Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Q-algebra (with many objects). 1.

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.

We proved in a previous article that the bar complex of an E ∞-algebra inherits a natural E ∞-algebra structure. As a consequence, a well-defined iterated bar construction B n (A) can be associated to any algebra over an E ∞-operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E ∞-algebras. We use this effective definition to prove that the n-fold bar construction admits an extension to categories of algebras over En-operads. Then we prove that the n-fold bar complex determines the homology theory associated to the category of algebras over an En-operad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γ-homology with trivial coefficients. 57T30; 55P48, 18G55, 55P35

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

Abstract. We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. Specifically, we show that for an E ∞ ring spectrum A, the classical construction of gl1A, the spectrum of units, is the right adjoint of the functor To a map of spectra Σ ∞ + Ω ∞ : ho(connective spectra) → ho(E ∞ ring spectra). f: b → bgl1A, we associate an E ∞ A-algebra Thom spectrum Mf, which admits an E ∞ A-algebra map to R if and only if the composition b → bgl1A → bgl1R is null; the classical case developed by [MQRT77] arises when A is the sphere spectrum. We develop the analogous theory for A ∞ ring spectra. If A is an A ∞ ring spectrum, then to a map of spaces f: B → BGL1A we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if

Abstract not found

We proved in a previous article that the bar complex of an E∞algebra inherits a natural E∞-algebra structure. As a consequence, a welldefined iterated bar construction Bn (A) can be associated to any algebra over an E∞-operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E∞-algebras. We use this effective definition to prove that the n-fold bar construction admits an extension to categories of algebras over En-operads. Then we prove that the n-fold bar complex determines the homology theory associated to the category of algebras over an En-operad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γ-homology with trivial coefficients.

Abstract. The stabilization of Hochschild homology of commutative algebras is Gamma homology. We describe a cyclic variant of Gamma homology and prove that the associated analogue of Connes ’ periodicity sequence becomes almost trivial, because the cyclic version coincides with the ordinary version from homological degree two on. We offer an alternative explanation for this by proving that the B-operator followed by the stabilization map is trivial from degree one on. 1.