Difference between revisions of "Equi-dimensional ideal"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
Line 1: | Line 1: | ||
− | + | <!-- | |
+ | e0359701.png | ||
+ | $#A+1 = 14 n = 0 | ||
+ | $#C+1 = 14 : ~/encyclopedia/old_files/data/E035/E.0305970 Equi\AAhdimensional ideal | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
− | + | {{TEX|auto}} | |
+ | {{TEX|done}} | ||
− | + | An ideal $ \mathfrak m $ | |
+ | of an integral domain $ R $( | ||
+ | finitely generated over some field $ k $) | ||
+ | having the following property: All prime ideals $ \mathfrak P _ {1} \dots \mathfrak P _ {s} $ | ||
+ | associated with the primary ideals $ \mathfrak Q _ {1} \dots \mathfrak Q _ {s} $ | ||
+ | from the primary decomposition | ||
− | + | $$ | |
+ | \mathfrak m = \ | ||
+ | \mathfrak Q _ {1} \cap \dots | ||
+ | \cap \mathfrak Q _ {s} $$ | ||
+ | have the same dimension, that is, the quotient rings $ R/ \mathfrak P _ {i} $ | ||
+ | have the same Krull dimension for all $ i $. | ||
+ | This common dimension is called the dimension of the equi-dimensional ideal $ \mathfrak m $. | ||
+ | If $ R $ | ||
+ | is the ring of regular functions on a certain affine variety $ X $, | ||
+ | then an ideal $ \mathfrak m $ | ||
+ | of it is equi-dimensional if and only if all irreducible components of the subvariety $ Y \subset X $ | ||
+ | defined by $ \mathfrak m $ | ||
+ | have the same dimension. | ||
====Comments==== | ====Comments==== |
Latest revision as of 19:37, 5 June 2020
An ideal $ \mathfrak m $
of an integral domain $ R $(
finitely generated over some field $ k $)
having the following property: All prime ideals $ \mathfrak P _ {1} \dots \mathfrak P _ {s} $
associated with the primary ideals $ \mathfrak Q _ {1} \dots \mathfrak Q _ {s} $
from the primary decomposition
$$ \mathfrak m = \ \mathfrak Q _ {1} \cap \dots \cap \mathfrak Q _ {s} $$
have the same dimension, that is, the quotient rings $ R/ \mathfrak P _ {i} $ have the same Krull dimension for all $ i $. This common dimension is called the dimension of the equi-dimensional ideal $ \mathfrak m $.
If $ R $ is the ring of regular functions on a certain affine variety $ X $, then an ideal $ \mathfrak m $ of it is equi-dimensional if and only if all irreducible components of the subvariety $ Y \subset X $ defined by $ \mathfrak m $ have the same dimension.
Comments
An equi-dimensional ideal is also called an unmixed ideal. Instead of the phrase "dimension of an equi-dimensional ideal" one also uses the term "equi-dimension of an idealequi-dimension" (of the ideal).
An integrally-closed Noetherian domain is an integral domain all principal ideals of which are equi-dimensional, [a1], p. 196.
References
[a1] | O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) |
Equi-dimensional ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equi-dimensional_ideal&oldid=46840