Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups

Devinatz, Ethan S.; Hopkins, Michael

doi:10.1016/s0040-9383(03)00029-6

Cited by 127 publications

(307 citation statements)

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“…Their localization functors are weakly equivalent, so that L K(2) X L K C X. As before, it follows from the work of Devinatz and Hopkins in [3] that for X a finite spectrum L K C X E hG C C ∧ X. Further, for any closed subgroup G of G C , there is a spectral sequence analogous to (1.1.1).…”

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confidence: 80%

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Towards the homotopy of the K(2)-local Moore spectrum at p= 2

Beaudry

2017

Advances in Mathematics

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Let V (0) be the mod 2 Moore spectrum and let C be the supersingular elliptic curve over F 4 defined by the Weierstrass equation y 2 +y = x 3 . Let F C be its formal group law and E C be the spectrum classifying the deformations of F C . The group of automorphisms of F C , which we denote by S C , acts on E C . Further, S C admits a surjective homomorphism to Z 2 whose kernel we denote by S 1 C . The cohomology of S 1 C with coefficients in (E C ) * V (0) is the E 2 -term of a spectral sequence converging to the homotopy groups of E hS 1. In this paper, we use the algebraic duality resolution spectral sequence to compute an associated graded for H * (S 1 C ; (E C ) * V (0)). These computations rely heavily on the geometry of elliptic curves made available to us at chromatic level 2.

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confidence: 80%

“…Then H * (C 6 ; (E C ) * V (0)) ∼ = F 4 [[u 3 1 ]][v 1 , v ±1 2 , h]/(v −1 2 v 3 1 = u 3 1 ), for a class h in H 1 (C 6 ; (E C ) * V (0)) satisfying η = hv 1 . In particular, {v n 2 h s } n∈Z is a set topological generators of H s (C 6 ; (E C ) * V (0)) as an F 4 [v 1 ]-module.…”

Section: ) First Note That (Ementioning

confidence: 99%

Towards the homotopy of the K(2)-local Moore spectrum at p= 2

Beaudry

2017

Advances in Mathematics

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“…As mentioned above, for any closed subgroup H of G n , Devinatz and Hopkins [9] constructed the commutative S-algebra E dhH n . Further, they showed that E dhH n behaves like a homotopy fixed point spectrum with respect to a continuous action of H. In more detail, [9] showed that E dhH n has the following properties: (a) there is a K(n)-local E n -Adams spectral sequence…”

Section: 32 Cor 233])mentioning

confidence: 99%

The homotopy fixed point spectra of profinite Galois extensions

Behrens

Davis

2010

Trans. Amer. Math. Soc.

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Abstract. Let E be a k-local profinite G-Galois extension of an E ∞ -ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show that the function spectrum

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“…where η ∈ KO 1 , y ∈ KO 4 and w ∈ KO 8 . By the 'Theorem of Reg Wood' [1, p. 206], multiplication by the non-zero element η ∈ KO 1 induces a cofibre sequence of KO-modules…”

Section: Is Faithful As An A-module Then B/a Is a G-galois Extensionmentioning

confidence: 99%

Realizability of algebraic Galois extensions by strictly commutative ring spectra

Baker

Richter²

2006

Trans. Amer. Math. Soc.

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Abstract. We discuss some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict our attention to the case of finite Galois groups and to global Galois extensions.We describe parts of the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones by applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative S-algebras. Examples such as the complex K-theory spectrum as a KO-algebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones, and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert's theorem 90 for the units associated with a Galois extension.

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Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups

Cited by 127 publications

References 13 publications

Towards the homotopy of the K(2)-local Moore spectrum at p= 2

Towards the homotopy of the K(2)-local Moore spectrum at p= 2

The homotopy fixed point spectra of profinite Galois extensions

Realizability of algebraic Galois extensions by strictly commutative ring spectra

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