On the Adams spectral sequence for<i>R</i>–modules

Baker, Andrew; Lazarev, Andrey

doi:10.2140/agt.2001.1.173

Cited by 34 publications

(84 citation statements)

References 12 publications

(40 reference statements)

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“…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…”

Section: Theorem 9 (Landweber) the Functor (−) ⊗ M U * M * On The Camentioning

confidence: 86%

Idempotents and Landweber exactness in brave new algebra

May

2001

Homology, Homotopy and Applications

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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

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“…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…”

Section: Theorem 9 (Landweber) the Functor (−) ⊗ M U * M * On The Camentioning

confidence: 86%

Idempotents and Landweber exactness in brave new algebra

May

2001

Homology, Homotopy and Applications

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“…is π * (K) by [BL,Corollary 5.11], hence using the commutative diagram coming from the Künneth isomorphism…”

Section: Lemma 72 If X Is Dualizable In D Ek Then For An Arbitramentioning

confidence: 99%

Galois extensions of Lubin-Tate spectra

Baker

Richter

2008

Homology, Homotopy and Applications

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Let E n be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E nr n whose coefficients are built from the coefficients of E n and contain all roots of unity whose order is not divisible by p. For odd primes p we show that E nr n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that there are no non-trivial connected Galois extensions of E nr n with Galois group a finite group G with cyclic quotient. Our results carry over to the K(n)-local context.

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“…Full details of the construction and related issues have been worked out by Baker and Lazarev in [2].…”

Section: An Adams Spectral Sequence For Tmfmentioning

confidence: 99%