Classical smooth critical-rank classifications

PDF p. 1: Introduction, first paragraph

Assume AC and the article's universes. Over an algebraically closed field $k$ of arbitrary characteristic, the ordinary integral Chow Chern maps are bijections $C_2:{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_2(X)\stackrel{\sim }{\to }{\mathrm{C}\mathrm{H}}^1(X)\times {\mathrm{C}\mathrm{H}}^2(X)$ for every smooth affine surface $X$, and $C_3:{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_3(Y)\stackrel{\sim }{\to }{\mathrm{C}\mathrm{H}}^1(Y)\times {\mathrm{C}\mathrm{H}}^2(Y)\times {\mathrm{C}\mathrm{H}}^3(Y)$ for every smooth affine threefold $Y$. In each case every tuple is realized, and its realizing bundle is unique up to isomorphism. These are the classical surface classification of Murthy–Swan and threefold classification of N. Mohan Kumar–M. P. Murthy. Here ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_r$ denotes bundle isomorphism classes, and $c_i$ is the coefficient of the standard quotient-projective-bundle Chern relation, with $c_1(\mathcal{O}(D))=[D]$.

The surface attribution is retained from Fasel–Yakerson, Vector bundles on singular surfaces and threefolds, p. 1, citing Murthy–Swan, Vector bundles over affine surfaces, Invent. Math. 36 (1976), 125–165, DOI 10.1007/BF01390007. The original Murthy–Swan proof was not read. The local proof uses a cylinder, the completed Serre splitting supplier and a separately namespaced cancellation import. Consequently the patch is independent of the uncompleted native cancellation target. Ordinary Chow classes are the interface here; no unnamed motivic comparison is assumed. The two smooth schemes are understood equidimensional, as surfaces and threefolds in the article; finitely many disjoint equidimensional components can equally be handled componentwise.