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32
Techniques of localization in the theory of algebraic cycles
 J. Algebraic Geom
"... Abstract. We extend the localization techniques of Bloch to simplicial spaces. As applications, we give an extension of Bloch’s localization theorem for the higher Chow groups to schemes of finite type over a regular scheme of dimension one (including mixed characteristic) and, relying on a fundamen ..."
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Cited by 25 (8 self)
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Abstract. We extend the localization techniques of Bloch to simplicial spaces. As applications, we give an extension of Bloch’s localization theorem for the higher Chow groups to schemes of finite type over a regular scheme of dimension one (including mixed characteristic) and, relying on a fundamental result of FriedlanderSuslin, we globalize the BlochLichtenbaum spectral sequence to give a spectral sequence converging to the Gtheory of a scheme X, of finite type over a regular scheme of dimension one, with E 1term the motivic BorelMoore homology (the same as the higher Chow groups of X, after a reindexing). 0.1. Bloch’s higher Chow groups. We recall Bloch’s definition of the higher Chow groups [1]. Fix a base field k. Let ∆N k denote the standard “algebraic Nsimplex”: = Spec k[t0,..., tN] / ∑
Open problems in the motivic stable homotopy theory, I
 In Motives, Polylogarithms and Hodge Theory, Part I
, 2002
"... ..."
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
Localization theorems in topological Hochschild homology and topological cyclic homology
, 2008
"... We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of ..."
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Cited by 15 (1 self)
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We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of ThomasonTrobaugh in Ktheory. We also deduce versions of Thomason’s blowup formula and the projective bundle formula for THH and TC.
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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Cited by 12 (4 self)
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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.
Geometric models for algebraic Ktheory
 Ktheory
"... Abstract. We show how algebraic Ktheory and related theories can be defined in terms of “algebraic mapping spaces ” to appropriate indschemes. Keywords: algebraic Ktheory, infinite loop space, Quotschemes, Γspace ..."
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Cited by 11 (7 self)
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Abstract. We show how algebraic Ktheory and related theories can be defined in terms of “algebraic mapping spaces ” to appropriate indschemes. Keywords: algebraic Ktheory, infinite loop space, Quotschemes, Γspace
On the Grayson spectral sequence
 Chisel, Algebra i Algebr. Geom.):218–253
, 2003
"... The main purpose of these notes is to show that Grayson’s motivic cohomology coincides with the usual definition of motivic cohomology see [V2, SV] for example and hence Grayson’s spectral sequence [Gr] for a smooth semilocal scheme X essentially of finite type over a field F takes the form ..."
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Cited by 10 (0 self)
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The main purpose of these notes is to show that Grayson’s motivic cohomology coincides with the usual definition of motivic cohomology see [V2, SV] for example and hence Grayson’s spectral sequence [Gr] for a smooth semilocal scheme X essentially of finite type over a field F takes the form
Motivic cohomology are isomorphic to higher Chow groups
, 1999
"... In this short paper we show that the motivic cohomology groups defined in [3],[4] are isomorphic to the motivic cohomology groups defined in [1, ] for smooth schemes over any field. In view of [1, Corollary 11.2] it implies that motivic cohomology of [3],[4] are isomorphic to higher Chow groups. Bec ..."
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Cited by 10 (4 self)
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In this short paper we show that the motivic cohomology groups defined in [3],[4] are isomorphic to the motivic cohomology groups defined in [1, ] for smooth schemes over any field. In view of [1, Corollary 11.2] it implies that motivic cohomology of [3],[4] are isomorphic to higher Chow groups. Because of the homotopy invariance property of higher Chow groups this implies that the Cancellation Theorem [4, Theorem 2.1] holds over any field. Both facts were previously known only under the resolution of singularities assumption. The only new element in the proof is Proposition 5. Recall that the motivic complex Z(q) of weight q was defined in [4],[3] as C∗(Ztr(G∧q m))[−q]. In [1, §11] Friedlander and Suslin defined complexes which we will denote ZSF tr (q) as C∗(zequi(Aq, 0))[−2q]. The hypercohomology with coefficients in ZSF tr (q) are shown in [1] to coincide with the higher Chow groups for smooth varieties over all fields. In this paper we prove the following result. Theorem 1 For any field k the complexes of sheaves with transfers Ztr(q)
SemiTopological KTheory of Real VARIETIES
"... . The semitopological Ktheory of real varieties, KR semi (), is an oriented multiplicative (generalized) cohomology theory which extends the authors' earlier theory, K semi (), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles ( ..."
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Cited by 8 (3 self)
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. The semitopological Ktheory of real varieties, KR semi (), is an oriented multiplicative (generalized) cohomology theory which extends the authors' earlier theory, K semi (), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles (sharpened to real semitopological equivalence), consideration of Z/2equivariant mapping spaces of morphisms of algebraic varieties to Grassmannian varieties, and consideration of the algebraic Ktheory of real varieties. The authors verify that the semitopological Ktheory of a real variety X interpolates between the algebraic Ktheory of X and Atiyah's Real Ktheory of the associated Real space of complex points, X R (C). The resulting natural maps of spectra K alg (X) # KR semi (X) # KRtop (X R (C)) satisfy numerous good properties: the first map is a modn equivalence for any projective real variety and any n > 0; the second map is an equivalence for smooth projective curves...