Euler-group comparison and the article's expectation

PDF p. 3: Opening paragraph, expected Euler class identification

Assume AC and the fixed universe conventions. The article's expectation of an appropriate Euler-group description and an interpretation of rank-three splitting is retained as historical mathematical context. Its concrete candidate, the oriented-ideal group $E(A)$, is already covered by Krishna's established comparison: for a reduced finite-type algebra $A$ of dimension $d\ge 2$ over an algebraically closed field, $E(A)\cong {\mathrm{CH}}_{\mathrm{L}\mathrm{W}}^d(A)\stackrel{\sim }{\to }{F}_{\mathrm{s}\mathrm{m},\mathrm{h}\mathrm{t}}^d{K}_0^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}(A)\mathrm{.}$ Here the last subgroup is generated by smooth maximal ideals of height $d$, and the second arrow is the actual cycle map. Thus this specific group-comparison candidate must not be labelled an unresolved problem. On a pure-dimensional normal affine threefold the affine perfect comparison identifies the target with the completed smooth-point subgroup $F_{\mathrm{s}\mathrm{m}}^3{K}_0(X)$. If components have different dimensions, retain the source height-three convention rather than adding smooth points of smaller components.

The original request to specify an Euler group, the actual comparison, its hypotheses and its compatibility with the rank-three splitting obstruction is preserved here. The cited group isomorphisms supply the comparison part for the explicit candidate. This remark neither verifies a determinant-twisted Schlichting comparison nor asserts a new unresolved formulation or a new proof of obstruction compatibility.

Euler class group presentation

Subgroup generated by smooth closed points

Affine perfect K zero and bundle K zero

Krishna's Corollary 7.6 and Theorem 7.7 are the precise established group-comparison inputs, with the hypotheses and reading limits recorded above. The local Euler presentation is exactly Definition 7.1. In the pure-dimensional normal affine case the point generators are the completed smooth closed points; the affine perfect bridge takes their classical finite-projective Euler classes to the actual perfect residue classes. This identifies the subgroup used here without equating coherent G-theory and perfect K-theory on a singular scheme. No local reconstruction of Krishna's deep comparison proof or current-status audit of other Euler theories is claimed.