We prove that the natural map , where is an algebraic torus over a field of dimension at most , a smooth proper geometrically irreducible variety over containing as an open subset and is the group of classes of zero-dimensional cycles on of degree zero, is an isomorphism. In particular, the group is finite if is finitely generated over the prime subfield, over the complex field, or over a -adic field.
Alexander Merkurjev. "R-equivalence on three-dimensional tori and zero-cycles." Algebra Number Theory 2 (1) 69 - 89, 2008. https://doi.org/10.2140/ant.2008.2.69