homogeneous elements of a graded ring
Let be a field, and let be a connected commutative -algebra graded (http://planetmath.org/GradedAlgebra) by . Then via the grading, we can decompose into a direct sum of vector spaces: , where .
For an arbitrary ring element , we define the homogeneous degree of to be the value such that , and we denote this by . (See also homogeneous ideal)
A set of some importance (ironically), is the irrelevant ideal of , denoted by , and given by
Finally, we often need to consider the elements of such a ring without using the grading, and we do this by looking at the homogeneous union of :
In particular, in defining a homogeneous system of parameters, we are looking at elements of .
References
- 1 Richard P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhauser Press. Boston, MA. 1986.
Title | homogeneous elements of a graded ring |
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Canonical name | HomogeneousElementsOfAGradedRing |
Date of creation | 2013-03-22 14:14:52 |
Last modified on | 2013-03-22 14:14:52 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 6 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 13A02 |
Related topic | HomogeneousIdeal |
Defines | homogeneous element |
Defines | homogeneous degree |
Defines | irrelevant ideal |
Defines | homogeneous union |