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Conductor of an integral closure

From Encyclopedia of Mathematics
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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)
How to Cite This Entry:
Conductor of an integral closure. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Conductor_of_an_integral_closure&oldid=35181
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article