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23
Stable model categories are categories of modules
 TOPOLOGY
, 2003
"... A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
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Cited by 78 (16 self)
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A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R.
Cyclic homology, cdhcohomology and negative Ktheory
, 2005
"... We prove a blowup formula for cyclic homology which we use to show that infinitesimal Ktheory satisfies cdhdescent. Combining that result with some computations of the cdhcohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic Ktheor ..."
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Cited by 19 (7 self)
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We prove a blowup formula for cyclic homology which we use to show that infinitesimal Ktheory satisfies cdhdescent. Combining that result with some computations of the cdhcohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic Ktheory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.
Morita theory in abelian, derived and stable model categories, Structured ring spectra
 London Math. Soc. Lecture Note Ser
, 2004
"... These notes are based on lectures given at the Workshop on Structured ring spectra and ..."
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Cited by 19 (0 self)
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These notes are based on lectures given at the Workshop on Structured ring spectra and
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 (4 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 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
Fibred sites and stack cohomology
 Math. Z
"... A stack G is traditionally defined to be a pseudofunctor on a Grothendieck ..."
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Cited by 7 (6 self)
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A stack G is traditionally defined to be a pseudofunctor on a Grothendieck
Classification of Stable Model Categories
"... A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
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Cited by 6 (5 self)
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A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a `ring spectrum with several objects', i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R. 1.
Periodicity of hermitian Kgroups
 In preparation
"... 0. Introduction and statements of main results By the fundamental work of Bott [10] it is known that the homotopy groups of classical Lie groups are periodic, of period 2 or 8. For instance, the general linear and symplectic groups satisfy the isomorphisms: πn(GL(R)) ∼ = πn+8(GL(R)) ..."
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Cited by 3 (3 self)
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0. Introduction and statements of main results By the fundamental work of Bott [10] it is known that the homotopy groups of classical Lie groups are periodic, of period 2 or 8. For instance, the general linear and symplectic groups satisfy the isomorphisms: πn(GL(R)) ∼ = πn+8(GL(R))
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.