Zero-cycles and \(K\)-theory on normal surfaces.
Let \(X\) be a normal quasi-projective surface over a field \(k\), and denote by \(F^2K_0(X)\) the subgroup of the Grothendieck group \(K_0(X)\) of vector bundles over \(X\) generated by the classes of smooth codimension-two points of \(X\). In the paper under review, the authors prove a conjecture stated by S. Bloch and V. Srinivas [cf.: \textit{V. Srinivas}, J. Reine Angew. Math. 362, 4--27 (1985; Zbl 0603.14007)] some years ago. More precisely, let \(X\) be irreducible and let \(\pi:\widetilde X\to X\) be a resolution of singularities of \(X\) with reduced exceptional divisor \(E\) on \(\widetilde X\). Denote by \(nE\) the \(n\)th infinitesimal neighborhood of \(E\) in \(\widetilde X\) and consider the relative \(K\)-groups \(F^2K_0(\widetilde X,nE)\) for each number \(n\geq 1\). Under these assumptions, the authors confirm the long-standing conjecture that, for sufficiently large numbers \(n\), the maps \[ F^2K_0 (\widetilde X,nE)\to F^2K_0 \bigl(\widetilde X,(n-1)E\bigr) \] and \[ F^2K_0(X)\to F^2K_0(\widetilde X,nE) \] are isomorphisms. This implies, in particular, that \[ F^2K_0(X)= \varprojlim_nF^2K_0(\widetilde X,nE) \] and leads to further new results on zero-cycles on singular surfaces, which are also derived in the present paper. An especially far-reaching application refers to the famous Bloch conjecture on the Albanese map \(\alpha_X:\text{CH}^n(X)_{\deg 0}\to \text{Alb}(X)\) for a normal projective variety \(X\) over an algebraically closed field, stating that \(\alpha_X\) is an isomorphism on torsion subgroups. This paper contains a wealth of new results on Chow groups of zero cycles, which makes it utmost fundamental and important.
scientific article; zbMATH DE number 1850542