Products on 𝑀𝑈-modules

Strickland, N.

doi:10.1090/S0002-9947-99-02436-8

Products on $MU$-modules
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by N. P. Strickland

Trans. Amer. Math. Soc. 351 (1999), 2569-2606

DOI: https://doi.org/10.1090/S0002-9947-99-02436-8

Published electronically: March 1, 1999

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Abstract:

Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of $MU$-modules 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 $MU[\frac {1}{2}]_*$ 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.

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