## IDEMPOTENTS AND LANDWEBER EXACTNESS IN BRAVE NEW ALGEBRA

### BibTeX

@MISC{_idempotentsand,

author = {},

title = {IDEMPOTENTS AND LANDWEBER EXACTNESS IN BRAVE NEW ALGEBRA},

year = {}

}

### OpenURL

### Abstract

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 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 R-

### Citations

61 | Equivariant orthogonal spectra and S-modules
- Mandell, May
(Show Context)
Citation Context ...ing point of [13, 14]. Interesting examples also arise in equivariant algebraic topology. The results above generalize directly to the equivariant setting of commutative SG-algebras and their modules =-=[6, 9, 12]-=-, where G is a compact Lie group and SG is the sphere G-spectrum. Here, for a commutative SG-algebra R, we take R∗ = π∗(RG ). In particular, (SG)∗ is the equivariant stable homotopy groups of spheres ... |

39 | properties of comodules over M - Landweber, Homological - 1976 |

39 |
with contributions by
- May
- 1977
(Show Context)
Citation Context ...able homotopy groups of spheres and (SG)0 is isomorphic to the Burnside ring A(G). The ring A(G), and more so its localizations at subrings of the rationals, usually does have non-trivial idempotents =-=[5, 8]-=-. The splittings of Theorem 4 give model theoretic refinements of splittings in equivariant stable homotopy theory that are discussed in [8, V] and [12, XVII§6]. Those sources describe splittings of h... |

36 | The relation of cobordism to K-theories - Conner, Floyd - 1966 |

26 | Multiplicative infinite loop space theory
- May
(Show Context)
Citation Context ...zation 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 simil... |

14 |
et al., Equivariant Homotopy and Cohomology Theory. Volume 91
- May
- 1996
(Show Context)
Citation Context ...ing point of [13, 14]. Interesting examples also arise in equivariant algebraic topology. The results above generalize directly to the equivariant setting of commutative SG-algebras and their modules =-=[6, 9, 12]-=-, where G is a compact Lie group and SG is the sphere G-spectrum. Here, for a commutative SG-algebra R, we take R∗ = π∗(RG ). In particular, (SG)∗ is the equivariant stable homotopy groups of spheres ... |

13 | On the Adams spectral sequence for R-modules
- Baker, Lazarev
- 2001
(Show Context)
Citation Context ...wicz map gives X∗ a structure of R∗R-comodule. Proof. This is proven by diagram chasing as in Adams [1]. It is the starting point of the development of an Adams spectral sequence in brave new algebra =-=[3]-=-. The main point is that R∗R ⊗R∗ X∗ ∼ = π∗((R ∧ R) ∧R X) ∼ = π∗(R ∧R X). Of course, this applies with R = MU. following conclusion. The previous three results imply the Proposition 12. Let M∗ be a Lan... |

11 |
A variant of E.H. Brown’s representability theorem. Topology 10
- Adams
- 1971
(Show Context)
Citation Context ...revious three results imply the Proposition 12. Let M∗ be a Landweber exact MU∗-module. Then the functor π∗(X) ⊗MU∗ M∗ specifies a homology theory on finite cell MU-modules X. Applying Adams’ variant =-=[2]-=- of Brown’s representability theorem, which applies since MU∗ is countable [6, III.2.13], we obtain the MU-module M promised in Theorem 8. The construction of M is non-uniquely functorial: given a map... |

11 |
with an appendix by
- Elmendorf, Kriz, et al.
(Show Context)
Citation Context ...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 ... |

10 |
with contributions by
- P
- 1977
(Show Context)
Citation Context ...plex 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, a... |

9 | Adjoining roots of unity to E∞ ring spectra in good cases—a remark. In Homotopy invariant algebraic structures
- Schwänzl, Vogt, et al.
- 1998
(Show Context)
Citation Context ... Primary 55N20, 55N91, 55P43. The author was partially supported by the NSF. 12 J.P. MAY the same idea occurred independently to Schwänzl, Vogt, and Waldhausen, who gave quite different applications =-=[13, 14]-=-. Definition 3. Let R be a cell commutative S-algebra and let e ∈ R0 be an idempotent element. As a matter of algebra, R∗[e −1 ] = eR∗. Define eR to be the cell commutative R-algebra R[e −1 ] of Theor... |

2 |
Lectures on generalized cohomology
- Adams
- 1969
(Show Context)
Citation Context ... 11. If X is an R-module, where R is a commutative S-algebra such that R∗R is R∗-flat, then the Hurewicz map gives X∗ a structure of R∗R-comodule. Proof. This is proven by diagram chasing as in Adams =-=[1]-=-. It is the starting point of the development of an Adams spectral sequence in brave new algebra [3]. The main point is that R∗R ⊗R∗ X∗ ∼ = π∗((R ∧ R) ∧R X) ∼ = π∗(R ∧R X). Of course, this applies wit... |

2 |
Idempotent elements in the Burnside ring
- Dieck
(Show Context)
Citation Context ...able homotopy groups of spheres and (SG)0 is isomorphic to the Burnside ring A(G). The ring A(G), and more so its localizations at subrings of the rationals, usually does have non-trivial idempotents =-=[5, 8]-=-. The splittings of Theorem 4 give model theoretic refinements of splittings in equivariant stable homotopy theory that are discussed in [8, V] and [12, XVII§6]. Those sources describe splittings of h... |

2 |
Topological Hoschschild homology
- Schwänzl, Vogt, et al.
(Show Context)
Citation Context ... Primary 55N20, 55N91, 55P43. The author was partially supported by the NSF. 12 J.P. MAY the same idea occurred independently to Schwänzl, Vogt, and Waldhausen, who gave quite different applications =-=[13, 14]-=-. Definition 3. Let R be a cell commutative S-algebra and let e ∈ R0 be an idempotent element. As a matter of algebra, R∗[e −1 ] = eR∗. Define eR to be the cell commutative R-algebra R[e −1 ] of Theor... |