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19
Morava Ktheories and localisation
 Mem. Amer. Math. Soc
, 1999
"... Abstract. We study the structure of the categories of K(n)local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)nilpoten ..."
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Cited by 63 (18 self)
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Abstract. We study the structure of the categories of K(n)local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)nilpotent spectra. We give a number of useful extensions to the theory of vn self maps of finite spectra, and to the theory of Landweber exactness. We show that certain rings of cohomology operations are left Noetherian, and deduce some powerful finiteness results. We study the Picard group of invertible K(n)local spectra, and the problem of grading homotopy groups over it. We prove (as announced by Hopkins and Gross) that the BrownComenetz dual of MnS lies in the Picard group. We give a detailed analysis of some examples when n =1 or 2, and a list of open problems.
Invertible spectra in the E(n)local stable homotopy category
 J. London Math. Soc
"... Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such inver ..."
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Cited by 28 (7 self)
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Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such invertible objects. The
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 19 (1 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.
Moduli problems for structured ring spectra
 DANIEL DUGGER AND BROOKE
, 2005
"... In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theore ..."
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Cited by 10 (0 self)
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In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theorem on the LubinTate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is nonempty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.
Hermitian Ktheory of the integers
 Amer. J. Math
"... Abstract. Rognes and Weibel used Voevodsky’s work on the Milnor conjecture to deduce the strong DwyerFriedlander form of the LichtenbaumQuillen conjecture at the prime 2. In consequence (the 2completion of) the classifying space for algebraic Ktheory of the integers Z[1/2] can be expressed as a ..."
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Cited by 10 (7 self)
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Abstract. Rognes and Weibel used Voevodsky’s work on the Milnor conjecture to deduce the strong DwyerFriedlander form of the LichtenbaumQuillen conjecture at the prime 2. In consequence (the 2completion of) the classifying space for algebraic Ktheory of the integers Z[1/2] can be expressed as a fiber product of wellunderstood spaces BO and BGL(F3) + over BU. Similar results are now obtained for Hermitian Ktheory and the classifying spaces of the integral symplectic and orthogonal groups. For the integers Z[1/2], this leads to computations of the 2primary Hermitian Kgroups and affirmation of the LichtenbaumQuillen conjecture in the framework of Hermitian Ktheory.
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.
On degeneration of onedimensional formal group laws and stable homotopy theory, AJM 125
, 2003
"... Abstract. In this note we study a certain formal group law over a complete discrete valuation ring F[[un−1]] of characteristic p> 0 which is of height n over the closed point and of height n − 1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on t ..."
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Cited by 5 (0 self)
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Abstract. In this note we study a certain formal group law over a complete discrete valuation ring F[[un−1]] of characteristic p> 0 which is of height n over the closed point and of height n − 1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law Hn−1 of height n − 1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group Sn−1 of Hn−1. We show that the automorphism group Sn of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of Sn−1 to the cohomology of Sn with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed. 1. Introduction. The ring MU∗(MU) of cooperations in complex cobordism theory has a wellknown interpretation in terms of onedimensional commutative formal group laws. The category C of plocal comodules over MU∗(MU) has a filtration C = C0 ⊃ C1 ⊃···⊃Cn ⊃···
Algebraic Ktheory of the twoadic integers
 J. Pure Appl. Algebra
, 1997
"... Abstract. We compute the twocompleted algebraic Kgroups K∗ ( ˆ Z2) ∧ 2 of the twoadic integers, and determine the homotopy type of the twocompleted algebraic Ktheory spectrum K ( ˆ Z2) ∧ 2. The natural map K(Z) ∧ 2 → K( ˆ Z2) ∧ 2 is shown to induce an isomorphism modulo torsion in degrees 4 ..."
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Cited by 5 (2 self)
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Abstract. We compute the twocompleted algebraic Kgroups K∗ ( ˆ Z2) ∧ 2 of the twoadic integers, and determine the homotopy type of the twocompleted algebraic Ktheory spectrum K ( ˆ Z2) ∧ 2. The natural map K(Z) ∧ 2 → K( ˆ Z2) ∧ 2 is shown to induce an isomorphism modulo torsion in degrees 4k + 1 with k ≥ 1.
A resolution of the K(2)local sphere
"... We develop a framework for displaying the stable homotopy theory of the sphere, at least when we are localized at the second Morava Ktheory K(2). At the prime 3, we write the spectrum LK(2) S as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E ..."
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Cited by 3 (0 self)
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We develop a framework for displaying the stable homotopy theory of the sphere, at least when we are localized at the second Morava Ktheory K(2). At the prime 3, we write the spectrum LK(2) S as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E where F is a finite subgroup of the Morava stabilizer group and E2 is the second Morava or LubinTate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n = 2 at p = 3 represents the edge of our current knowledge: n = 1 is classical and at n = 2, the prime 3 is the largest prime where the Morava stabilizer group has a ptorsion subgroup, so that the homotopy theory is not entirely algebraic. The problem of understanding the homotopy groups of spheres has been central to algebraic topology ever since the field emerged as a distinct area of mathematics. A period of calculation beginning with Serre's computation of the cohomology of EilenbergMacLane spaces and the advent of the Adams spectral sequence culminated, in the late 1970s, with the work of Miller, Ravenel, and Wilson on periodic phenomena in the homotopy groups of spheres and Ravenel's nilpotence conjectures. The solutions to most of these conjectures by Devinatz, Hopkins, and Smith in the middle 1980s established the primacy of the "chromatic" point of view and there followed a period in which the community absorbed these results and extended the qualitative picture of stable homotopy theory. Computations passed from center stage, to some extent, although there has been steady work in the wings  most notably by Shimomura and his coworkers, and Ravenel, and more lately by Hopkins and his coauthors in their work on topological modular forms. The amount of interest generat...
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.