Torsion zero-cycles and étale homology of singular schemes
The étale homology theory is brought to bear with the \(K\)-theoretic and cohomological methods used by several authors in the last years in a way that yields more powerful tools to deal with \(A_ 0(X)_{tor}\), the torsion part of the group of rational classes of zero-cycles on an algebraic scheme \(X\) over a field \(k\). The main result establishes that for any prime number \(\ell\) which is different from the characteristic of \(k\) there is a canonical surjective map \(\varphi_ X:H_ 1(X,\mathbb{Q}_ \ell/\mathbb{Z}_ \ell)\to A_ 0(X)\{\ell\}\) from the étale homology group of \(X\) onto the group of elements in \(A_ 0(X)\) of finite order not divisible by \(\ell\), and that \(\varphi_ X\) is an isomorphism if there is an open immersion \(X\hookrightarrow Z\) with \(Z\) an algebraic proper scheme over \(k\) such that \(Z-X\) is connected and with the property (let us call it \(S)\) that an element in \(H^ 1(X,\mathbb{Q}_ \ell)\) is zero if its restriction to any regular subscheme of \(Z\) is zero. The main corollary is that if \(X\) is complete and satisfies \(S\), then \(A_ 0(X)\{\ell\}\simeq\pi_ 1^{ab}(X)\otimes\mathbb{Q}_ \ell/\mathbb{Z}_ \ell\). If \(X\) is in addition smooth, then this isomorphism yields Roitman's isomorphism \(A_ 0(X)\{\ell\}\simeq\text{Alb}(X)\{\ell\}\). The relation of M. Levine's generalization of Roitman's isomorphism to the main corollary is also clarified.
torsion part of the group of rational classes of zero-cycles
scientific article; zbMATH DE number 24260
scientific article