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

Merkurjev, Alexander

doi:10.2140/ant.2008.2.69

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

Alexander Merkurjev

Vol. 2 (2008), No. 1, 69–89

DOI: 10.2140/ant.2008.2.69

Abstract

We prove that the natural map $T (F) ∕ R \to A_{0} (X)$ , 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 $A_{0} (X)$ is the group of classes of zero-dimensional cycles on $X$ of degree zero, is an isomorphism. In particular, the group $A_{0} (X)$ is finite if $F$ is finitely generated over the prime subfield, over the complex field, or over a $p$ -adic field.

Keywords

algebraic tori, $R$-equivalence, $K\!$-cohomology, zero-dimensional cycle

Mathematical Subject Classification 2000

Primary: 19E15

Milestones

Received: 28 June 2007

Revised: 23 October 2007

Accepted: 20 November 2007

Published: 1 February 2008

Authors

Alexander Merkurjev
Department of Mathematics University of California Los Angeles, CA 90095-1555 United States

Vol. 2, No. 1, 2008