Smooth rank, first Chern, determinant and supported positive leading cycle

PDF p. 9: Determinant normalization; PDF p. 10: Smooth rank and first Chern normalization; PDF p. 16: Final supported generic-length paragraph

Assume AC and the compatible universe and Nisnevich-site size bounds of the supplied constructions, and use the specified perfect-complex nonconnective K-spectrum and its actual EM motivic filtration. For a smooth finite-type scheme $X$ over a field $k$, the weight-zero projection is the locally constant rank function. The first-stage projection gives ${\tilde{K}}_0(X)={\pi }_0{F}^{1}K(X)⟶H^{2,1}(X)=\mathrm{Pic}(X),x⟼c_1(x)=\det (x),$ and $K_1(X)={\pi }_1{F}^{1}K(X)⟶H^{1,1}(X)=\mathrm{\Gamma }(X,{\mathcal{O}}_X{)}^{\times },z⟼\det (z)\mathrm{.}$ Here reduced $K_0$ means the kernel of the locally constant rank map, component by component; determinant on perfect complexes uses its alternating graded-line convention.

For the supported assertion let $k$ be perfect (in particular the article's algebraically closed field qualifies), let $U$ be smooth of finite type and pure dimension three over $k$, and let $i:C\hookrightarrow U$ be an integral codimension-two closed curve. Neither smoothness nor normality of $C$, nor global quasi-projectivity of $U$, is required. Write $K_C(U)=\mathrm{fib}(K(U)\to K(U\setminus C)),F_C^{j}K(U)=\mathrm{fib}(F^{j}K(U)\to F^{j}K(U\setminus C))\mathrm{.}$ The actual augmentation ${ϵ}_2:F_C^{2}K(U)\to K_C(U)$ is an equivalence, and the supported layer is canonically ${\pi }_0{\mathrm{gr}}_C^{2}K(U)=H_C^{4,2}(U)=H_0{z}_C^2(U,\bullet )=\mathbb{Z}[C]\mathrm{.}$ This is the supported cycle group, not the image of $[C]$ in ambient Chow cohomology. For $\xi \in {\pi }_0{F}_C^{2}K(U)$ let $E$ denote its augmented class in the actual supported $K_0$, using support-category localization. Then its specified stage-to-layer projection is $p_2(\xi )=({\sum }_n(-1{)}^n{\mathrm{\ell }}_{{\mathcal{O}}_{U,\eta }}{H}^n(E{)}_{\eta })[C],\eta =\mathrm{generic}(C)\mathrm{.}$ The notation for a virtual class $E$ means the additive extension of this formula from supported perfect complexes. Equivalently $p_2\circ ({\pi }_0{ϵ}_2{)}^{-1}$ on supported $K_0$ is alternating generic length times the positive cycle. In particular $i_{\ast }{\mathcal{O}}_C[0]$ maps to $+[C]$, and every closed-point-supported class maps to zero.

The supporting normalization was checked in the published Kahn–Levine final PDF, printed p.487, Proposition2.2.2 p.496, Lemmas2.3.1–2.3.2 pp.499–500, Definition3.1.5 p.503, Definitions3.1.9–3.1.15 pp.505–507, Theorem3.2.4 and proof pp.508–510, section5.9 pp.531–532, Lemma6.2.1 p.535, Lemmas6.4.1–6.4.3 pp.539–540, and Theorem6.5.5 and proof pp.542–543. The maps are restriction-to-support/generic-coefficient maps with ordinary $A=k$ dimension normalization. Only the affine generic neighbourhood is required to be quasi-projective. The stable augmentation comparison was checked in Levine arXiv:math/0510334v1, model conventions pp.44–45, section8.3 and Theorem9.0.3/Remark9.0.4 pp.46–47; its deep proof is imported. EM arXiv:2309.08463v1 was checked at section3.1, Theorem3.5 and Remark3.6, Constructions4.34/4.36, Theorem6.1, Remark6.3, Corollary6.4/proof and footnote12. In particular no published proof of the then-in-preparation comparison is claimed. This reading concerns application-critical passages and their hypotheses, not full-paper or whole-library review. The article's original locations on pp.9,10,16 and all existing coverage are retained.