Graded Modules

We now define the notion of a graded module.

Graded Modules

Definition 1 For a commutative monoid \(I\), let \(A=\bigoplus_{i\in I}A_i\) be an \(I\)-graded ring and let \(M\) be a left \(A\)-module. Then \(M\) is called an \(I\)-graded left \(A\)-module if for any \(i,j\in I\),

\[A_iM_j\subseteq M_{i+j}\]

holds.

Similarly, we define an \(I\)-graded right \(A\)-module. In particular, if we regard \(A\) as a left \(A\)-module over itself, by Definition 1, every graded ring is a graded (left) \(A\)-module over itself. If every element of \(I\) is cancellable with respect to addition, then by §Graded Rings, ⁋Proposition 2, \(A_0\) is a ring. Then from the above formula, it is clear that each \(M_j\) becomes an \(A_0\)-module.

Definition 2 For two \(I\)-graded left \(A\)-modules \(M,M'\), an \(A\)-linear map \(u:M \rightarrow M'\) is called a graded homomorphism if \(u(M_i)\subseteq M_i'\) always holds.

Through this, we can define the category \(\bgr_I\lMod{A}\) of \(I\)-graded left \(A\)-modules. More generally, we define the following.

Definition 3 For two \(I\)-graded left \(A\)-modules \(M,M'\), an \(A\)-linear map \(u:M \rightarrow M'\) is called a graded homomorphism of degree \(i\) if \(u(M_j)\subseteq M_{i+j}'\) always holds.

Then the graded homomorphisms in Definition 2 are all graded homomorphisms of degree \(0\). If every element of \(I\) is cancellable, we can also define a graded homomorphism of degree \(-i\) by the condition

\[u(M_{i+j})\subseteq M_j',\qquad u(M_j)=0\text{ if $j-i\not\in I$}\]

However, note that when defining in this way, a bijective graded homomorphism of degree \(i\) is generally not considered an isomorphism between \(I\)-graded left \(A\)-modules when \(i\neq 0\).

This type of generalization is discussed in more detail in homological algebra.

Graded Submodules

Proposition 4 Let an \(I\)-graded left \(A\)-module \(M=\bigoplus_{i\in I} M_i\) be given. Then for a submodule \(N\) of \(M\), the following are all equivalent:

1. \(N\) is the sum of the \(N\cap M_i\).
2. When any element of \(N\) is decomposed into homogeneous elements, each of these elements also belongs to \(N\).
3. \(N\) is generated by homogeneous elements.

This proposition is a generalization of §Graded Rings, ⁋Proposition 6, and its proof is identical. Submodules satisfying this equivalent condition are called graded submodules. The following proposition is also a generalization of §Graded Rings, ⁋Proposition 7.

Proposition 5 For a graded \(A\)-homomorphism \(u:M \rightarrow N\) of degree \(d\), the following holds:

1. \(\im(u)\) is a graded submodule of \(N\).
2. If \(d\) is cancellable, then \(\ker(u)\) is a graded submodule of \(M\).
3. If \(d=0\), the canonical bijection \(M/\ker(u)\cong\im(u)\) defines an isomorphism between graded modules.