Galois extensions of structured ring spectra. Stably dualizable groups

Rognes, John

doi:10.1090/memo/0898

Cited by 128 publications

(286 citation statements)

References 39 publications

(75 reference statements)

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“…We can now give the key definition of a Galois extension essentially due to Rognes [26]. Definition 1.4.4.…”

Section: Proposition 141 Let X Be An A-module Then X Is Strongly mentioning

confidence: 99%

“…Proofs that these are actually Galois extensions can be found in [26]. whose action on KU 2n = Zu n satisfies…”

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

confidence: 99%

“…Using Proposition 1.4.5(c) (or results of [26]) it is enough to check that B ∧ A ϕ * B is an H-Galois extension of B.…”

Section: And There Is An Isomorphism Ofmentioning

confidence: 99%

“…Both of these are topological manifestations of properties of algebraic Galois extensions. We stress that this material is known to Rognes and is systematically described in [26], and the presentation reflects our approach to understanding his results rather than being original.…”

Section: Introductionmentioning

confidence: 99%

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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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“…We can now give the key definition of a Galois extension essentially due to Rognes [26]. Definition 1.4.4.…”

Section: Proposition 141 Let X Be An A-module Then X Is Strongly mentioning

confidence: 99%

“…Proofs that these are actually Galois extensions can be found in [26]. whose action on KU 2n = Zu n satisfies…”

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

confidence: 99%

“…Using Proposition 1.4.5(c) (or results of [26]) it is enough to check that B ∧ A ϕ * B is an H-Galois extension of B.…”

Section: And There Is An Isomorphism Ofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Realizability of algebraic Galois extensions by strictly commutative ring spectra

Baker

Richter²

2006

Trans. Amer. Math. Soc.

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“…In [34], John Rognes develops a Galois theory of commutative S-algebras which mimics the Galois theory of commutative rings. Let k be an S-module, and let (−) k denote Bousfield localization with respect to k. Given a k-local cofibrant Let G be a profinite group.…”

Section: Introductionmentioning

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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