#### Vol. 2, No. 1, 2008

 Download this article For screen For printing
 Recent Issues
 The Journal Cover Editorial Board Editors' Addresses Editors' Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)
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