Homology, Homotopy and Applications

Volume 4 (2002)

Number 1

Brave new Hopf algebroids and extensions of $MU$-algebras

Pages: 163 – 173

DOI: http://dx.doi.org/10.4310/HHA.2002.v4.n1.a9


Andrew Baker (Department of Mathematics, University of Glasgow, Scotland, United Kingdom)

Alain Jeanneret (Mathematisches Institut, Universität Bern, Switzerland)


We apply recent work of A. Lazarev which develops an obstruction theory for the existence of $R$-algebra structures on $R$-modules, where $R$ is a commutative $S$-algebra. We show that certain $MU$-modules have such $A_\infty$ structures. Our results are often simpler to state for the related $BP$-modules under the currently unproved assumption that $BP$ is a commutative $S$-algebra. Part of our motivation is to clarify the algebra involved in Lazarev’s work and to generalize it to other important cases. We also make explicit the fact that $BP$ admits an $MU$-algebra structure as do $E(n)$ and $\widehat{E(n)}$, in the latter case rederiving and strengthening older results of U. Würgler and the first author.


$R$-algebra, Hopf algebroid, obstruction theory

2010 Mathematics Subject Classification

55N20, 55N45, 55P43, 55S35, 55T25

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