Difference between revisions of "Conductor of an integral closure"
From Encyclopedia of Mathematics
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| − | The ideal of a commutative integral ring | + | 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|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><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> |
| + | <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> | ||
Revision as of 18:38, 31 August 2014
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=33221
Conductor of an integral closure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conductor_of_an_integral_closure&oldid=33221
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article