Graded Modules

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Now we define the notion of a graded module.

Graded Modules

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, viewing \(A\) as a left \(A\)-module over itself, we see by Definition 1 that every graded ring is a graded (left) \(A\)-module over itself. If every element of \(I\) is cancellable under addition, then by §Graded Rings, ⁋Proposition 2, \(A_0\) is a ring. From the above inclusion it is then immediate that each \(M_j\) is 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 \(u(M_i)\subseteq M_i'\) for all \(i\).

This allows us to define the category \(\bgr_I\lMod{A}\) of \(I\)-graded left \(A\)-modules. More generally, we make the following definition.

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 \(u(M_j)\subseteq M_{i+j}'\) for all \(j\).

Thus the graded homomorphisms of Definition 2 are precisely the 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 must be careful: a bijective graded homomorphism of degree \(i\) with \(i\neq 0\) is generally not regarded as an isomorphism of \(I\)-graded left \(A\)-modules.

This sort of generalization is treated in greater detail in homological algebra.

Graded Submodules

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

1. \(N\) is the sum of the \(N\cap M_i\).
2. Whenever an element of \(N\) is decomposed into homogeneous components, each component also lies in \(N\).
3. \(N\) is generated by homogeneous elements.

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

Proposition 5 Let \(u:M \rightarrow N\) be a graded \(A\)-homomorphism of degree \(d\). Then 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 of graded modules.