View source for Conductor of an integral closure

{{TEX|done}}{{MSC|13B}}

The ideal of a commutative [[integral ring]] $A$ which is the [[annihilator]] of the $A$-module $\bar A / A$, where $\bar A$ is the [[integral closure]] of $A$ in its [[field of fractions]]. Sometimes the conductor is regarded as an ideal of $\bar A$. If $\bar A$ is an $A$-module of finite type (e.g., if $A$ is a [[geometric ring]]), a [[prime ideal]] $\mathfrak P$ of $A$ contains the conductor if and only if the localization $A_{\mathfrak{P}}$ is not an integrally-closed [[local ring]]. In geometrical terms this means that the conductor determines a [[closed subscheme]] of the [[affine scheme]] $\mathrm{Spec}\,A$, consisting of the points that are not normal.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  O. Zariski,   P. Samuel,   "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR>
</table>