Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations Eâˆ—E is flat over Eâˆ—. We wish to address the following question: given a commutative Eâˆ—-algebra A in Eâˆ—E-comodules,â€¦ (More)

A Î -algebra A is a graded group {An}nâ‰¥1 with all of the primary algebraic structure possessed by the collection of homotopy groups of a pointed topological space. In particular, An is abelian for n â‰¥â€¦ (More)

There are many ways to present model categories, each with a different point of view. Here weâ€™d like to treat model categories as a way to build and control resolutions. This an historical approach,â€¦ (More)

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 theâ€¦ (More)

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. Whileâ€¦ (More)

The homology of a homotopy inverse limit can be studied by a spectral sequence with has E2 term the derived functors of limit in the category of coalgebras. These derived functors can be computedâ€¦ (More)

There are two practical ways to complete an abelian group A at a prime p. One could complete A with respect to the neighborhood base of zero given by the subgroups pA. This completion, calledâ€¦ (More)

We develop a framework for displaying the stable homotopy theory of the sphere, at least when we are localized at the second Morava K-theory K(2). At the prime 3, we write the spectrum LK(2)S 0 asâ€¦ (More)