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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 38 (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 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
Motivic cohomology over Dedekind rings
 Math. Z
, 2004
"... Abstract. 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 th ..."
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Abstract. 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. 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
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
The E2–term of the descent spectral sequence for continuous G–spectra
 J. of Math
"... Abstract. Let {Xi} be a tower of discrete Gspectra, each of which is fibrant as a spectrum, so that X = holimi Xi is a continuous Gspectrum, with homotopy fixed point spectrum X hG. The E2term of the descent spectral sequence for π∗(X hG) cannot always be expressed as continuous cohomology. Howev ..."
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Cited by 4 (1 self)
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Abstract. Let {Xi} be a tower of discrete Gspectra, each of which is fibrant as a spectrum, so that X = holimi Xi is a continuous Gspectrum, with homotopy fixed point spectrum X hG. The E2term of the descent spectral sequence for π∗(X hG) cannot always be expressed as continuous cohomology. However, we show that the E2term is always built out of a certain complex of spectra, that, in the context of abelian groups, is used to compute the continuous cochain cohomology of G with coefficients in limi Mi, where {Mi} is a tower of discrete Gmodules. 1.
Class Groups and General Linear Group Cohomology for a Ring of Algebraic Integers
, 1994
"... this paper; in effect we work entirely in the category H ..."
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this paper; in effect we work entirely in the category H
Total cofibres of diagrams of spectra
 New York Journal of Mathematics
"... If Y is a diagram of spectra indexed by an arbitrary poset C together with a specified subposet D, we define the total cofibre Γ(Y) of Y as cofibre(hocolimD(Y) ✲ hocolimC(Y)). We construct a comparison map ˆ ΓY:holimCY ✲ hom(Z, ˆ Γ(Y)) to a mapping spectrum of a fibrant replacement of Γ(Y) where ..."
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If Y is a diagram of spectra indexed by an arbitrary poset C together with a specified subposet D, we define the total cofibre Γ(Y) of Y as cofibre(hocolimD(Y) ✲ hocolimC(Y)). We construct a comparison map ˆ ΓY:holimCY ✲ hom(Z, ˆ Γ(Y)) to a mapping spectrum of a fibrant replacement of Γ(Y) where Z is a simplicial set obtained from C and D, and characterise those poset pairs D ⊂ C for which ˆΓY is a stable equivalence. The characterisation is given in terms of stable cohomotopy of spaces related to Z. For example, if C is a finite polytopal complex with C  ∼ = B m a ball with boundary sphere D, then Z  ∼ =PL S m, and ˆ Γ(Y) and holimC(Y) agree up to mfold looping and up to stable equivalence. As an application of the general result we give a spectral sequence for π∗(Γ(Y)) with E2term involving higher derived inverse limits of π∗(Y), generalising earlier constructions for spacevalued diagrams indexed by the