Vol. 2, No. 1, 2008

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R-equivalence on three-dimensional tori and zero-cycles

Alexander Merkurjev

Vol. 2 (2008), No. 1, 69–89

We prove that the natural map T(F)R A0(X), where T is an algebraic torus over a field F of dimension at most 3, X a smooth proper geometrically irreducible variety over F containing T as an open subset and A0(X) is the group of classes of zero-dimensional cycles on X of degree zero, is an isomorphism. In particular, the group A0(X) is finite if F is finitely generated over the prime subfield, over the complex field, or over a p-adic field.

algebraic tori, $R$-equivalence, $K\!$-cohomology, zero-dimensional cycle
Mathematical Subject Classification 2000
Primary: 19E15
Received: 28 June 2007
Revised: 23 October 2007
Accepted: 20 November 2007
Published: 1 February 2008
Alexander Merkurjev
Department of Mathematics
University of California
Los Angeles, CA 90095-1555
United States