# Brave new Hopf algebroids and extensions of MU-algebras

## Andrew Baker and Alain Jeanneret

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 $\En$ and $\hEn$, in the latter case rederiving and strengthening older results of U.~W\"urgler and the first author.

Homology, Homotopy and Applications, Vol. 4(2002), No. 1, pp. 163-173

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