Gross–Hopkins duality

Strickland, Neil P.

doi:10.1016/s0040-9383(99)00049-x

Cited by 61 publications

(63 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [GHMR12], the homotopy fibre of the rightmost arrow was named S det , and its Morava module was shown to be E ∨ n * (S det ) = E n * det . In [HG94,Str00] it was shown that this is the same Morava module as for Σ n−n 2 I n , where I n is the Brown-Comenetz dual of M n S 0 , the n th monochromatic layer of the sphere spectrum. It was also shown that there and in [HMS94] that for 2p − 2 ≥ max{n 2 , 2n + 2}, an invertible spectrum is determined by its Morava module.…”

Section: Gross-hopkins Dualitymentioning

confidence: 86%

A higher chromatic analogue of the image of J

Westerland

2017

Geom. Topol.

View full text Add to dashboard Cite

We prove a higher chromatic analogue of Snaith's theorem which identifies the K-theory spectrum as the localisation of the suspension spectrum of CP ∞ away from the Bott class; in this result, higher Eilenberg-MacLane spaces play the role of CP ∞ = K(Z, 2). Using this, we obtain a partial computation of the part of the Picard-graded homotopy of the K(n)-local sphere indexed by powers of a spectrum which for large primes is a shift of the Gross-Hopkins dual of the sphere. Our main technical tool is a K(n)-local notion generalising complex orientation to higher Eilenberg-MacLane spaces. As for complex-oriented theories, such an orientation produces a one-dimensional formal group law as an invariant of the cohomology theory. As an application, we prove a theorem that gives evidence for the chromatic redshift conjecture.

show abstract

Section: Gross-hopkins Dualitymentioning

confidence: 86%

A higher chromatic analogue of the image of J

Westerland

2017

Geom. Topol.

View full text Add to dashboard Cite

show abstract

“…We recall briefly that there is a reduced determinant map G n → Z × p , and if M is a (E n ) * -module with a compatible action of G n , then we write M det for the module with G n action twisted by the determinant map. Then Hopkins and Gross in fact prove the stronger statement (see also [Str00]) that…”

Section: Applications To the K(1)-local Picard Groupmentioning

confidence: 92%

K-theory, reality, and duality

Heard

Stojanoska

2014

J. K-Theory

View full text Add to dashboard Cite

Abstract. We show that the real K-theory spectrum KO is Anderson selfdual using the method previously employed in the second author's calculation of the Anderson dual of T mf . Indeed the current work can be considered as a lower chromatic version of that calculation. Emphasis is given to an algebrogeometric interpretation of this result in spectrally derived algebraic geometry. We finish by applying the result to a calculation of 2-primary Gross-Hopkins duality at height 1, and obtain an independent calculation of the group of exotic elements of the K(1)-local Picard group.

show abstract

“…Finally we indicate how Gross-Hopkins duality fits into the picture. This paper could not have been written without [16] and [20]; a lot of what we do is to give a different slant to the material in [20]. Although our treatment has an intrinsic interest, it can also be viewed as an extended example of the theory of [8], an example which highlights the importance of orientability issues.…”

Section: Z=pmentioning

confidence: 99%

“…The purpose of this section is to recall some material from [16] and [20]. As in 1.17, let I denote Cell S K Hom.S;I/, where I is the ordinary Brown-Comenetz dual of the sphere.…”

Section: Chromatic Ingredientsmentioning

confidence: 99%