Normalization defect delta of a reduced curve

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field and let $C$ be a reduced curve of finite type over $k$: a $k$-scheme of finite type with $C=C_{\mathrm{r}\mathrm{e}\mathrm{d}}$ (The reduction of a scheme) and pure dimension one, proper over $k$ in the applications. Let

$\nu \text{\hspace{0.17em}⁣}:\tilde{C}⟶C$

be the finite normalization of Normalization of a reduced curve is finite, and let

${\mathcal{Q}}_C:=\mathrm{coker}({\mathcal{O}}_C⟶{\nu }_{\ast }{\mathcal{O}}_{\tilde{C}})$

be the normalization defect sheaf, a coherent ${\mathcal{O}}_C$-module (Coherent module sheaves). The normalization defect of $C$ is

${\delta }_k(C):={\dim }_k{H}^0(C,{\mathcal{Q}}_C),$

the $k$-dimension of its space of global sections, a nonnegative integer.

The following comments record why the definition is meaningful. Since $\nu$ is finite, ${\nu }_{\ast }{\mathcal{O}}_{\tilde{C}}$ is a coherent ${\mathcal{O}}_C$-module and so is its quotient ${\mathcal{Q}}_C$; this is the finite-pushforward case of Coherent higher direct images under proper morphisms, and also follows directly from the affine-local description of a finite morphism. The stalks of ${\mathcal{Q}}_C$ vanish exactly at the points $p$ at which ${\mathcal{O}}_{C,p}\to ({\nu }_{\ast }{\mathcal{O}}_{\tilde{C}}{)}_p$ is an isomorphism, so ${\mathcal{Q}}_C$ is supported on the non-normal locus of $C$: a closed subset of the Noetherian one-dimensional space $C$ containing no generic point of $C$, because the local ring of the reduced curve $C$ at a generic point is a field and hence normal. Such a closed set is a finite set of closed points; componentwise this is the finiteness of Proper closed subsets of a curve are finite. Consequently $H^0(C,{\mathcal{Q}}_C)$ is the direct sum of the finitely many stalk contributions ${\mathcal{Q}}_{C,p}$ over the support of ${\mathcal{Q}}_C$, each of which is a finite-dimensional $k$-vector space because ${\mathcal{Q}}_{C,p}$ has finite length over the Noetherian local ring ${\mathcal{O}}_{C,p}$ and the residue field $\kappa (p)$ is finite over $k$. Thus ${\delta }_k(C)$ is a well-defined nonnegative integer. When $C$ is proper over $k$, the same finiteness is the statement of Euler characteristic of a coherent sheaf for the coherent sheaf ${\mathcal{Q}}_C$.

• The definition uses only the finite normalization $\nu$, its pushforward ${\nu }_{\ast }{\mathcal{O}}_{\tilde{C}}$, and $H^0$ of the cokernel; no choice of a resolution of singularities or of a blowup sequence enters.
• The defect can be read off pointwise as a sum of local contributions; the identity with the Euler characteristic difference $\chi ({\mathcal{O}}_{\tilde{C}})-\chi ({\mathcal{O}}_C)$ and the weighted sum of local lengths are proved later on this page, and are not part of the definition.