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
Abstract

We prove that the natural map $T\left(F\right)∕R\to {A}_{0}\left(X\right)$, 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 ${A}_{0}\left(X\right)$ is the group of classes of zero-dimensional cycles on $X$ of degree zero, is an isomorphism. In particular, the group ${A}_{0}\left(X\right)$ 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
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