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43
The BlochKato conjecture and a theorem of SuslinVoevodsky
 J. reine angew. Math
"... Abstract. We give a new proof of the theorem of SuslinVoevodsky which shows that the BlochKato conjecture implies a portion of the BeilinsonLichtenbaum conjectures. Our proof does not rely on resolution of singularities, and thereby extends the SuslinVoevodsky to positive characteristic. 1. ..."
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Abstract. We give a new proof of the theorem of SuslinVoevodsky which shows that the BlochKato conjecture implies a portion of the BeilinsonLichtenbaum conjectures. Our proof does not rely on resolution of singularities, and thereby extends the SuslinVoevodsky to positive characteristic. 1.
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 13 (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.
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
Weilétale cohomology over finite fields
"... Abstract. We calculate the derived functors Rγ ∗ for the base change γ from the Weilétale site to the étale site for a variety over a finite field. For smooth and proper varieties, we apply this to express Tate’s conjecture and Lichtenbaum’s conjecture on special values of ζfunctions in terms of W ..."
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Cited by 8 (3 self)
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Abstract. We calculate the derived functors Rγ ∗ for the base change γ from the Weilétale site to the étale site for a variety over a finite field. For smooth and proper varieties, we apply this to express Tate’s conjecture and Lichtenbaum’s conjecture on special values of ζfunctions in terms of Weilétale cohomology of the motivic complex Z(n). 1.
A sheaf theoretic reformulation of the Tate conjecture
, 1997
"... 2. Continuous étale cohomology 6 ..."
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
Duality via cycle complexes
"... Summary. We show that Bloch’s complex of relative zerocycles can be used as a dualizing complex over perfect fields and number rings. This leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over algebraically closed fields, finite fields, local fields of mix ..."
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Cited by 6 (4 self)
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Summary. We show that Bloch’s complex of relative zerocycles can be used as a dualizing complex over perfect fields and number rings. This leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over algebraically closed fields, finite fields, local fields of mixed characteristic, and rings of integers in number rings, generalizing results which so far have only been known for smooth schemes or in low dimensions, and unify the padic and ladic theory. 1
The GeisserLevine method revisited and algebraic cycles over a finite field
 Math. Ann
"... Abstract. We reformulate part of the arguments of T. Geisser and M. Levine relating motivic cohomology with finite coefficients to truncated étale cohomology with finite coefficients [9, 10]. This reformulation amounts to a uniqueness theorem for motivic cohomology, and shows that the GeisserLevine ..."
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Cited by 6 (3 self)
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Abstract. We reformulate part of the arguments of T. Geisser and M. Levine relating motivic cohomology with finite coefficients to truncated étale cohomology with finite coefficients [9, 10]. This reformulation amounts to a uniqueness theorem for motivic cohomology, and shows that the GeisserLevine method can be applied generally to compare motivic cohomology with other types of cohomology theories. We apply this to prove an equivalence between conjectures of Tate and Beilinson on cycles in characteristic p and a vanishing conjecture for continuous étale cohomology. Contents 1. Functoriality of motivic cohomology 3 2. The GeisserLevine method revisited 17
Algebraic Ktheory and trace invariants
 Proceedings of the International Congress of Mathematicians
, 2002
"... The cyclotomic trace of BökstedtHsiangMadsen, the subject of Bökstedt’s lecture at the congress in Kyoto, is a map of proabelian groups K∗(A) tr − → TR · ∗(A; p) from Quillen’s algebraic Ktheory to a topological refinement of Connes ’ cyclic homology. Over the last decade, our understanding of t ..."
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Cited by 5 (1 self)
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The cyclotomic trace of BökstedtHsiangMadsen, the subject of Bökstedt’s lecture at the congress in Kyoto, is a map of proabelian groups K∗(A) tr − → TR · ∗(A; p) from Quillen’s algebraic Ktheory to a topological refinement of Connes ’ cyclic homology. Over the last decade, our understanding of the target and its relation to Ktheory has been significantly advanced. This and possible future development is the topic of my lecture. The cyclotomic trace takes values in the subset fixed by an operator F called the Frobenius. It is known that the induced map K∗(A, Z/p v) tr − → TR · ∗(A; p, Z/p v) F=1 is an isomorphism, for instance, if A is a regular local Fpalgebra, or if A is a henselian discrete valuation ring of mixed characteristic (0, p) with a separably closed residue field. It is possible to evaluate Ktheory by means of the cyclotomic trace for a wider class of rings, but the precise connection becomes slightly more complicated to spell out. The proabelian groups TR · ∗(A; p) are typically very large. But they come equipped with a number of operators, and the combined algebraic structure is quite rigid. There is a universal example of this structure — the de RhamWitt complex — which was first considered by BlochDeligneIllusie in connection with Grothendieck’s crystalline cohomology. In general, the canonical map W · Ω q A → TR · q(A;p) is an isomorphism, if q ≤ 1, and the higher groups, too, can often be expressed in terms of the de RhamWitt groups. This is true, for example, if A is a regular Fpalgebra, or if A is a smooth algebra over the ring of integers in a local number field. The calculation in the latter case verifies the LichtenbaumQuillen conjecture for local number fields, or more generally, for henselian discrete valuation fields of geometric type.