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Open problems in the motivic stable homotopy theory, I
 In Motives, Polylogarithms and Hodge Theory, Part I
, 2002
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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 41 (8 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.
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 38 (12 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] / ∑
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 29 (8 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
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.
Higher algebraic Ktheory (after Quillen, . . . )
, 2007
"... We give a short introduction (with a few proofs) to higher algebraic Ktheory (mainly of schemes) based on the work of Quillen, Waldhausen, Thomason and others. ..."
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Cited by 13 (0 self)
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We give a short introduction (with a few proofs) to higher algebraic Ktheory (mainly of schemes) based on the work of Quillen, Waldhausen, Thomason and others.
Milnor Ktheory of rings, higher Chow groups and . . .
 INVENT. MATH.
, 2002
"... If R is a smooth semilocal algebra of geometric type over an infinite field, we prove that the Milnor Kgroup K M n (R) surjects onto the higher Chow group CHn (R, n) for all n � 0. Our proof shows moreover that there is an algorithmic way to represent any admissible cycle in CHn (R, n) modulo eq ..."
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Cited by 13 (1 self)
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If R is a smooth semilocal algebra of geometric type over an infinite field, we prove that the Milnor Kgroup K M n (R) surjects onto the higher Chow group CHn (R, n) for all n � 0. Our proof shows moreover that there is an algorithmic way to represent any admissible cycle in CHn (R, n) modulo equivalence as a linear combination of “symbolic elements” defined as graphs of units in R. As a byproduct we get a new and entirely geometric proof of results of Gabber, Kato and Rost, related to the Gersten resolution for the Milnor Ksheaf. Furthermore it is also shown that in the semilocal PID case we have, under some mild assumptions, an isomorphism. Some applications are also given.
Multiplicative Properties of the Slice Filtration
"... Let S be a Noetherian separated scheme of finite Krull dimension, and let SH(S) denote the motivic stable homotopy category of Morel and Voevodsky. In order to get a motivic version of the Postnikov tower, Voevodsky [Voe02] constructs a filtered family of triangulated subcategories of SH(S): ..."
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Cited by 13 (5 self)
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Let S be a Noetherian separated scheme of finite Krull dimension, and let SH(S) denote the motivic stable homotopy category of Morel and Voevodsky. In order to get a motivic version of the Postnikov tower, Voevodsky [Voe02] constructs a filtered family of triangulated subcategories of SH(S):
Techniques, computations, and conjectures for semitopological Ktheory
 MATH. ANN
, 2004
"... We establish the existence of an “AtiyahHirzebruchlike” spectral sequence relating the morphic cohomology groups of a smooth, quasiprojective complex variety to its semitopological Kgroups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence tha ..."
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Cited by 12 (2 self)
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We establish the existence of an “AtiyahHirzebruchlike” spectral sequence relating the morphic cohomology groups of a smooth, quasiprojective complex variety to its semitopological Kgroups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that connects the motivic cohomology and algebraic Ktheory of varieties, and it is also compatible with the classical AtiyahHirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the BorelMoore (singular) cohomology of complex varieties introduced by H. Gillet and C. Soulé — to compute the semitopological Ktheory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational threefolds, and related varieties, the semitopological Kgroups and topological Kgroups are isomorphic in all degrees permitted by cohomological considerations. We also