A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R.

Roughly speaking, the stable homotopy category of algebraic topology is obtained from the

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 .

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.

. This is the first of two interconnected parts: Part I contains the geometric theory of generalized modular forms and their connections with the cooperation algebra for elliptic cohomology, E" E", while Part II is devoted to the more algebraic theory associated with Hecke algebras and stable operations in elliptic cohomology. We investigate the structure of the stable operation algebra E" E" by first determining the dual cooperation algebra E" E". A major ingredient is our identification of the cooperation algebra E" E" with a ring of generalized modular forms whoses exact determination involves understanding certain integrality conditions; this is closely related to a calculation by N. Katz of the ring of all `divided congruences' amongst modular forms. We relate our present work to previous constructions of Hecke operators in elliptic cohomology. We also show that a well known operator on modular forms used by Ramanujan, Swinnerton-Dyer, Serre and Katz cannot extend to a stabl...

This paper is a continuation of [7]. Since so much time has lapsed since its publication a recasting of the context is probably in order. In [15] Ravenel described a series of conjectures getting at the structure of stable homotopy theory in the large. The theory was organized around a family of "higher periodicities" generalizing Bott periodicity, and depended on being able to determine the nilpotent and non-nilpotent maps in the category of spectra. There are three senses in which a map of spectra can be nilpotent: Definition 1.

Abstract. We discuss some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative S-algebras. Examples such as the complex K-theory spectrum as a KO-algebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert’s theorem 90 for the units associated with a Galois extension.

Abstract. We investigate Γ-cohomology of some commutative cooperation algebras E∗E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Γ-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E ∞ structures. As a consequence we obtain an E∞ structure for the connective Adams summand. For the Johnson-Wilson spectrum E(n) with n � 1 we establish the existence of a unique E ∞ structure for its In-adic completion.

Abstract. Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a two-cell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hochschild cohomology rings of resulting A∞-algebras have an interpretation as totally ramified extensions of discrete valuation rings. All A∞-algebras are supposed to be unital and we give a detailed analysis of unital structures which is of independent interest. Keywords: A∞-algebra, derivation, Hochschild cohomology, formal power series. 1.