Abstract. We give a new proof of the theorem of Suslin-Voevodsky which shows that the Bloch-Kato conjecture implies a portion of the Beilinson-Lichtenbaum conjectures. Our proof does not rely on resolution of singularities, and thereby extends the Suslin-Voevodsky to positive characteristic. 1.

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 Bloch-Kato conjecture implies the Beilinson-Lichtenbaum 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 quasi-isomorphism provided the Bloch-Kato conjecture holds. 1.

2. Continuous étale cohomology 6

The main purpose of these notes is to show that Grayson’s motivic cohomology coincides with the usual definition of motivic cohomology- see [V2, S-V] 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

Abstract not found

The cyclotomic trace of Bökstedt-Hsiang-Madsen, the subject of Bökstedt’s lecture at the congress in Kyoto, is a map of pro-abelian groups K∗(A) tr − → TR · ∗(A; p) from Quillen’s algebraic K-theory to a topological refinement of Connes ’ cyclic homology. Over the last decade, our understanding of the target and its relation to K-theory 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 Fp-algebra, 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 K-theory by means of the cyclotomic trace for a wider class of rings, but the precise connection becomes slightly more complicated to spell out. The pro-abelian 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 Rham-Witt complex — which was first considered by Bloch-Deligne-Illusie 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 Rham-Witt groups. This is true, for example, if A is a regular Fp-algebra, 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 Lichtenbaum-Quillen conjecture for local number fields, or more generally, for henselian discrete valuation fields of geometric type.

In recent years, the topological cyclic homology functor of [4] has been used to study and to calculate higher algebraic K-theory. It is known that for finite algebras over the ring of Witt vectors of a perfect field of characteristic p, the p-adic K-theory and topological cyclic homology agree in non-negative degrees, [20]. This has been

Abstract. We calculate the derived functors Rγ ∗ for the base change γ from the Weil-étale 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 ζ-functions in terms of Weil-étale cohomology of the motivic complex Z(n). 1.

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 Geisser-Levine 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 Geisser-Levine method revisited 17

We discuss several constructions of homomorphisms from SK1 and SK2 of central simple algebras to subquotients of Galois cohomology groups.