graded module (original) (raw)
Definition
Whenever we speak of a graded module, the module is always assumed to be over a graded ring. As any ring R is trivially a graded ring (where Ri=R if i=0 and Ri=0 otherwise), every module M is trivially a graded module with Mi=M if i=0 and Mi=0 otherwise. However, it is customary to regard a graded module (or a graded ring) non-trivially.
If R is a graded ring, then clearly it is a graded module over itself, by setting Mi=Ri (M=R in this case). Furthermore, if M is graded over R, then so is Mz for any indeterminate z.
Example. To see a concrete example of a graded module, let us first construct a graded ring. For convenience, take any ring R, the polynomial ring S=R[x] is a graded ring, as
with Si:=Rxi. Then SiSj=(Rxm)(Rxn)⊆Rxm+n=Si+j.
Therefore, S is a graded module over S. Similarly, the submodules
Sxi of S are also graded over S.
It is possible for a module over a graded ring to be graded in more than one way. Let S be defined as in the example above. Then S[y] is graded over S. One way to grade S[y] is the following:
| S[y]=⊕k=0∞Ak, where Ak=R[y]xk, |
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since SpAq=(Rxp)(R[y]xq)⊆R[y]xp+q=Ap+q. Another way to grade S[y] is:
| S[y]=⊕k=0∞Bk, where Bk=∑i+j=kRxiyj, |
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since
| SpBq=(Rxp)(∑i+j=qRxiyj)=∑i+j=qRxi+pyj⊆∑i+j=p+qRxiyj=Bp+q. |
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Graded homomorphisms and graded submodules
Let N be a submodule of a graded module M (over R). We can turn N into a graded module by defining Ni=N∩Mi. Of course, N may already be a graded module in the first place. But the two gradings on N may not be isomorphic. A submodule N of a graded module M (over R) is said to be a graded submodule of M if its grading is defined by Ni=N∩Mi. If N is a graded submodule of M, then the injection N↦M is a graded homomorphism.