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The homotopy fixed point spectra of profinite Galois extensions
 Trans. Amer. Math. Soc
"... Abstract. 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 for ..."
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Cited by 10 (8 self)
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Abstract. 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 that the function spectrum FA((E hH)k, (E hK)k) is equivalent to the localized 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 in terms of the derived functor of fixed points.
Iterated homotopy fixed points for the LubinTate spectrum, submitted for publication, available online as arXiv:math.AT/0610907
"... Abstract. 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 = ̂ ..."
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Cited by 8 (4 self)
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Abstract. 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
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 5 (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.
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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Cited by 4 (1 self)
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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.
EXPLICIT FIBRANT REPLACEMENT FOR DISCRETE GSPECTRA
"... Abstract. If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is wellknown that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in ..."
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Cited by 2 (2 self)
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Abstract. If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is wellknown that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in the model category of discrete Gspectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dimension and X is a discrete Gspectrum, then there is an explicit fibrant model for X. Also, we give several applications of this concrete model related to closed subgroups of G. 1.
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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Cited by 2 (2 self)
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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.
BUILDINGS, ELLIPTIC CURVES, AND THE K(2)LOCAL SPHERE
, 2005
"... We investigate a dense subgroup Γ of the second Morava stabilizer group given by a certain group of quasiisogenies of a supersingular elliptic curve in characteristic p. The group Γ acts on the BruhatTits building for GL2(Qℓ) through its action on the ℓadic Tate module. This action has finite sta ..."
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We investigate a dense subgroup Γ of the second Morava stabilizer group given by a certain group of quasiisogenies of a supersingular elliptic curve in characteristic p. The group Γ acts on the BruhatTits building for GL2(Qℓ) through its action on the ℓadic Tate module. This action has finite stabilizers, giving a small resolution for the homotopy fixed point spectrum (EhΓ 2)hGal by spectra of topological modular forms. Here, E2 is a version of
Equivariant Homotopy . . .
, 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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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