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THE SPECTRAL SEQUENCE RELATING ALGEBRAIC KTHEORY TO MOTIVIC COHOMOLOGY
"... The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilins ..."
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Cited by 45 (5 self)
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The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilinson [Be] as a natural analogue of the AtiyahHirzebruch spectral sequence from the singular cohomology to the topological Ktheory of a topological space. The expectation of such a spectral sequence has provided much of the impetus for the development of motivic cohomology (e.g., [B1], [V2]) and should facilitate many computations in algebraic Ktheory. In the special case in which X equals SpecF, this spectral sequence was established by S. Bloch and S. Lichtenbaum [BL]. Our construction depends crucially upon the main result of [BL], the existence of an exact couple relating the motivic cohomology of the field F to the multirelative Ktheory of coherent sheaves on standard simplices over F (recalled as Theorem 5.5 below). A major step in generalizing the work of Bloch and Lichtenbaum is our reinterpretation of their spectral sequence in terms of the “topological filtration ” on the Ktheory of the standard cosimplicial scheme ∆ • over F. We find that the spectral sequence arises from a tower of Ωprespectra K( ∆ • ) = K 0 ( ∆ • ) ← − K 1 ( ∆ • ) ← − K 2 ( ∆ • ) ← − · · · Thus, even in the special case in which X equals SpecF, we obtain a much clearer understanding of the BlochLichtenbaum spectral sequence which is essential for purposes of generalization. Following this reinterpretation, we proceed using techniques introduced by V. Voevodsky in his study of motivic cohomology. In order to do this, we provide an equivalent formulation of Ktheory spectra associated to coherent sheaves on X with conditions on their supports K q ( ∆ • × X) which is functorial in X. We then Partially supported by the N.S.F. and the N.S.A.
Universal homotopy theories
 Adv. Math
"... Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy the ..."
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Cited by 37 (3 self)
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Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy theory of schemes. Contents
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
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A closed model structure for ncategories, internal Hom, nstacks and generalized SeifertVan Kampen
, 1997
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Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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Cited by 14 (0 self)
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
An AtiyahHirzebruch spectral sequence for KRtheory
, 2001
"... 3. Equivariant Postnikovsection functors 10 ..."
Excision for simplicial sheaves on the Stein site and Gromov’s Oka principle
 MR1966772 (2004b:32041), Zbl 1078.32017
, 2003
"... Abstract. A complex manifold X satisfies the OkaGrauert property if the inclusion O(S, X) ֒ → C(S, X) is a weak equivalence for every Stein manifold S, where the spaces of holomorphic and continuous maps from S to X are given the compactopen topology. Gromov’s Oka principle states that if X has a ..."
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Cited by 8 (3 self)
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Abstract. A complex manifold X satisfies the OkaGrauert property if the inclusion O(S, X) ֒ → C(S, X) is a weak equivalence for every Stein manifold S, where the spaces of holomorphic and continuous maps from S to X are given the compactopen topology. Gromov’s Oka principle states that if X has a spray, then it has the OkaGrauert property. The purpose of this paper is to investigate the OkaGrauert property using homotopical
Algebraic (geometric) nstacks
"... In the introduction of LaumonMoretBailly ([LMB] p. 2) they refer to a possible theory of algebraic nstacks: Signalons au passage que Grothendieck propose d’élargir à son tour le cadre précédent en remplaçant les 1champs par des nchamps (grosso modo, des faisceaux en ncatégories sur (Aff) ou su ..."
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Cited by 8 (2 self)
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In the introduction of LaumonMoretBailly ([LMB] p. 2) they refer to a possible theory of algebraic nstacks: Signalons au passage que Grothendieck propose d’élargir à son tour le cadre précédent en remplaçant les 1champs par des nchamps (grosso modo, des faisceaux en ncatégories sur (Aff) ou sur un site arbitraire) et il ne fait guère de doute qu’il existe une notion utile de nchamps algébriques.... The purpose of this paper is to propose such a theory. I guess that the main reason why Laumon and MoretBailly didn’t want to get into this theory was for fear of getting caught up in a horribly technical discussion of nstacks of groupoids over a general site. In this paper we simply assume that a theory of nstacks of groupoids exists. This is not an unreasonable assumption, first of all because there is a relatively good substitute—the theory of simplicial presheaves or presheaves of spaces ([Bro] [BG] [Jo] [Ja] [Si3] [Si2])— which should be equivalent, in an appropriate sense, to any eventual theory of nstacks; and second of all because it seems likely that a real theory of nstacks of ngroupoids could be developped in the near future ([Br2], [Ta]).