# Conductor of an integral closure

2010 Mathematics Subject Classification: *Primary:* 13B [MSN][ZBL]

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

[1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |

[2] | O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) |

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Conductor of an integral closure.

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