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 that is simultaneously an $I$-graded abelian group $M=\bigoplus_{i\in I}M_i$. 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 also define an $I$-graded right $A$-module. In particular, if we regard $A$ as a left $A$-module over $A$ itself, then by Definition 1 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. It then follows immediately from the above equation 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 $u(M_i)\subseteq M_i'$ always holds.

Through this, we can define the category of $I$-graded left $A$-modules, $\bgr_I\lMod{A}$. 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 $u(M_j)\subseteq M_{i+j}'$ always holds.

Then the graded homomorphisms in Definition 2 are nothing but graded homomorphisms of degree $0$. If all elements of $I$ are cancellable, we can also define a graded homomorphism of degree $-i$ by the following condition:

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

However, a point of caution when defining in this way is that a bijective graded homomorphism of degree $i$, in the case $i\neq 0$, is generally not considered 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 of $M$, $N$, the following are all equivalent.

$N$ is the sum of the $N\cap M_i$.
Whenever an element of $N$ is decomposed into homogeneous elements, each of those homogeneous elements also belongs to $N$.
$N$ is generated by homogeneous elements.

This proposition is a generalization of §Graded Rings, ⁋Proposition 6, and its proof is identical as well. Submodules satisfying these equivalent conditions are called graded submodules. On the other hand, for a graded submodule $N$, the proof of §Graded Rings, ⁋Proposition 7 carries over verbatim, so the quotient module $M/N$ becomes a graded module via the decomposition

\[M/N=\bigoplus_{i\in I}M_i/(N\cap M_i)\]

Then the following holds.

Proposition 5 For a degree $d$ graded $A$-homomorphism $u:M \rightarrow N$, the following hold.

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

Proof

For 1, since $M=\bigoplus_j M_j$, the image $\im(u)$ is generated by the homogeneous elements $u(x_j)\in N_{d+j}$, and hence is a graded submodule by the third condition of Proposition 4.

For 2, let $x=\sum_j x_j\in\ker(u)$. Then $0=u(x)=\sum_j u(x_j)$ and each $u(x_j)$ lies in $N_{d+j}$. Since $d$ is cancellable, the map $j\mapsto d+j$ is injective, so the terms of this sum lie in distinct degrees, and we obtain $u(x_j)=0$ for each component. Thus each $x_j$ belongs to $\ker(u)$, so the second condition of Proposition 4 holds.

For 3, for any $y\in\im(u)\cap N_i$, if we take $y=u(x)$ with $x=\sum_j x_j$, then since $d=0$, the component of $u(x)$ in $N_i$ is $u(x_i)$, and hence $\im(u)\cap N_i=u(M_i)$. Then the canonical bijection $M/\ker(u)\rightarrow\im(u)$ sends $M_i/(\ker(u)\cap M_i)$ to $\im(u)\cap N_i$, so it preserves degree with respect to the grading on $M/\ker(u)$ given above.

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Graded Modules

Graded Modules

Graded Submodules

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