Secondary power operations and the Brown--Peterson spectrum at the prime 2

Lawson, Tyler

doi:10.4007/annals.2018.188.2.3

Cited by 33 publications

(34 citation statements)

References 61 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 6.22. By a result of Basterra and Mandell [BM13], BP admits the structure of an E 4 -ring spectrum, but Lawson [Law17] proved recently that BP does not admit the structure of an E 12 -ring spectrum at the prime 2. Consequently, the stable ∞-category Mod BP of BP -module spectra is only known to be E 3 -monoidal, hence is not a stable category as defined in Section 2.…”

Section: Local Duality In Stable Homotopy Theorymentioning

confidence: 99%

Local duality in algebra and topology

Barthel

Heard

Valenzuela

2018

Advances in Mathematics

View full text Add to dashboard Cite

The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable ∞-category C together with a collection of compact objects K ⊂ C we construct local cohomology and local homology functors satisfying an abstract version of local duality. When specialized to the derived category of a commutative ring A and a suitable ideal in A, we recover the classical local duality due to Grothendieck as well as generalizations by Greenlees and May. More generally, applying our result to the derived category of quasi-coherent sheaves on a quasi-compact and separated scheme X implies the local duality theorem of Alonso Tarrío, Jeremías López, and Lipman.As a second objective, we establish local duality for quasi-coherent sheaves over many algebraic stacks, in particular those arising naturally in stable homotopy theory. After constructing an appropriate model of the derived category in terms of comodules over a Hopf algebroid, we show that, in familiar cases, the resulting local cohomology and local homology theories coincide with functors previously studied by Hovey and Strickland. Furthermore, our framework applies to global and local stable homotopy theory, in a way which is compatible with the algebraic avatars of these theories. In order to aid computability, we provide spectral sequences relating the algebraic and topological local duality contexts.

show abstract

Section: Local Duality In Stable Homotopy Theorymentioning

confidence: 99%

Local duality in algebra and topology

Barthel

Heard

Valenzuela

2018

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…Remark The proof essentially says that the

E_{\infty}

structure on

M U P

extends the classical

H_{\infty}^{2}

structure on

M U

[8, Corollary VIII.5.3]. In the language of [14, Problem 1.3.7], the map

prefixdouble-struckZ \to prefix Pic (M U)

defining

M U P

refines the

H_{\infty}^{2}

structure on

M U

to an

E_{\infty}^{2}

structure.…”

Section: Preliminariesmentioning

confidence: 99%

Exotic multiplications on periodic complex bordism

Hahn

Yuan

2020

Journal of Topology

View full text Add to dashboard Cite

Victor Snaith gave a construction of periodic complex bordism by inverting the Bott element in the suspension spectrum of BU. This presents an E∞ structure on periodic complex bordism by different means than the usual Thom spectrum definition of the E∞-ring MUP. Here, we prove that these two E∞-rings are in fact different, though the underlying E2-rings are equivalent. Nonetheless, we prove that both rings E∞-orient KU ∧ 2 and other forms of K-theory.

show abstract

“…While this gives a rough description of the main result of [Law17], and the size of this relation provides some mild amusement value, most people who see this relation are inclined to ask where on earth it came from. There are no obvious indications why we should focus attention on this particular secondary operation.…”

Section: Introductionmentioning

confidence: 99%

Calculating obstruction groups for 𝐸_{∞} ring spectra

Lawson¹

2019

Homotopy Theory: Tools And Applications

Self Cite

View full text Add to dashboard Cite

We describe a special instance of the Goerss-Hopkins obstruction theory, due to Senger, for calculating the moduli of E ∞ ring spectra with given mod-p homology. In particular, for the 2-primary Brown-Peterson spectrum we give a chain complex that calculates the rst obstruction groups, locate the rst potential genuine obstructions, and discuss how some of the obstruction classes can be interpreted in terms of secondary operations.

show abstract

Secondary power operations and the Brown--Peterson spectrum at the prime 2

Cited by 33 publications

References 61 publications

Local duality in algebra and topology

Local duality in algebra and topology

Exotic multiplications on periodic complex bordism

Calculating obstruction groups for 𝐸_{∞} ring spectra

Contact Info

Product

Resources

About