Levine–Weibel cycle image with pure-dimensional and component conventions

PDF p. 3: Levine–Weibel image after TheoremC; PDF p. 14: Definition6.5 pure-dimensional smooth-point subgroup

Assume AC and the completed size conventions. In the article's pure-dimensional normal affine finite-type threefold setting over an algebraically closed field, $\mathrm{im}({\mathrm{cyc}}_X:{\mathrm{CH}}_0^{LW}(X)\to K_0(X))=F_{\mathrm{s}\mathrm{m}}^3{K}_0(X)\mathrm{.}$ The map sends $[x]$ to the actual perfect residue-sheaf class $[i_{\ast }\kappa (x)]={\phi }_x(1)$ and kills the specified Cartier-curve principal-divisor relations. Thus it is a surjection onto this image subgroup; no injectivity or identification of the whole Chow group with its image is asserted. The pure-dimensional interpretation is explicit in the source Definition6.5, so it does not require treating the customary word threefold/variety as an original-paper error.

For a normal affine finite-type scheme of dimension three over the same field with lower-dimensional clopen components, put $X_3$ for the union of its three-dimensional components. The exact source dimension-three convention gives the stronger, unambiguous componentwise formula ${\mathrm{CH}}_{\mathrm{K}\mathrm{r}}^3(X)\cong {\mathrm{CH}}_0^{LW}(X_3),\mathrm{im}({\mathrm{cyc}}_X^{\mathrm{K}\mathrm{r}})=F_{\mathrm{s}\mathrm{m},\mathrm{t}\mathrm{o}\mathrm{p}}^3{K}_0(X)=(i_3{)}_{\ast }{F}_{\mathrm{s}\mathrm{m}}^3{K}_0(X_3)\subseteq F_{\mathrm{m}\mathrm{o}\mathrm{t}}^3{K}_0(X)\mathrm{.}$ Here $F_{\mathrm{s}\mathrm{m},\mathrm{t}\mathrm{o}\mathrm{p}}^3$ is the zero-extended top-dimensional point image defined and justified in the local prerequisite. Lower-dimensional strata have their own same-dimension Levine–Weibel point images (and a free weight-zero point group in dimension zero). The subgroup generated by all geometrically smooth closed points across these strata is their direct sum; it is not in general a subgroup of weight three. The explicit ${\mathbb{A}}^3⨿\mathrm{Spec}k$ counterexample prevents that overbroad generated interpretation. None of these clarifications edits the completed pure-dimensional definitions.