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

Abstract

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.