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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

14C05 Parametrization (Chow and Hilbert schemes)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14G20 Local ground fields in algebraic geometry