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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.
Local projective model structures on simplicial presheaves
 Ktheory
"... Abstract. We give a model structure on the category of simplicial presheaves on some essentially small Grothendieck site T. When T is the Nisnevich site it specializes to a proper simplicial model category with the same weak equivalences as in [MV], but with fewer cofibrations and consequently more ..."
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Cited by 35 (0 self)
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Abstract. We give a model structure on the category of simplicial presheaves on some essentially small Grothendieck site T. When T is the Nisnevich site it specializes to a proper simplicial model category with the same weak equivalences as in [MV], but with fewer cofibrations and consequently more fibrations. This allows a simpler proof of the comparison theorem of [V2], one which makes no use of ∆closed classes. The purpose of this note is to introduce different model structures on the categories of simplicial presheaves and simplicial sheaves on some essentially small Grothendieck site T and to give some applications of these simplified model categories. In particular, we prove that the stable homotopy categories SH((Sm/k)Nis, A 1) and SH((Sch/k)cdh, A 1) are equivalent. This result was first proven by Voevodsky in [V2] and our proof uses many of his techniques, but it does not use his theory of ∆closed classes developed in [V3]. 1. The local projective model structure on presheaves We first recall some of the other wellknown model structures on simplicial presheaves. Definition 1.1. A map f: X → Y of simplicial presheaves (or sheaves) is a local weak equivalence if f ∗ : π0(X) → π0(Y) induces an isomorphism of associated sheaves and, for all U ∈ T, f ∗ : πn(X, x) → πn(Y, f(x)) induces an isomorphism of associated sheaves on T/U for any choice of basepoint x ∈ X(U). The map f is a sectionwise weak equivalence (respectively sectionwise fibration) if for all U ∈ T, the map f(U) : X(U) → Y (U) is a weak equivalence (respectively Kan fibration) of simplicial sets. Heller [He] discovered a model structure on simplicial presheaves whose weak equivalences are the sectionwise weak equivalences. We will refer to his model structure as the injective model structure. Date: January 11, 2001. I would like to thank Dan Isaksen for his many helpful suggestions, and I thank my adviser Peter May for his encouragement and careful reading of many drafts. I am also grateful to Vladimir Voevodsky for noticing an error in an earlier version and for his work that inspired this note. 1 2 BENJAMIN BLANDER
Classical motivic polylogarithm according to Beilinson and Deligne
 DOC. MATH. J. DMV
, 1998
"... ..."
Hypercohomology spectra and Thomason’s descent theorem
 IN ALGEBRAIC KTHEORY, FIELDS INSTITUTE COMMUNICATIONS
, 1997
"... The celebrated LichtenbaumQuillen conjectures predict that for a sufficiently nice scheme and given prime ℓ, the ℓadic algebraic Kgroups of X are closely related to the ℓadic étale cohomology groups of X. More precisely, one version of the conjectures asserts that there is a descent spectral se ..."
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Cited by 20 (2 self)
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The celebrated LichtenbaumQuillen conjectures predict that for a sufficiently nice scheme and given prime ℓ, the ℓadic algebraic Kgroups of X are closely related to the ℓadic étale cohomology groups of X. More precisely, one version of the conjectures asserts that there is a descent spectral sequence of AtiyahHirzebruch type H p q ét (X; Zℓ ()) ⇒ πq−p(KX 2 ℓ) but with the convergence only valid in sufficiently high degrees. Here the coefficient sheaves are Tate twists of the ℓadic integers, and are to be interpreted as zero if q is odd. Throughout this paper, étale cohomology is continuous étale cohomology [19], and the indicated abutment of the spectral sequence consists of the homotopy groups of the Bousfield ℓadic completion of the spectrum KX, not the naive ℓadic completion of the Kgroups. In a remarkable paper [42], Thomason proved the LichtenbaumQuillen conjectures for a certain localized form of Ktheory socalled “Bottperiodic” Ktheory. The first step was the development of an elaborate theory of hypercohomology spectra H ·(X; E) associated to étale presheaves of spectra E or more generally, to presheaves of spectra on a Grothendieck site. These hypercohomology spectra are by their very construction naturally Supported by a grant from the National Science Foundation 1 equipped with a suitable descent spectral sequence, and there is a natural map E(X)−→H · (X; E). In particular, the Ktheory of a scheme X maps to its associated hypercohomology spectrum H · ét(X; K). Fix a prime ℓ, which we will assume is odd in order to simplify the discussion.Let L(−) denote Bousfield localization with respect to complex Ktheory, and let ˆ L(−) denote its ℓadic completion. The main theorem of [42] can be stated as follows. Theorem 0.1 Let X be a separated noetherian regular scheme of finite Krull dimension, with sufficiently nice residue fields of characteristic different from ℓ. Then the natural map KX−→H · ét(X; K) induces a weak equivalence
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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These notes are based on lectures given at the Workshop on Structured ring spectra and
The Homotopy Coniveau Tower
, 2005
"... We examine the “homotopy coniveau tower ” for a general cohomology theory on smooth kschemes and give a new proof that the layers of this tower for Ktheory agree with motivic cohomology. In addition, the homotopy coniveau tower agrees with Voevodsky’s slice tower for S 1spectra, giving a proof ..."
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Cited by 16 (5 self)
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We examine the “homotopy coniveau tower ” for a general cohomology theory on smooth kschemes and give a new proof that the layers of this tower for Ktheory agree with motivic cohomology. In addition, the homotopy coniveau tower agrees with Voevodsky’s slice tower for S 1spectra, giving a proof of a connectedness conjecture of Voevodsky. The homotopy coniveau tower construction extends to a tower of functors on the MorelVoevodsky stable homotopy category, and we identify this P 1stable homotopy coniveau tower with Voevodsky’s slice filtration for P 1spectra. We also show that the 0th layer for the motivic sphere spectrum is the motivic cohomology spectrum, which gives the layers for a general P 1spectrum the structure of a module over motivic cohomology. This recovers and extends recent results of Voevodsky on the 0th layer of the slice filtration, and yields a spectral sequence that is reminiscent of the
(Pre)sheaves of Ring Spectra over the Moduli Stack of Formal Group Laws
, 2004
"... In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem. ..."
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Cited by 12 (1 self)
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In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem.
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.
Topological Ktheory of Algebraic Ktheory Spectra
 J. Algebraic KTheory
, 1999
"... Introduction One of the central problems of algebraic Ktheory is to compute the Kgroups K n X of a scheme X. Since these groups are, by denition, the homotopy groups of a spectrum KX, it makes sense to analyze the homotopytype of the spectrum, rather than just the disembodied homotopy groups. ..."
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Cited by 10 (3 self)
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Introduction One of the central problems of algebraic Ktheory is to compute the Kgroups K n X of a scheme X. Since these groups are, by denition, the homotopy groups of a spectrum KX, it makes sense to analyze the homotopytype of the spectrum, rather than just the disembodied homotopy groups. In addition to facilitating the computation of the Kgroups themselves, knowledge of the spectrum KX can be applied to the study of other topological invariants. For example, if X = Spec R, then the homology groups of the zeroth space 1 KX are of interest since they are the homology groups of the innite general linear group GLR; but they are not determined by the homotopy groups of KX alone. Topological complex Ktheory is another important invariant. Let K denote the periodic complex Ktheory spectrum, and let ^ K denote its Bouseld `adic completion
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.