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42
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 46 (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
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 34 (3 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
Motivic cohomology over Dedekind rings
 MATH. Z
, 2004
"... We study properties of Bloch’s higher Chow groups on smooth varieties over Dedekind rings. We prove a conditional Gersten resolution, which implies that Hi (Z(n)) = 0 for i> n and that there is a Gersten resolution for Hi (Z/pr(n)), if the residue characteristic is p. We also show that the Blo ..."
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Cited by 22 (4 self)
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We study properties of Bloch’s higher Chow groups on smooth varieties over Dedekind rings. We prove a conditional Gersten resolution, which implies that Hi (Z(n)) = 0 for i> n and that there is a Gersten resolution for Hi (Z/pr(n)), if the residue characteristic is p. We also show that the BlochKato conjecture implies the BeilinsonLichtenbaum conjecture, an identification Z/m(n)ét ∼ = µ ⊗n m, for m invertible on the scheme, and a Gersten resolution with (arbitrary) finite coefficients. Over a discrete valuation ring of mixed characteristic (0, p), we construct a map from motivic cohomology to syntomic cohomology, which is a quasiisomorphism provided the BlochKato conjecture holds.
Topological cyclic homology of schemes
 Preprint 1997 28 THOMAS GEISSER AND MARC LEVINE
, 2001
"... In recent years, the topological cyclic homology functor of [4] has been used to study and to calculate higher algebraic Ktheory. It is known that for finite algebras over the ring of Witt vectors of a perfect field of characteristic p, the padic Ktheory and topological cyclic homology agree in n ..."
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In recent years, the topological cyclic homology functor of [4] has been used to study and to calculate higher algebraic Ktheory. It is known that for finite algebras over the ring of Witt vectors of a perfect field of characteristic p, the padic Ktheory and topological cyclic homology agree in nonnegative degrees, [20]. This has been
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 14 (4 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
Infinitesimal cohomology and the Chern character to negative cyclic homology
, 2008
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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 7 (6 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.
Stable étale realization and étale cobordism
 Adv. Math
"... We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A 1homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realizati ..."
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Cited by 6 (5 self)
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We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A 1homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero. On the other hand we get a natural setting for étale cohomology theories. In particular, we define and discuss an étale topological cobordism theory for schemes. It is equipped with an AtiyahHirzebruch spectral sequence starting from étale cohomology. Finally, we construct maps from algebraic to étale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element. 1