It is a well-known fact that a polynomial (over, say, ) can be written as a sum of monomials in a unique way. A monomial is a special kind of a polynomial. Unlike polynomials, the monomials can be partitioned so that the sum of any two monomial within a partition, and the product of any two monomials, are again monomials. As one may have guessed, one would partition the monomials by their degree. The above notion can be generalized, and the general notion is that of a graded ring (and a graded module).
Let be a graded ring. A module over is said to be a graded module if
Whenever we speak of a graded module, the module is always assumed to be over a graded ring. As any ring is trivially a graded ring (where if and otherwise), every module is trivially a graded module with if and otherwise. However, it is customary to regard a graded module (or a graded ring) non-trivially.
If is a graded ring, then clearly it is a graded module over itself, by setting ( in this case). Furthermore, if is graded over , then so is for any indeterminate .
with . Then .
Therefore, is a graded module over . Similarly, the submodules of are also graded over .
It is possible for a module over a graded ring to be graded in more than one way. Let be defined as in the example above. Then is graded over . One way to grade is the following:
since . Another way to grade is:
Graded homomorphisms and graded submodules
Let be graded modules over a (graded) ring . A module homomorphism is said to be graded if . is a graded isomorphism if it is a graded module homomorphism and an isomorphism. If is a graded isomorphism , then
. Suppose and . Write where , and for all but finitely many . Then each . Since if , if . Therefore .
is graded. If , then by the previous fact. Then for some , so since is one-to-one.
Suppose a graded module has two gradings: . The two gradings on are said to be isomorphic if there is a graded isomorphism on with . In the example above, we see that the two gradings of are non-isomorphic.
Let be a submodule of a graded module (over ). We can turn into a graded module by defining . Of course, may already be a graded module in the first place. But the two gradings on may not be isomorphic. A submodule of a graded module (over ) is said to be a graded submodule of if its grading is defined by . If is a graded submodule of , then the injection is a graded homomorphism.
The above definition can be generalized, and the generalization comes from the subscripts. The set of subscripts in the definition above is just the set of all non-negative integers (sometimes denoted ) with a binary operation . It is reasonable to extend the set of subscripts from to an arbitrary set with a binary operation . Normally, we require that is associative so that is a semigroup. An -module is said to be -graded if
Examples of -graded modules are mainly found in modules over a semigroup ring .
|Date of creation||2013-03-22 11:45:05|
|Last modified on||2013-03-22 11:45:05|
|Last modified by||CWoo (3771)|
|Defines||graded module homomorphism|
|Defines||homogeneous of degree|