Abstract not found

Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# -algebra A in E#E-comodules, is there an E# -ring spectrum X with E#X = A as comodule algebras? We will formulate this as a moduli problem, and give a way -- suggested by work of Dwyer, Kan, and Stover -- of dissecting the resulting moduli space as a tower with layers governed by appropriate Andre-Quillen cohomology groups. A special case is A = E#E itself. The final section applies this to discuss the Lubin-Tate or Morava spectra En .

Given a good homology theory E and a topological space X, E∗X is not just an E∗-module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A, Γ). That is, we construct

In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem.

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 G be a closed subgroup of Gn, the extended Morava stabilizer group. Let En be the Lubin-Tate spectrum, X an arbitrary spectrum with trivial G-action, and let ˆ L = L K(n). We prove that ˆ L(En ∧ X) is a continuous G-spectrum with homotopy fixed point spectrum ( ˆ L(En ∧ X)) hG, defined with respect to the continuous action. Also, we construct a descent spectral sequence whose abutment is π∗( ( ˆ L(En∧X)) hG). We show that the homotopy fixed points of ˆ L(En ∧ X) come from the K(n)-localization of the homotopy fixed points of the spectrum (Fn ∧ X). 1.

Abstract. We show that any closed model category of simplicial algebras over an algebraic theory

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

There are many interesting situations in which algebraic structure can be described

Abstract. We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads. 1.