Equivariant orthogonal spectra and 𝑆-modules

Mandell, Michael A.; May, J. P.

doi:10.1090/memo/0755

Cited by 203 publications

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“…An orthogonal G -spectrum is a G -space [31,II.4.3], an enriched functor from (a skeleton of) G to based Gspaces. That is, an orthogonal G -spectrum T consists of a based G -space T .V / for each orthogonal G -representation V together with based G -maps…”

Section: Details Of the Category Of Cyclotomic Spectramentioning

confidence: 99%

Localization theorems in topological Hochschild homology and topological cyclic homology

Blumberg

Mandell

2012

Geom. Topol.

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We construct localization cofibration sequences for the topological Hochschild homology (THH ) and topological cyclic homology (TC ) of small spectral categories. Using a global construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofibration sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason-Trobaugh in K -theory. We also deduce versions of Thomason's blow-up formula and the projective bundle formula for THH and TC .

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Section: Details Of the Category Of Cyclotomic Spectramentioning

confidence: 99%

Localization theorems in topological Hochschild homology and topological cyclic homology

Blumberg

Mandell

2012

Geom. Topol.

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“…We follow the treatment of a topological model structure given in [35,Section III.1]. Note that the G -fixed points of the mapping spaces in T G are the mapping spaces in GT .…”

Section: Lemma 215mentioning

confidence: 99%

“…Most of the results extend easily from the theory developed for compact Lie groups by Michael Mandell and Peter May [35]. We include enough details to make our presentation readable, and provide new proofs when the generalizations to our context are not immediate.…”

Section: Introductionmentioning

confidence: 97%

Equivariant homotopy theory for pro–spectra

Fausk

2008

Geom. Topol.

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We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The G -homotopy theory is "pieced together" from the G=U -homotopy theories for suitable quotient groups G=U of G ; a motivation is the way continuous group cohomology of a profinite group is built out of the cohomology of its finite quotient groups. In the model category of equivariant spectra Postnikov towers are studied from a general perspective. We introduce pro-G -spectra and construct various model structures on them. A key property of the model structures is that pro-spectra are weakly equivalent to their Postnikov towers. We discuss two versions of a model structure with "underlying weak equivalences". One of the versions only makes sense for pro-spectra. In the end we use the theory to study homotopy fixed points of pro-G -spectra. 55P91; 18G55

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“…The results above generalize directly to the equivariant setting of commutative S G -algebras and their modules [6,9,12], where G is a compact Lie group and S G is the sphere G-spectrum. Here, for a commutative S G -algebra R, we take R * = π * (R G ).…”

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confidence: 75%

“…Those sources describe splittings of homology and cohomology theories, and it is now apparent that these splittings arise from splittings of corresponding equivariant stable categories. The splittings involve change of group functors, and these are discussed model theoretically in the contexts both of S G -modules and of orthogonal G-spectra in [9]. Briefly, by [9, VI.1.2], for an inclusion ι : H ⊂ G, there is a Quillen adjoint pair (G + ∧ H (−), ι * ) relating HM to GM .…”

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confidence: 99%

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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Equivariant orthogonal spectra and 𝑆-modules

Cited by 203 publications

References 0 publications

Localization theorems in topological Hochschild homology and topological cyclic homology

Localization theorems in topological Hochschild homology and topological cyclic homology

Equivariant homotopy theory for pro–spectra

Idempotents and Landweber exactness in brave new algebra

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