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| Title of object: |
homogeneous ideal |
| Canonical Name: |
HomogeneousIdeal |
| Type: |
Definition |
| Created on: |
2001-10-15 18:35:49 |
| Modified on: |
2004-02-16 00:29:01 |
| Classification: |
msc:13A15 |
| Keywords: |
commutative algebra, algebraic geometry |
| Defines: |
homogeneous, homogeneous element |
Revision comment (for changes between this and next version):
| For some reason PlanetMath added a surplus single quote to the pronunciation. |
Preamble:
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Content:
Let $R = \oplus_{g\in G} R_g$ be a graded ring. Then an element $r$ of $R$ is said to be \emph{homogeneous} if it is an element of some $R_g$. An ideal $I$ of $R$ is said to be homogeneous if it can be generated by a set of homogeneous elements, or equivalently if it is the ideal generated by the set of elements $\bigcup_{g\in G} I\cap R_g$.
One observes that if $I$ is a homogeneous ideal and $r=\sum_i r_{g_i}$ is the sum of homogeneous elements $r_{g_i}$ for distinct $g_i$, then each $r_{g_i}$ must be in $I$.
To see some examples, let $k$ be a field, and take $R=k[X_1,X_2,X_3]$ with the usual grading by total degree. Then the ideal generated by $X_1^n+X_2^n-X_3^n$ is a homogeneous ideal. It is also a radical ideal. One reason homogeneous ideals in $k[X_1,\ldots,X_n]$ are of interest is because (if they are radical) they define projective varieties; in this case the projective variety is the \PMlinkname{Fermat}{FermatsLastTheorem} curve. For contrast, the ideal generated by $X_1+X_2^2$ is not homogeneous. |
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