Generic rank and point-supported classes on a curve

Every nonempty open $V\subseteq C$ is integral and Noetherian, has dimension at most one, and contains $\eta$. Every point $x\mathrm{\ne }\eta$ of $V$ is closed in $V$: a further proper specialization would give a chain of length at least two starting at $\eta$. Hence every proper nonempty integral closed subscheme of $V$ is the reduced point at such an $x$. The whole integral closed subscheme is $V$ itself. The possibility that $V=\{\eta \}$ is open is allowed; then $V=\mathrm{Spec}k(\eta )$ and there are no points $x\mathrm{\ne }\eta$ in $V$.

Let $H_V\subseteq G_0(V)$ be generated by $[{\mathcal{O}}_V]$ and $[\kappa (x)]$ for $x\in V\setminus \{\eta \}$. These point sheaves are coherent: on an affine neighbourhood $W=\mathrm{Spec}A$ of $x$, the global closed-point direct image restricts to $\widetilde{A\mathrm{/}{\mathfrak{m}}_x}$, which is coherent by [F2]; on $V\setminus \{x\}$ it is zero. These two opens cover $V$, so locality of coherence gives the global assertion. Apply [F1] under [A1] to the property $P(F):[F]\in H_V$. It holds for zero and is two-out-of-three by the exact relations in $G_0(V)$. For the integral closed subscheme $V$, the witness is ${\mathcal{O}}_V$, with generic stalk $k(\eta )$ annihilated by the zero maximal ideal of ${\mathcal{O}}_{V,\eta }$. For each proper integral closed subscheme from step 1.1, use $\kappa (x)$; its stalk at $x$ is one-dimensional over the residue field and annihilated by the maximal ideal. All witness hypotheses hold, so $H_V=G_0(V)$, including when $V=\{\eta \}$.

Taking stalks at $\eta$ is exact and gives finite-dimensional $k(\eta )$-vector spaces. Dimension therefore defines the generic-rank homomorphism. It sends $[{\mathcal{O}}_V]$ to $1$ and each $[\kappa (x)]$ with $x\mathrm{\ne }\eta$ to zero. By step 2.1, an element of its kernel is an integral linear combination of precisely these point classes. Conversely all such point classes have generic rank zero. For $V=C$, the generic point is not closed because $\dim C=1$, so these are exactly the closed-point classes in the Statement. Coherent sheaves supported at closed points have generic rank zero, and each residue sheaf is such a sheaf. Finally dimension identifies the Grothendieck group of finite-dimensional vector spaces with $\mathbb{Z}$, giving the stated $K_0(k(\eta ))$ normalization.

The nonempty opens are filtered under reverse inclusion, since every finite intersection contains $\eta$. Restriction is exact and preserves the generic stalk and structure-sheaf class. For $x\in V\setminus \{\eta \}$, the open $V\setminus \{x\}$ still contains $\eta$ and kills the residue-sheaf class. Each element in step 2.1 involves only finitely many such points, so one nonempty open removes them all. Thus every colimit element is an integer multiple of the compatible structure-sheaf class, and generic rank sends that class to $1$. This proves the colimit isomorphism. If $V=\{\eta \}$, its only generator in this argument is $[{\mathcal{O}}_V]$; the generic point is never treated as a removable point class. For $V=C$, step 3.1 identifies the kernel of localization with the stated point-supported subgroup.

Under the additional bounds in [A1], apply [F3] and let $\mathcal{I}$ define $C$ in $U$. The image of $[E]$ in $G_0(C)$ is the finite sum of the classes of ${\mathcal{I}}^r{H}^n(E)\mathrm{/}{\mathcal{I}}^{r+1}{H}^n(E)$ with signs $(-1{)}^n$. At $\eta$ each quotient is a finite-dimensional vector space over ${\mathcal{O}}_{U,\eta }\mathrm{/}{\mathcal{I}}_{\eta }=k(\eta )$, so its dimension is its length as an ${\mathcal{O}}_{U,\eta }$-module. The finite exhaustive filtration and additivity of length identify the sum of these dimensions, for each $n$, with the length of $H^n(E{)}_{\eta }$. Only finitely many $n$ contribute. Applying generic rank gives exactly the alternating generic-length formula, without requiring regularity or smoothness of $C$.