Reported alternative argument in arbitrary dimension

PDF p. 16: Remark 6.8

The article reports that Matthew Morrow communicated an alternative proof of its Theorem 6.7, working in arbitrary dimension $d$ without the normality assumption. Theorem 6.7 is the article's comparison between the indicated motivic filtration subgroup of $K_0$ and the subgroup generated by classes of smooth closed points, in its stated normal affine threefold setting.

The reported argument identifies the motivic cohomology group $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{2d,d}(X,\mathbb{Z})$ with the Nisnevich cohomology group $H_{\mathrm{N}\mathrm{i}\mathrm{s}}^d(X,{\mathcal{K}}_d^M)$ and invokes a result of Kato and Saito that the latter group is generated by cycle classes of smooth closed points. Motivic cohomology here is the Elmanto–Morrow cohomology described in Motivic complexes and cohomology; Nisnevich cohomology is sheaf cohomology on the Nisnevich site, and ${\mathcal{K}}_d^M$ denotes the Milnor $K$-theory sheaf intended in the communication. The article says that the argument will appear in ongoing work of Elden Elmanto and Matthew Morrow.

The article's authors also express the belief that the ideas in their proof of Theorem 6.7 might generalize to higher dimensions under a stronger restriction on singularities, giving normality as an example. This is reported as their expectation, not as a proved generalization.

Remark 6.8 does not give complete hypotheses for the communicated comparison, a precise Kato–Saito locator, or the convention for the Milnor $K$-sheaf. In particular, Milnor K symbols and the Nisnevich sheaf defines a naive convention but does not identify it with the convention intended in the report. This item records the communication as reported; it supplies neither a proof nor an independently established arbitrary-dimensional comparison and is not a logical premise for that comparison.