|
Let  be a field, and let  be a connected commutative  -algebra graded 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
 .
- 1
- Richard P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhauser Press. Boston, MA. 1986.
|
