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Torsion algebraic cycles on varieties over local fields. (English) Zbl 0709.14005
Algebraic $$K$$-theory: Connections with geometry and topology, Proc. Meet., Lake Louise/Can. 1987, NATO ASI Ser., Ser. C 279, 343-388 (1989).
[For the entire collection see Zbl 0685.00007.]
Let A be a Henselian discrete valuation ring with finite or separably closed residue field F of characteristic $$p,$$ $${\mathcal X}$$ a smooth proper scheme over Spec(A) with special fibre Y and let $$CH^ 2({\mathcal X}), CH^ 2(Y)$$ be the corresponding second Chow groups.
Then it is proved tat the map $$CH^ 2({\mathcal X})\{\ell \}\to CH^ 2(Y)\{\ell \}$$ is injective for all primes $$\ell \neq p$$. - As a corollary it is proved for instance that if $${\mathcal X}/A$$ is projective and F is finite then the prime to p part of $$CH^ 2({\mathcal X})_{tors}$$ is finite. Similar results are proved for K-cohomology.
Reviewer: A.Buium

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14G20 Local ground fields in algebraic geometry
##### Keywords:
Henselian discrete valuation ring; second Chow groups