Arithmetic homology and an integral version of Katos conjecture

We define an integral Borel-Moore homology theory over finite fields, called arithmetic homology, and an integral version of Kato homology. Both types of groups are expected to be finitely generated, and sit in a long exact sequence with higher Chow groups of zero-cycles.Comment: improved version, to appear in Journal fuer die reine und angewandte Mathemati

Parshin's conjecture revisited

We show that Pashin's conjecture on the vanishing of rational higher K-groups of smooth, projective varieties over finite fields can be thought of as a combination of three weaker conjectures.Comment: to appear in proceedings of a conference on K-theory and non-commutative geometry, Valladolid 200

Comparing the Brauer group and the Tate Shafarevich group

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate-Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a smooth and proper surface over a finite field is a square if it is finite.Comment: Rewrote the proof to make it more readable. To appear in Journal of the Institute of Mathematics of Jussie

Rojtman's theorem for normal schemes

We show that Rojtman's theorem holds for normal schemes: For any reduced normal scheme of finite type over an algebraically closed field, the torsion of the zero'th Suslin homology group agrees with the torsion of the albanese variety (the universal object for maps to semi-abelian varieties). The proof uses proper hypercovers to reduce to the smooth case, which was previously proven by Spiess-Szamuely.Comment: Improved and corrected, similar to the version to appear in Mathematical Research Letter

Duality via cycle complexes

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 unify the p-adic and l-adic theory. As an application, we generalize Rojtman's theorem to normal, projective schemes.Comment: Updated and improved version; accepted at Annals of Mathematic

Parshin's conjecture and motivic cohomology with compact support

We discuss Parshin's conjecture on rational K-theory over finite fields and its implications for motivic cohomology with compact support

Duality for integral motivic cohomology

We discuss duality pairings on integral \'etale motivic cohomology groups of regular and proper schemes over algebraically closed fields, local fields, finite fields, and arithmetic schemes.Comment: To appear in: Proceedings of the International Colloquium in K-theory held at TIFR in January, 201

The cyclotomic trace map and values of zeta-functions

We show that the cyclotomic trace map for smooth varieties over number rings can be interpreted as a regulator map and hence are related to special values of $\zeta$-functions

Suslin's singular homology and cohomology

We discuss Suslin's singular homology and cohomology. In the first half we examine the p-part in characteristic p, and the situation over non-algebraically closed fields. In the second half we focus on finite base fields. We study finite generation properties, and give a modified definition which behaves like a homology theory: in degree zero it is a copy of Z for each connected component, in degree one it is related to the abelianized (tame) fundamental group, even for singular schemes, and it is expected to be finitely generated in general

Weil-etale cohomology over finite fields

We calculate the total derived functor for the map from the Weil-etale site introduced by Lichtenbaum to the etale site for varieties over finite fields. In particular, there is a long exact sequence relating Weil-etale cohomology and etale cohomology. In the second half of the paper, we apply this to study the Weil-etale cohomology of the motivic complex for smooth and projective varieties. These groups are expected to be finitely generated, to give an integral model for l-adic cohomology, and to be related to special values of the zeta function. We give necessary and sufficient conditions for this to hold, and examples.Comment: Revised versio