Results 1  10
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19
Galois extensions of structured ring spectra
, 2005
"... We introduce the notion of a Galois extension of commutative Salgebras (E ∞ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg–MacLane spectra of commutative rings, real and complex topological Ktheory, Lubin–Tate ..."
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Cited by 58 (3 self)
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We introduce the notion of a Galois extension of commutative Salgebras (E ∞ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg–MacLane spectra of commutative rings, real and complex topological Ktheory, Lubin–Tate spectra and cochain Salgebras. We establish the main theorem of Galois theory in this generality. Its proof involves the notions of separable (and étale) extensions of commutative Salgebras, and the Goerss–Hopkins–Miller theory for E ∞ mapping spaces. We show that the global sphere spectrum S is separably closed (using Minkowski’s discriminant theorem), and we estimate the separable closure of its localization with respect to each of the Morava Ktheories. We also define Hopf–Galois extensions of commutative Salgebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin–Tate Galois extensions.
The homotopy fixed point spectra of profinite Galois extensions
"... Let E be a klocal profinite GGalois extension of an E∞ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete Gspectrum. Also, we prove that if E is a profaithful klocal profinite extension which satisfies certain extra conditions, then the forward dir ..."
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Cited by 22 (15 self)
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Let E be a klocal profinite GGalois extension of an E∞ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete Gspectrum. Also, we prove that if E is a profaithful klocal profinite extension which satisfies certain extra conditions, then the forward direction of Rognes’s Galois correspondence extends to the profinite setting. We show the function spectrum FA((EhH)k, (EhK)k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]]) hK)k where H and K are closed subgroups of G. Applications to Morava Etheory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points.
EQUIVARIANT HOMOTOPY THEORY FOR ProSpectra
, 2006
"... We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The G−homotopy theory is “pieced together” from the G/U−homotopy theories for suitable quotient groups G/U of G; a motivation is th ..."
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Cited by 15 (1 self)
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We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The G−homotopy theory is “pieced together” from the G/U−homotopy theories for suitable quotient groups G/U of G; a motivation is the way continuous group cohomology of a profinite group is built out of the cohomology of its finite quotient groups. In this category Postnikov towers are studied from a general perspective. We introduce pro−G−spectra and construct various model structures on them. A key property of the model structures is that prospectra are weakly equivalent to their Postnikov towers. We give a careful discussion of two versions of a model structure with “underlying weak equivalences”. One of the versions only makes sense for pro−spectra. In the end we use the theory to study homotopy fixed points of pro−Gspectra.
Iterated homotopy fixed points for the LubinTate spectrum
, 2006
"... When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not always possible to form the iterated homotopy fixed point spectrum (ZhH) hK/H, where Z is a continuous Gspectrum. However, we show that, if G = Gn, the extended Morava stabilizer group, and Z = ̂ L(En ∧ X ..."
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Cited by 12 (9 self)
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When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not always possible to form the iterated homotopy fixed point spectrum (ZhH) hK/H, where Z is a continuous Gspectrum. However, we show that, if G = Gn, the extended Morava stabilizer group, and Z = ̂ L(En ∧ X), where ̂ L is Bousfield localization with respect to Morava Ktheory, En is the LubinTate spectrum, and X is any spectrum with trivial Gnaction, then the iterated homotopy fixed point spectrum can always be constructed. Also, we show that (EhH n of Devinatz and Hopkins.) hK/H is just E hK
The LubinTate spectrum and its homotopy fixed point spectra
 NORTHWESTERN UNIVERSITY
, 2003
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CONTINUOUS HOMOTOPY FIXED POINTS FOR LUBINTATE SPECTRA
"... Abstract. We construct a stable model structure on profinite spectra with a continuous action of an arbitrary profinite group. This provides a natural framework for a new and conceptually simplified construction of continuous homotopy fixed point spectra and of continuous homotopy fixed point spectr ..."
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Abstract. We construct a stable model structure on profinite spectra with a continuous action of an arbitrary profinite group. This provides a natural framework for a new and conceptually simplified construction of continuous homotopy fixed point spectra and of continuous homotopy fixed point spectral sequences for LubinTate spectra under the action of the extended Morava stabilizer group. 1.
Deltadiscrete Gspectra and iterated homotopy fixed points
 MR2846911 (2012j:55009), Zbl 1230.55006, arXiv:1006.2762, doi: 10.2140/agt.2011.11.2775
"... Abstract. Let G be a profinite group with finite virtual cohomological dimension and let X be a discrete Gspectrum. If H and K are closed subgroups of G, with H C K, then, in general, the K/Hspectrum XhH is not known to be a continuous K/Hspectrum, so that it is not known (in general) how to def ..."
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Abstract. Let G be a profinite group with finite virtual cohomological dimension and let X be a discrete Gspectrum. If H and K are closed subgroups of G, with H C K, then, in general, the K/Hspectrum XhH is not known to be a continuous K/Hspectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum (XhH)hK/H. To address this situation, we define homotopy fixed points for deltadiscrete Gspectra and show that the setting of deltadiscrete Gspectra gives a good framework within which to work. In particular, we show that by using deltadiscrete K/Hspectra, there is always an iterated homotopy fixed point spectrum, denoted (XhH)hδK/H, and it is just XhK. Additionally, we show that for any deltadiscrete Gspectrum Y, (Y hδH)hδK/H ' Y hδK. Furthermore, if G is an arbitrary profinite group, there is a deltadiscrete Gspectrum Xδ that is equivalent to X and, though XhH is not even known in general to have a K/Haction, there is always an equivalence (Xδ) hδH
EVERY K(n)LOCAL SPECTRUM IS THE HOMOTOPY FIXED POINTS OF ITS MORAVA MODULE
"... Abstract. Let n ≥ 1 and let p be any prime. Also, let En be the LubinTate spectrum, Gn the extended Morava stabilizer group, and K(n) the nth Morava Ktheory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spe ..."
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Abstract. Let n ≥ 1 and let p be any prime. Also, let En be the LubinTate spectrum, Gn the extended Morava stabilizer group, and K(n) the nth Morava Ktheory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization L K(n)(X) is equivalent to the homotopy fixed point spectrum (L K(n)(En ∧ X)) hGn, which is formed with respect to the continuous action of Gn on L K(n)(En ∧ X). In this note, we show that this equivalence holds for any (Scofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adamstype spectral sequence abutting to π∗(L K(n)(X)) is isomorphic to the descent spectral sequence that abuts to π∗((L K(n)(En ∧ X)) hGn). 1.
Homotopy groups of homotopy fixed point spectra associated to En
 In Proceedings of the Nishida Fest, Kinosaki, 2003, in Geom. Topol. Monogr
, 2007
"... We compute the mod.p / homotopy groups of the continuous homotopy fixed point ..."
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We compute the mod.p / homotopy groups of the continuous homotopy fixed point