Namespaces
Variants
Actions

Difference between revisions of "Conductor of an integral closure"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(convert to LaTeX)
Line 1: Line 1:
The ideal of a commutative integral ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c0245801.png" /> which is the annihilator of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c0245802.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c0245803.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c0245804.png" /> is the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c0245805.png" /> in its field of fractions. Sometimes the conductor is regarded as an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c0245806.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c0245807.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c0245808.png" />-module of finite type (e.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c0245809.png" /> is a [[Geometric ring|geometric ring]]), a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c02458010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c02458011.png" /> contains the conductor if and only if the localization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c02458012.png" /> is not an integrally-closed local ring. In geometrical terms this means that the conductor determines a closed subscheme of the affine scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024580/c02458013.png" />, 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|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
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article