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 M U -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 −→ A M ofA at M can be constructed as the inclusion of a subcomplex in a cell commutative R-algebra. In particular, the commutative R-algebra A M 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]. 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

Motivic Landweber Exactness

“…Next we formulate the corresponding results for (highly structured) MGL-modules. In stable homotopy theory this viewpoint is emphasized in [20] and it plays an important role in this paper, cf. Section 9.…”

The Galois group of a stable homotopy theory

“…as a product of two objects in CAlg(C), as observed in [May01]. To see this, we may reduce to the case when A = 1, by replacing C by Mod C (A).…”

An overview of abelian varieties in homotopy theory

“…We refer to such an object as a Landweber exact theory. May showed that such theories can be realized by MU -modules [33,Theorem 8], and Hovey-Strickland showed that there is a functorial lifting from the category of Landweber exact theories to the homotopy category of MU -modules [26]. In addition, there are results for L-algebras rather than L-modules.…”