Extending the geometric support method to higher dimensions

PDF p. 16: Paragraph following Remark 6.8

Assume the Axiom of Choice and the size conventions of the smooth-point construction. Let $k$ be algebraically closed and let $X$ be an affine $k$-variety of pure dimension $d>3$: its coordinate algebra is finitely generated and reduced, and every irreducible component has dimension $d$. Under what precise restrictions on its singularities, for example normality, can the geometric support method in the proof of the article’s Theorem 6.7 be generalized to establish $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^d{K}_0(X)=F_{\mathrm{s}\mathrm{m}}^d{K}_0(X)?$ Normality means that the local rings are integrally closed domains. The left side is the image in $K_0(X)$ of the degree-zero homotopy group of the $d$th stage of the motivic filtration; the right side is the subgroup generated by the classes of the residue sheaves of closed points at which $X$ is smooth over $k$. Here $K_0(X)$ is the Grothendieck group of perfect complexes.

In particular, formulate hypotheses under which the proof’s reduction to support inside the smooth locus, correction by smooth closed points, and analysis of the remaining supported classes can be carried out in dimension $d$. The problem asks for a method and exact hypotheses; it does not assert that normality suffices.

Motivic filtration and image groups

Subgroup generated by smooth closed points

The article proves the corresponding third-piece statement for normal affine threefolds and expresses the belief that its geometric proof may generalize to higher dimensions for sufficiently mild singularities, mentioning normality as an example. Remark 6.8 separately reports a communicated alternative argument without normality and in arbitrary dimension, to appear in ongoing work; that report is not a proof of the requested geometric extension. No independent current-status review or solution is claimed, and this question is not a logical premise.