The conductor of a normalization

Assume the Axiom of Choice. Let $\nu \text{\hspace{0.17em}⁣}:X^{\nu }\to X$ be the normalization of an irreducible classical variety over an algebraically closed field (Normalization of a classical variety by gluing affine normalizations). Let $U\subseteq X$ be an affine chart with coordinate ring $A=k[U]$, and let $B=k[{\nu }^{-1}(U)]$ be the coordinate ring of its preimage, so that $A\subseteq B\subseteq k(X)$, $B$ is the integral closure of $A$ in $k(X)$, and $B$ is a finite $A$-module (The normalization of an irreducible affine variety, The normalization of an irreducible affine variety is finite).

The conductor of the normalization over $U$ is the ideal ${\mathfrak{c}}_U:={\mathrm{Ann}}_A(B\mathrm{/}A)=\{a\in A:aB\subseteq A\}$ of $A$ (Annihilators, torsion elements and the torsion subset of a module).

Compatibility. For $0\mathrm{\ne }f\in A$ and the principal open $D(f)\subseteq U$, the preimage ${\nu }^{-1}(D(f))$ has coordinate ring $B_f$, the integral closure of $A_f$ in $k(X)$ (Finite normalization commutes with principal localization), and ${\mathfrak{c}}_{D(f)}={\mathrm{Ann}}_{A_f}(B_f\mathrm{/}{A}_f)=({\mathfrak{c}}_U{)}_f$, because annihilators of finite modules commute with localization: if a localized scalar annihilates each of finitely many module generators, multiplying the finitely many denominator-clearing factors yields an element of the original annihilator. The reverse inclusion is immediate (Annihilators, torsion elements and the torsion subset of a module). If $f=0$, then $D(f)$ and its preimage are empty, $A_f=B_f=0$, and the conductor and its localized ideal are both zero (Principal localisation $R_f=\{1,f,f^2,\dots {\}}^{-1}R$); no integral closure inside $k(X)$ is asserted in this case. Since nonempty principal opens form a basis and the chartwise ideals agree on overlaps under these identifications, the ideals ${\mathfrak{c}}_U$ glue to an ideal sheaf $\mathfrak{c}\subseteq {\mathcal{O}}_X$, the conductor of the normalization (Integral classical varieties in the compatible affine-atlas register).

Support. The support of ${\mathcal{O}}_X\mathrm{/}\mathfrak{c}$, equivalently the closed locus defined by $\mathfrak{c}$, is exactly the non-normal locus of $X$, equivalently the set of points at which $\nu$ is not an isomorphism. At a point $x$ with maximal ideal $\mathfrak{m}$, the localisation $(B\mathrm{/}A{)}_{\mathfrak{m}}$ vanishes exactly when $B_{\mathfrak{m}}=A_{\mathfrak{m}}$, which is exactly normality of $x$; and $\nu$ is an isomorphism over the normal locus (The normalization is an isomorphism over the normal locus). Since $B\mathrm{/}A$ is finite, its support is $V({\mathrm{Ann}}_A(B\mathrm{/}A))=V({\mathfrak{c}}_U)$ (For a finite module, support is the set of primes containing the annihilator). Thus $\mathfrak{c}\subseteq {\mathcal{O}}_X$ is an ideal sheaf whose quotient has support equal to the non-normal locus, and $\mathfrak{c}={\mathcal{O}}_X$ exactly when $\nu$ is already an isomorphism.