Products on 𝑀𝑈-modules

Abstract

Abstract. Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of MUmodules such as BP , K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over MU[ ] * that are concentrated in degrees divisible by 4; this guarantees that various obstruction groups are trivial. We extend these results to the cases where 2 = 0 or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising 2-local MU * -modules as MU-modules.

Topological Hochschild homology and cohomology ofA∞ring spectra

“…For example, Corollary 3.2 strengthens Proposition 3.1(1) of Strickland [34], which says that for R even commutative and x a nonzero divisor, any multiplication on R=x is homotopy associative, to saying that any multiplication on R=x can be extended to an A 1 structure. Corollary 3.7 strengthens various results in Lazarev [24] and Baker and Jeanneret [6] about the associativity of M U =I for certain regular ideals to all regular ideals.…”

“…We need to go into more detail about how to calculate the actual obstructions. We first do the n D 2 case, following Strickland [34]. In this case we want to know how to determine whether or not an A 2 structure on A is (co)cyclic, or in other words whether or not A has a homotopy commutative multiplication.…”

On the Adams spectral sequence forR–modules

“…We will use the following terminology of Strickland [16]. If the homotopy ring R * = π * R is concentrated in even degrees, a localized quotient of R will be an R ring spectrum of the form R/I We will make use of the language and ideas of algebraic derived categories of modules over a commutative ring, mildly extended to deal with evenly graded rings and their modules.…”

“…By comparing the two Künneth Spectral Sequences we find that τ i ∈ E R * E can be chosen to be the image of τ i under the evident ring homomorphism [16,7].…”

“…We consider the Adams Spectral Sequence for R-modules based on localized regular quotient ring spectra over a commutative S -algebra R in the sense of [11,16], making systematic use of ideas and notation from those two sources. This work grew out of a preprint [4] and the work of [6]; it is also related to ongoing collaboration with Alain Jeanneret on Bockstein operations in cohomology theories defined on R-modules [7].…”