Idempotents and Landweber exactness in brave new algebra

Abstract

We explain how idempotents in homotopy groups give rise to splittings of homotopy categories of modules over commutative S-algebras, and we observe that there are naturally occurring equivariant examples involving idempotents in Burnside rings. We then give a version of the Landweber exact functor theorem that applies to MU-modules. In 1997, not long after [6] was written, I gave an April Fool’s talk on how to prove that BP is an E1 ring spectrum or equivalently, in the language of [6], a commutative S-algebra. Unfortunately, the problem of whether or not BP is an E1 ring spectrum remains open. However, two interesting remarks emerged and will be presented here. One concerns splittings along idempotents and the other concerns the Landweber exact functor theorem. One of the nicest things in [6] is its one line proof that KO and KU are commutative S-algebras. This is an application of the following theorem [6, VIII.2.2], or rather the special case that follows. Theorem 1. Let R be a cell commutative S-algebra, A be a cell commutative Ralgebra, and M be a cell R-module. Then the BousÞeld localization i : A A! AM of A at M can be constructed as the inclusion of a subcomplex in a cell commutative R-algebra. In particular, the commutative R-algebra AM is a commutative S-algebra by neglect of structure.