to periodic homotopy as in my work and the work of Arone–Mahowald. 1.

The aim of this paper is to give an overview of some of the existing homology theories for commutative (S-)algebras. We do not claim any originality; nor do we pretend to give a complete account. But the results in that field are widely spread in the literature, so for someone who does not actually work in that subject, it can be difficult to trace all the relationships between the different homology theories. The theories we aim to compare are • topological André-Quillen homology • Gamma homology • stable homotopy of Γ-modules • stable homotopy of algebraic theories • the André-Quillen cohomology groups which arise as obstruction groups in the Goerss-Hopkins approach As a comparison between stable homotopy of Γ-modules and stable homotopy of algebraic theories is not explicitly given in the literature, we will give a proof of Theorem 2.1 which says that both homotopy theories are isomorphic

These notes contain a brief account of some basic results on topological André-Quillen homology and cohomology for CW commutative A-algebras, where A is a commutative S-algebra. The main goal is to develop arguments based on skeletal filtrations with a view to continuing the work begun in [7] by extending results of [1] to the case of CW commutative S-algebras;

Abstract not found

pagination and layout may vary from GTM published version Goodwillie towers and chromatic homotopy: an overview