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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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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 E2–term of the descent spectral sequence for continuous G–spectra
 J. of Math
"... Abstract. Let {Xi} be a tower of discrete Gspectra, each of which is fibrant as a spectrum, so that X = holimi Xi is a continuous Gspectrum, with homotopy fixed point spectrum X hG. The E2term of the descent spectral sequence for π∗(X hG) cannot always be expressed as continuous cohomology. Howev ..."
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Abstract. Let {Xi} be a tower of discrete Gspectra, each of which is fibrant as a spectrum, so that X = holimi Xi is a continuous Gspectrum, with homotopy fixed point spectrum X hG. The E2term of the descent spectral sequence for π∗(X hG) cannot always be expressed as continuous cohomology. However, we show that the E2term is always built out of a certain complex of spectra, that, in the context of abelian groups, is used to compute the continuous cochain cohomology of G with coefficients in limi Mi, where {Mi} is a tower of discrete Gmodules. 1.
LOWER CENTRAL SERIES OBSTRUCTIONS TO HOMOTOPY SECTIONS OF CURVES OVER NUMBER FIELDS
"... 1.1. Some background and motivation 2 1.2. Summary of results 10 2. δn and Massey products 13 ..."
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1.1. Some background and motivation 2 1.2. Summary of results 10 2. δn and Massey products 13
A DESCENT SPECTRAL SEQUENCE FOR ARBITRARY K(n)LOCAL SPECTRA WITH EXPLICIT E2TERM
"... Abstract. Let n be any positive integer and p any prime. Also, let X be any spectrum and let K(n) denote the nth Morava Ktheory spectrum. Then we construct a descent spectral sequence with abutment π∗(LK(n)(X)) and E2term equal to the continuous cohomology of Gn, the extended Morava stabilizer gro ..."
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Abstract. Let n be any positive integer and p any prime. Also, let X be any spectrum and let K(n) denote the nth Morava Ktheory spectrum. Then we construct a descent spectral sequence with abutment π∗(LK(n)(X)) and E2term equal to the continuous cohomology of Gn, the extended Morava stabilizer group, with coefficients in a certain discrete Gnmodule that is built from various homotopy fixed point spectra of the Morava module of X. This spectral sequence can be contrasted with the K(n)local EnAdams spectral sequence for π∗(LK(n)(X)), whose E2term is not known to always be equal to a continuous cohomology group. 1.
OBTAINING INTERMEDIATE RINGS OF A LOCAL PROFINITE GALOIS EXTENSION WITHOUT LOCALIZATION
 JOURNAL OF HOMOTOPY AND RELATED STRUCTURES
"... Let En be the LubinTate spectrum and let Gn be the nth extended Morava stabilizer group. Then there is a discrete Gnspectrum Fn, with LK(n)(Fn) ' En, that has the property that (Fn)hU ' EhUn, for every open subgroup U of Gn. In particular, (Fn)hGn ' LK(n)(S0). More generally, for ..."
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Let En be the LubinTate spectrum and let Gn be the nth extended Morava stabilizer group. Then there is a discrete Gnspectrum Fn, with LK(n)(Fn) ' En, that has the property that (Fn)hU ' EhUn, for every open subgroup U of Gn. In particular, (Fn)hGn ' LK(n)(S0). More generally, for any closed subgroup H of Gn, there is a discrete Hspectrum Zn,H, such that (Zn,H)hH ' EhHn. These conclusions are obtained from results about consistent klocal profinite GGalois extensions E of finite vcd, where Lk(−) is LM (LT (−)), with M a finite spectrum and T smashing. For example, we show that Lk(EhH) ' EhH, for every open subgroup H of G.
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.
PROFINITE GSPECTRA
 HOMOLOGY, HOMOTOPY AND APPLICATIONS, VOL. 15(1), 2013, PP.151–189
, 2013
"... We construct a stable model structure on profinite spectra with a continuous action of an arbitrary profinite group. The motivation is to provide a natural framework in a subsequent paper for a new and conceptually simplified construction of continuous homotopy fixed point spectra and of continuous ..."
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We construct a stable model structure on profinite spectra with a continuous action of an arbitrary profinite group. The motivation is to provide a natural framework in a subsequent paper 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.