Graded Modules

This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.

Now we define the notion of a graded module.

Definition

Definition 1 Let \(I\) be a commutative monoid, 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 view \(A\) as a left \(A\)-module over itself, then 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 equation it is obvious 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 nothing but 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, one point to be careful about when defining in this way is that a bijective graded homomorphism of degree \(i\) with \(i\neq 0\) is generally not regarded as an isomorphism between \(I\)-graded left \(A\)-modules.

This kind of generalization is treated 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. If an arbitrary element of \(N\) is decomposed into homogeneous elements, then each of those 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 also the same. 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 hold.

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 \(d=0\), then the canonical bijection \(M/\ker(u)\cong\im(u)\) defines an isomorphism between graded modules.