2,645 research outputs found
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
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
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
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
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
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 -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
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