Top cycles supported on an integral curve

PDF p. 13: Divisor presentation for supported degree six

Assume AC. Let $k$ be algebraically closed, $U$ a smooth finite-type $k$-scheme of pure dimension three, and $T\subset U$ an integral closed curve. Put $K=k(T)$ and define ${\mathrm{div}}_T(f)={\sum }_{x\in T\text{ closed}}{\mathrm{ord}}_{{\mathcal{O}}_{T,x}}(f)[x],{\mathrm{ord}}_R(a\mathrm{/}b)={\mathrm{\ell }}_R(R\mathrm{/}aR)-{\mathrm{\ell }}_R(R\mathrm{/}bR)\mathrm{.}$ For the fixed-support Bloch complex with alternating face differential, ${\mathrm{C}\mathrm{H}}_T^3(U)\cong \mathrm{coker}\text{\hspace{0.17em}⁣}(K^{\times }\stackrel{{\mathrm{div}}_T}{\to }{⨁}_{x\in T\text{ closed}}\mathbb{Z}[x])\mathrm{.}$ When $T$ is affine, normalization $\nu :\tilde{T}\to T$ induces a surjection $\mathrm{Pic}(\tilde{T})={\mathrm{C}\mathrm{H}}^1(\tilde{T})\twoheadrightarrow {\mathrm{C}\mathrm{H}}_T^3(U),$ induced on point cycles by $[y]\mapsto [\nu (y)]$. For $f\in K^{\times }$, ${\mathrm{ord}}_x(f)={\sum }_{y\in {\nu }^{-1}(x)}{\mathrm{ord}}_y(f)$, with DVR valuations upstairs and the displayed length orders downstairs. No normality or smoothness of $T$ is assumed.