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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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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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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.
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
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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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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Prop groups and towers of rational homology spheres
 Geom. Topol
"... Abstract In the preceding paper, Calegari and Dunfield exhibit a sequence of hyperbolic 3manifolds which have increasing injectivity radius, and which, subject to some conjectures in number theory, are rational homology spheres. We prove unconditionally that these manifolds are rational homology s ..."
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Abstract In the preceding paper, Calegari and Dunfield exhibit a sequence of hyperbolic 3manifolds which have increasing injectivity radius, and which, subject to some conjectures in number theory, are rational homology spheres. We prove unconditionally that these manifolds are rational homology spheres, and give a sufficient condition for a tower of hyperbolic 3manifolds to have first Betti number 0 at each level. The methods involved are purely prop group theoretical.
GrossHopkins duality
 Topology
"... In [8] Hopkins and Gross state a theorem revealing a profound relationship between two different kinds of duality in stable homotopy theory. A proof of a related but weaker result is given in [3], and we understand that Sadofsky is preparing a proof that works in general. Here we present a proof tha ..."
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In [8] Hopkins and Gross state a theorem revealing a profound relationship between two different kinds of duality in stable homotopy theory. A proof of a related but weaker result is given in [3], and we understand that Sadofsky is preparing a proof that works in general. Here we present a proof that seems rather different and complementary to Sadofsky’s. We thank IChiau Huang for help with Proposition 18, and John Greenlees for helpful discussions. We first indicate the context of the HopkinsGross theorem. Cohomological duality theorems have been studied in a number of contexts; they typically say that H k (X ∗ ) = H d−k (X) ∨ for some class of objects X with some notion of duality X ↔ X ∗ and some type of cohomology groups H k (X) with some notion of duality A ↔ A ∨ and some integer d. For example, if M is a compact smooth oriented manifold of dimension d we have a Poincaré duality isomorphism H k (M; Q) = Hom(H d−k (M; Q), Q) (so here we just have M ∗ = M). For another example, let S be a smooth complex projective variety of dimension d, and let Ω d be the sheaf of topdimensional differential forms. Then for any coherent sheaf F on S we have a Serre duality isomorphism H k (S; Hom(F, Ω d)) = Hom(H d−k (S; F), C). This can be seen as a special case of the Grothendieck duality theorem for a proper morphism [7], which is formulated in terms of functors between derived categories. There is a wellknown analogy
Double coset formulas for profinite groups, preprint http://www.ma.umist.ac.uk/pas/preprints
 School of Mathematics, University of Manchester, P.O. Box 88, Manchester M60 1QD, England Email address: Peter.Symonds@manchester.ac.uk
"... Abstract. We show that in certain circumstances there is a sort of double coset formula for induction followed by restriction for representations of profinite groups. 1. ..."
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Abstract. We show that in certain circumstances there is a sort of double coset formula for induction followed by restriction for representations of profinite groups. 1.
Tate Cohomology In Axiomatic Stable Homotopy Theory.
"... Contents 1. Introduction 1 2. Axiomatic Tate cohomology in a stable homotopy category. 2 3. Finite localizations. 4 4. The category of Gspectra 6 5. The derived category of a commutative ring. 7 6. The category of modules over a highly structured ring 10 7. The derived category of kG 11 8. Chromat ..."
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Contents 1. Introduction 1 2. Axiomatic Tate cohomology in a stable homotopy category. 2 3. Finite localizations. 4 4. The category of Gspectra 6 5. The derived category of a commutative ring. 7 6. The category of modules over a highly structured ring 10 7. The derived category of kG 11 8. Chromatic categories. 12 9. Splittings of the Tate construction. 13 10. Calculation by comparison. 14 11. Calculations by associative algebra. 16 12. Calculations by commutative algebra. 19 13. Gorenstein localizations. 22 References 23 1. Introduction The purpose of the present note is to show how the axiomatic approach to Tate cohomology of [18, Appendix B] can be implemented in the axiomatic stable homotopy theory of HoveyPalmieriStrickland [32]. Much of the work consists of collecting known results in a single language and a single framework. The very effortlessness of the process is an effective advertisement for the language, and a call fo