Removing normality from the article results

PDF p. 3: Finally, we expect normality to be an artifact; PDF p. 16: Remark 6.8

Assume the Axiom of Choice and the size conventions of the supplied motivic constructions. Let $k$ be an algebraically closed field. An affine $k$-scheme here means the spectrum of a finitely generated commutative $k$-algebra; its dimension is the supremum of the lengths of chains of distinct irreducible closed subsets. No normality assumption is imposed. A rank-$n$ vector bundle is a locally free ${\mathcal{O}}_X$-module locally isomorphic to ${\mathcal{O}}_X^n$, and ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n(X)$ is its set of isomorphism classes. Write $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{a,b}(X,\mathbb{Z})$ for the supplied integral motivic cohomology and $c_i$ for its motivic Chern classes.

Determine whether the following extensions hold:

1. For every affine surface $X$ over $k$, meaning an affine $k$-scheme of dimension $2$, is $(c_1,c_2):{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_2(X)\to H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{2,1}(X,\mathbb{Z})\times H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{4,2}(X,\mathbb{Z})$ a bijection?
2. For every affine threefold $X$ over $k$, meaning an affine $k$-scheme of dimension $3$, is $(c_1,c_2,c_3):{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_3(X)\to H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{2,1}(X,\mathbb{Z})\times H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{4,2}(X,\mathbb{Z})\times H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{6,3}(X,\mathbb{Z})$ a bijection?
3. For every such threefold of pure dimension $3$, is $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^3{K}_0(X)=F_{\mathrm{s}\mathrm{m}}^3{K}_0(X)$? Here $K_0(X)$ is degree zero of perfect-complex K-theory, $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^3{K}_0(X)$ is the image of degree zero of the third motivic filtration stage in $K_0(X)$, and $F_{\mathrm{s}\mathrm{m}}^3{K}_0(X)$ is the subgroup generated by the classes of residue sheaves at closed points where $X$ is smooth over $k$. Pure dimension means that every irreducible component has dimension $3$. For threefolds without this purity hypothesis, determine the appropriate formulation and whether the comparison extends.
4. For every affine threefold $X$ over $k$, is $(c_1,c_2):{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_2(X)\to H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{2,1}(X,\mathbb{Z})\times H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{4,2}(X,\mathbb{Z})$ surjective?

The questions include schemes that are not reduced; an affirmative answer in a narrower class alone would not settle the formulations above. The purity qualification in the third question is the domain of the supplied smooth-point filtration definition.

The Axiom of Choice

Isomorphism classes of vector bundles

Motivic Chern classes

Motivic filtration and image groups

Subgroup generated by smooth closed points

As of 2026-11-09, this status record is limited to the supplied article arXiv:2610.22679v1. No later-publication search or independent verification of the reported work in progress was performed. The article proves the corresponding results under its normality hypotheses and states on p. 3 that the authors expect normality to be an artifact of their proofs. Its Remark 6.8 reports a communicated alternative approach to the smooth-point comparison without normality and in arbitrary dimension, to appear in work in progress; the present record does not treat that communication as a proof. The questions above record the proposed extensions, including the scope of nonreduced and non-pure schemes, without claiming an independent resolution or a comprehensive current literature-status review.