Second stage with integral curve support

PDF p. 16: Proof of Theorem 6.7, supported second-stage exact sequence

Assume the Axiom of Choice and the fixed universe bounds of the completed comparison suppliers. Let $k$ be a perfect field, $U$ a smooth finite-type $k$-scheme of pure dimension three, and $C\subset U$ an integral closed curve of codimension two. In particular, this includes the curve in the smooth locus used in Theorem 6.7 over an algebraically closed field. Neither smoothness nor normality of $C$ is assumed.

Use the actual Elmanto–Morrow motivic filtration $F^{\bullet }K$ and write $K_C(U)=\mathrm{fib}(K(U)\to K(U\setminus C)),F_C^{j}K(U)=\mathrm{fib}(F^{j}K(U)\to F^{j}K(U\setminus C))\mathrm{.}$ Here $\mathrm{fib}$ is the homotopy fiber of the restriction map, and $K_0(U\mathrm{/}(U\setminus C))$ means ${\pi }_0{K}_C(U)$. The actual augmentation ${ϵ}_2:F_C^{2}K(U)\to K_C(U)$ is an equivalence. Thus its induced isomorphism identifies ${\pi }_0{F}_C^{2}K(U)$ with $K_0(U\mathrm{/}(U\setminus C))$, as in the article. Under this identification there is an exact sequence ${\pi }_0{F}_C^{3}K(U)⟶K_0(U\mathrm{/}(U\setminus C))⟶H_{\mathrm{m}\mathrm{o}\mathrm{t},C}^{4,2}(U,\mathbb{Z})\cong \mathbb{Z}[C]\cong \mathbb{Z}\mathrm{.}$ The second arrow is the actual stage-to-layer projection composed with $({\pi }_0{ϵ}_2{)}^{-1}$. The generator $[C]$ is the positive fundamental cycle in the supported cycle group. On a supported perfect complex $E$, the arrow sends its class to ${\sum }_n(-1{)}^n{\mathrm{\ell }}_{{\mathcal{O}}_{U,\eta }}{H}^n(E{)}_{\eta }$ times $[C]$, where $\eta$ is the generic point of $C$ and $\mathrm{\ell }$ denotes module length. It extends additively to virtual classes. This group is supported cohomology, not the image of $[C]$ in the Chow group of $U$.