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| \subsubsection*{Definition} |
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| Let $R = R_0 \oplus R_1 \oplus \cdots$ be a graded ring. A module $M$ over $R$ is said to be a \emph{graded module} if |
Let $R = R_0 \oplus R_1 \oplus \cdots$ be a graded ring. A module $M$ over $R$ is said to be a \emph{graded module} if |
| $$M = M_0 \oplus M_1 \oplus \cdots$$ |
$$M = M_0 \oplus M_1 \oplus \cdots$$ |
| where $M_i$ are abelian subgroups of $M$, such that $R_i M_j \subseteq M_{i+j}$ for all $i,j$. Elements of $M_i$ are said to be \emph{homogeneous of degree} $i$. |
where $M_i$ are abelian subgroups of $M$, such that $R_i M_j \subseteq M_{i+j}$ for all $i,j$. |
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| 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 $R_i=R$ if $i=0$ and $R_i=0$ otherwise), every module $M$ is trivially a graded module with $M_i=M$ if $i=0$ and $M_i=0$ otherwise. However, it is customary to regard a graded module (or a graded ring) non-trivially. |
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| If $R$ is a graded ring, then clearly it is a graded module over itself, by setting $M_i=R_i$ ($M=R$ in this case). Furthermore, if $M$ is graded over $R$, then so is $Mz$ for any indeterminate $z$. |
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| \textbf{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 $$S=S_0\oplus S_1 \oplus S_2 \oplus \cdots \oplus S_n \oplus \cdots,$$ with $S_i:=Rx^i$. Then $S_iS_j=(Rx^m)(Rx^n)\subseteq Rx^{m+n}=S_{i+j}$. |
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| Therefore, $S$ is a graded module over $S$. Similarly, the submodules $Sx^i$ of $S$ are also graded over $S$. |
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| It is possible for a module over a graded ring to be graded in more than one way. For example, 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: |
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| $$S[y]=S\oplus Sy \oplus Sy^2 \oplus \cdots \oplus Sy^n \oplus \cdots.$$ |
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| Another way to grade $S[y]$ is: |
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| $$S[y]=R\oplus \sum_{i+j=1}Rx^iy^j \oplus \sum_{i+j=2}Rx^iy^j \oplus \cdots \oplus \sum_{i+j=n}Rx^iy^j \oplus \cdots,$$ |
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| since $$(S_p)(\sum_{i+j=q}Rx^iy^j)=(Rx^p)(\sum_{i+j=q}Rx^iy^j)=\sum_{i+j=q}Rx^{i+p}y^j\subseteq \sum_{i+j=p+q}Rx^iy^j.$$ |
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| \subsubsection*{Graded homomorphisms and graded submodules} |
To see an example of a graded module, let us first construct a graded ring. For convenience, take any commutative ring $R$, the polynomial ring $R[x]$ is a graded ring, as $$R[x]=R\oplus Rx \oplus Rx^2 \oplus \cdots \oplus Rx^n \oplus \cdots,$$ and that $(Rx^m)(Rx^n)\subseteq Rx^{m+n}$. |
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| Let $M,N$ be graded modules over a (graded) ring $R$. A module homomorphism $f:M\to N$ is said to be \emph{graded} if $f(M_i)\subseteq N_i$. $f$ is a \emph{graded isomorphism} if it is a graded module homomorphism and an isomorphism. If $f$ is a graded isomorphism $M\to N$, then |
An easy example of a graded module over $R[x]$ is $R[x]$ itself. Each $Rx^i$ serves as the abelian subgroup $M_i$ of $M:=R[x]$. Evidently, $R_iM_j=R_iR_j\subseteq R_{i+j}=M_{i+j}$. |
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| \item $f(M_i)=N_i$. Suppose $a\in N_i$ and $f(b)=a$. Write $b=\sum b_j$ where $b_j\in M_i$, and $b_j=0$ for all but finitely many $j$. Then each $f(b_j)\in N_j$. Since $N_j\cap N_i$ if $i\ne j$, $f(b_j)=0$ if $j\ne i$. Therefore $b=b_i\in N_i$. |
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| \item $f^{-1}$ is graded. If $a\in f^{-1}(N_i)$, then $f(a)\in N_i$, which means $a\in M_i$ by the previous fact. |
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| \end{enumerate} |
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| Suppose a graded $M$ has two gradings: $M=\sum A_i=\sum B_i$. The two gradings on $M$ are said to be \emph{isomorphic} if there is a graded isomorphism $f$ on $M$ with $f(A_j)=B_j$. In the example above, we see that the two gradings of $S[j]$ are non-isomorphic. |
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Let $N$ be a submodule of a graded module $M$ (over $R$). We can turn $N$ into a graded module by defining $N_i=N\cap M_i$. 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 \emph{graded submodule} of $M$ if its grading is defined by $N_i=N\cap M_i$. If $N$ is a graded submodule of $M$, then the injection $N\mapsto M$ is a graded homomorphism.
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If $M$ is graded over $R$, then so is $Mz$ for any indeterminate $z$. As a result, $R[x]x^i$, and indeed any module of the form $R[x]z$ is a graded $R[x]$-module.
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