A transchromatic proof of Strickland's theorem

Schlank, Tomer M.; Stapleton, Nathaniel

doi:10.1016/j.aim.2015.07.025

Cited by 21 publications

(2 citation statements)

References 16 publications

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“…by topological K-theory and rational cohomology. It turns out that the connection is surprisingly strong, an observation that has already been exploited in [SS15,BS16].…”

Section: Brown-peterson Cohomology From Morava E-theorymentioning

confidence: 89%

Brown–Peterson cohomology from Morava -theory

Barthel¹,

Stapleton²

2017

Compositio Math.

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We prove that the p-completed Brown-Peterson spectrum is a retract of a product of Morava E-theory spectra. As a consequence, we generalize results of Ravenel-Wilson-Yagita and Kashiwabara from spaces to spectra and deduce that the notion of good group is determined by Brown-Peterson cohomology. Furthermore, we show that rational factorizations of the Morava E-theory of certain finite groups hold integrally up to bounded torsion with height-independent exponent, thereby lifting these factorizations to the rationalized Brown-Peterson cohomology of such groups.2010 Mathematics Subject Classification 55N20 (primary), 55N22, 55R40 (secondary) Keywords: Brown-Peterson spectrum, Morava E-theory, transchromatic character theory. Brown-Peterson cohomology from Morava E-theoryTheorem. Let A be a finite abelian group, then the exponent of the torsion in the cokernel of the natural mapis bounded independent of the height n.In order to prove this, we construct a variant of the transchromatic character maps of [Sta13, BS16] from E-theory at height n to height 1, which allows for tight control over the torsion. Since products of torsion abelian groups with a common torsion exponent are torsion as well, it follows that the natural mapis a rational isomorphism. It is possible to apply the retract theorem to immediately deduce a similar decomposition for the BP -cohomology of finite abelian groups.Corollary. Let A be a finite abelian group and let I denote the transfer ideal, then the natural mapSimilarly, the scheme corepresented by E * n (BΣ m ) decomposes rationally into a product of subgroup schemes Γ Sub λ⊢m (G En ), indexed by partitions λ ⊢ m of m. Again, we prove that this statement holds integrally up to globally bounded integral torsion.Theorem. The exponent of the torsion in the cokernel of the natural mapis bounded independent of the height n. Consequently, there is a rational isomorphismwhere I λ is a certain transfer ideal.Finally, we give a further illustration of the methods of the paper by proving a well-known version of Artin induction for the BP -cohomology of good groups. Relation to the literatureThe question of when and how E * (X) can be computed algebraically from BP * (X) or, conversely, what kind of information about BP * (X) is retained in E * (X) has a long history. After pioneering work of Johnson and Wilson [JW73, JW75] and Landweber [Lan76], these problems have been studied systematically in a series of papers by Ravenel, Wilson, and Yagita [RWY98, Wil99] and Kashiwabara [Kas98, Kas01]. Their methods are based on a careful study of the associated Atiyah-Hirzebruch spectral sequences. As a consequence, the results contained in these papers are unstable, i.e., only valid for spaces rather than arbitrary spectra.The results of our paper generalize the main structural theorems of [RWY98, Wil99] from spaces to spectra, by replacing the Atiyah-Hirzebruch spectral sequence arguments by the above

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“…by topological K-theory and rational cohomology. It turns out that the connection is surprisingly strong, an observation that has already been exploited in [SS15,BS16].…”

Section: Brown-peterson Cohomology From Morava E-theorymentioning

confidence: 89%

Brown–Peterson cohomology from Morava -theory

Barthel¹,

Stapleton²

2017

Compositio Math.

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“…In view of the fact that the second Morava E-theory is a form of elliptic cohomology, the question arises of whether the finite subgroups of an elliptic curve are classified by the quotient of the corresponding elliptic cohomology of the symmetric group by a transfer ideal. Results suggesting an affirmative answer to this question come from generalized Morava E-theory [SS14] and quasi-elliptic cohomology and Tate K-theory [Hua18a]. Moreover, transfer plays an important role in understanding additive properties of power operations and interacts nicely with Hopkins-Kuhn-Ravenel character theory [HKR00].…”

Section: Künneth Maps Let πmentioning

confidence: 99%

Quasi-elliptic cohomology I

Huan

2018

Advances in Mathematics

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Quasi-elliptic cohomology is a variant of elliptic cohomology theories. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. Thus, the constructions on it can be made in a neat way. This theory reflects the geometric nature of the Tate curve. In this paper we provide a systematic introduction of its construction and definition.

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Level structures on p-divisible groups from the Morava E-theory of abelian groups

Huan

Stapleton²

2023

Math. Z.

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A transchromatic proof of Strickland's theorem

Cited by 21 publications

References 16 publications

Brown–Peterson cohomology from Morava -theory

Brown–Peterson cohomology from Morava -theory

Quasi-elliptic cohomology I

Level structures on p-divisible groups from the Morava E-theory of abelian groups

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