Difference between revisions of "Conductor of an integral closure"
From Encyclopedia of Mathematics
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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. | 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==== | ====References==== | ||
<table> | <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> | + | <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> | </table> | ||
Latest revision as of 08:20, 30 November 2014
2020 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://encyclopediaofmath.org/index.php?title=Conductor_of_an_integral_closure&oldid=33222
Conductor of an integral closure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conductor_of_an_integral_closure&oldid=33222
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article