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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.
Homotopy fixed points for LK(n)(En ∧X) using the continuous action
 J. Pure Appl. Algebra
"... Abstract. Let K(n) be the nth Morava Ktheory spectrum. Let En be the LubinTate spectrum, which plays a central role in understanding LK(n)(S 0), the K(n)local sphere. For any spectrum X, dene E_(X) to be the spectrum LK(n)(En ^ X). Let G be a closed subgroup of the pronite group Gn, the group of ..."
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Cited by 20 (14 self)
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Abstract. Let K(n) be the nth Morava Ktheory spectrum. Let En be the LubinTate spectrum, which plays a central role in understanding LK(n)(S 0), the K(n)local sphere. For any spectrum X, dene E_(X) to be the spectrum LK(n)(En ^ X). Let G be a closed subgroup of the pronite group Gn, the group of ring spectrum automorphisms of En in the stable homotopy category. We show that E_(X) is a continuous Gspectrum, with homotopy xed point spectrum (E_(X))hG. Also, we construct a descent spectral sequence with abutment ((E_(X))hG): 1.
Homotopy fixed points for L K(n)(En∧ X) using the continuous action
 J. Pure Appl. Algebra
"... Abstract. Let G be a closed subgroup of Gn, the extended Morava stabilizer group. Let En be the LubinTate spectrum, X an arbitrary spectrum with trivial Gaction, and let ˆ L = L K(n). We prove that ˆ L(En ∧ X) is a continuous Gspectrum with homotopy fixed point spectrum ( ˆ L(En ∧ X)) hG, define ..."
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Cited by 12 (5 self)
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Abstract. Let G be a closed subgroup of Gn, the extended Morava stabilizer group. Let En be the LubinTate spectrum, X an arbitrary spectrum with trivial Gaction, and let ˆ L = L K(n). We prove that ˆ L(En ∧ X) is a continuous Gspectrum with homotopy fixed point spectrum ( ˆ L(En ∧ X)) hG, defined with respect to the continuous action. Also, we construct a descent spectral sequence whose abutment is π∗( ( ˆ L(En∧X)) hG). We show that the homotopy fixed points of ˆ L(En ∧ X) come from the K(n)localization of the homotopy fixed points of the spectrum (Fn ∧ X). 1.
The LubinTate spectrum and its homotopy fixed point spectra
 NORTHWESTERN UNIVERSITY
, 2003
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Quasicoherent sheaves on the moduli stack of formal groups
"... For years I have been echoing my betters, especially Mike Hopkins, and telling anyone who would listen that the chromatic picture of stable homotopy theory is dictated and controlled by the geometry of the moduli stack Mfg of smooth, onedimensional formal groups. Specifically, I would say that the ..."
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Cited by 9 (1 self)
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For years I have been echoing my betters, especially Mike Hopkins, and telling anyone who would listen that the chromatic picture of stable homotopy theory is dictated and controlled by the geometry of the moduli stack Mfg of smooth, onedimensional formal groups. Specifically, I would say that the height filtration of Mfg dictates a canonical and natural decomposition of a quasicoherent sheaf on Mfg, and this decomposition predicts and controls the chromatic decomposition of a finite spectrum. This sounds well, and is even true, but there is no single place in the literature where I could send anyone in order for him or her to get a clear, detailed, unified, and linear rendition of this story. This document is an attempt to set that right. Before going on to state in detail what I actually hope to accomplish here, I should quickly acknowledge that the opening sentences of this introduction and, indeed, this whole point of view is not original with me. I have already mentioned Mike Hopkins, and just about everything I’m going to say here is encapsulated in the table in section 2 of [15] and can be gleaned from the notes
THE BROWNCOMENETZ DUAL OF THE K(2)LOCAL SPHERE AT THE PRIME 3
"... Abstract. We calculate the homotopy type of the BrownComenetz dual I2 of the K(2)local sphere at the prime 3 and show that there is an equivalence in the K(2)local ..."
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Abstract. We calculate the homotopy type of the BrownComenetz dual I2 of the K(2)local sphere at the prime 3 and show that there is an equivalence in the K(2)local
THE HOMOTOPY ORBIT SPECTRUM FOR PROFINITE GROUPS
"... Abstract. Let G be a profinite group. We define an S[[G]]module to be a Gspectrum X that satisfies certain conditions, and, given an S[[G]]module X, we define the homotopy orbit spectrum XhG. When G is countably based and X satisfies a certain finiteness condition, we construct a homotopy orbit s ..."
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Abstract. Let G be a profinite group. We define an S[[G]]module to be a Gspectrum X that satisfies certain conditions, and, given an S[[G]]module X, we define the homotopy orbit spectrum XhG. When G is countably based and X satisfies a certain finiteness condition, we construct a homotopy orbit spectral sequence whose E2term is the continuous homology of G with coefficients in the graded profinite bZ[[G]]module pi∗(X). Let Gn be the extended Morava stabilizer group and let En be the LubinTate spectrum. As an application of our theory, we show that the function spectrum F (En, LK(n)(S 0)) is an S[[Gn]]module with an associated homotopy orbit spectral sequence. 1.