Idempotents and Landweber Exactness in Brave New Algebra

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 E∞ ring spectrum or equivalently, in the language of [6], a commutative S-algebra. Unfortunately, the problem of whether or not BP is an E∞ 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 Bousfield localization λ : A −→ AM of A at M can be constructed as the inclusion of a subcomplex in a cell commutative Ralgebra. In particular, the commutative R-algebra AM is a commutative S-algebra by neglect of structure. The cell assumptions can always be arranged by use of the cofibrant replacement constructions in [6], so they result in no loss of generality. The theorem specializes as follows to algebraic localizations at elements of R∗ = π∗(R) [6, VIII.4.2]. Theorem 2. Let R be a cell commutative S-algebra and X a set of elements of R∗. The localization λ : R −→ R[X−1] that induces the algebraic localization R∗ −→ R∗[X−1] can be constructed as the unit of a cell commutative R-algebra. The connective real K-theory spectrum ko is a commutative S-algebra by multiplicative infinite loop space theory [11], and KO is the localization ko[β−1] obtained by inverting the Bott class. Therefore KO is a commutative ko-algebra and thus a commutative S-algebra. That’s the one line. Complex K-theory works similarly. As a matter of algebra, idempotents give localizations. Since MU arises in nature as an E∞ ring spectrum, that being the paradigmatic example that led to the definition [10], one might try to prove that BP is a Bousfield localization of MU and thus a commutative MU -algebra. That is April Fool’s nonsense, but the basic idea has a correct version with other applications, as we shall explain. Essentially Date: July 4, 2001. 1991 Mathematics Subject Classification. Primary 55N20, 55N91, 55P43. The author was partially supported by the NSF.