Abstract. The purpose of this paper is to study the p-part of motivic cohomology and algebraic K-theory in characteristic p (we use higher Chow groups as our definition of motivic cohomology). The main theorem states that for a field k of characteristic p, H i (k, Z(n)) is uniquely p-divisible for i = n. This implies that the natural map K M n (k) − → Kn(k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and that Kn(k) is p-torsion free.As a consequence, one can calculate the K-theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example Kn(X, Z/p r ) = 0 for n > dimX. Another consequence is Gersten's conjecture with mod p-coefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch's cycle complexes localized at p satisfy all BeilinsonLichtenbaum-Milne axioms for motivic complexes, except the vanishing conjecture.

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

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 H i (Z(n)) = 0 for i > n and that there is a Gersten resolution for H i (Z/p r (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)é t ∼ = 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.

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Summary. We construct cohomology groups with compact support H i c (Xar, Z(n)) for separated schemes of finite type over a finite field, which generalize Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds, and rational and numerical equivalence agree up to torsion, then the groups H i c (Xar, Z(n)) are finitely generated, form an integral model of ladic cohomology with compact support, and admit a formula for the special values of the ζ-function of X.

We show that Bloch's complex of relative zero-cycles 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 unifying the p-adic and l-adic theory. As an application, we generalize Rojtman's theorem to normal, projective schemes.

We calculate the derived functors Rγ * for the base change γ from the Weil-étale site to theétale 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).

It is well-known that algebraic K-theory preserves products of rings. However, in general, algebraic K-theory does not preserve fiber-products of rings, and bi-relative algebraic K-theory measures the deviation. It was proved by Cortinas that,rationally, bi-relative algebraic K-theory and bi-relative cyclic homology agree. In this paper, we show that, with finite coefficients, bi-relative algebraic K-theory and bi-relative topological cyclic homology agree. As an application, we show that for a, possibly singular, curve over a perfect field of positive characteristic p, the cyclotomic trace map induces an isomorphism of the p-adic algebraic K-groups and the p-adic topological cyclic homology groups in non-negative degrees. As a further application, we show that the difference between the p-adic K-groups of the integral group ring of a finite group and the p-adic K-groups of a maximal Z-order in the rational group algebra can be expressed entirely in terms of topological cyclic homology