📂Abstract Algebra

Definition of a Grading Module

빌드업

Let’s denote it as $n,m,i \in \mathbb{Z}$.

Graded Ring

A Graded Ring equipped with a direct sum $\left( R , \otimes \right) \simeq \bigoplus_{i} R_{i}$ of an Abelian group $R$ that Link $\left( R , + , \cdot \right)$ is defined by the multiplication $\otimes$ between $R_{i}$ being $$ R_{n} \otimes R_{m} \to R_{n+m} $$ Elements within each part of the direct sum, $R_{i}$, are called Homogeneous and have a Degree $i$. According to this definition, the degree of all $e \in R_{i}$ is $\deg e = i$. For example, considering a polynomial ring like $Z = \mathbb{Z}$, $Z_{n} = \mathbb{Z} t^{n}$, and $2 t^{6} \in Z_{6}$ is homogeneous of degree $6$ in $Z_{6}$, and $7 t^{3} \in Z_{3}$ is homogeneous of degree $3$ in $Z_{3}$. However, their sum $2 t^{6} + 7 t^{3} \in Z[t]$ is not homogeneous, and their product $$ 2 t^{6} \otimes 7 t^{3} = \left( 2 \cdot 7 \right) \left( t^{6} \cdot t^{3} \right) = 14 t^{9} \in Z_{6+3} $$ is homogeneous in $Z_{9}$ with degree $9$. As the example shows, in cases where $n \ge 0$, it is said to have a Standard Grading, and although the expression might seem difficult, one familiar with the polynomial ring would find it quite intuitive. Perhaps the most unfamiliar term might be $\otimes$, hence the use of the term Grade. Though we could think of even more complex and abstract examples, let’s move on to the definition of graded modules for now.

Definition ¹

A Graded Module equipped with a direct sum $M \simeq \bigoplus_{i} M_{i}$ on a $R$-Module $M$ is defined by the action $\otimes$ of $R$ on $M$ as follows. $$ R_{n} \otimes M_{m} \to M_{n+m} $$

Explanation

Even though the definition might seem puzzling at first, essentially, a Graded Module is similar to a graded ring but is a module, and importantly, it might possess the necessary properties within a polynomial ring as well. Especially when the Base Ring $R$ is a PID (Principal Ideal Domain) $D$, the following theorem, which characterizes its structure similarly to the Fundamental Theorem of Finite generated Abelian groups, is known.

The Structure of Graded Modules: On a PID $D$, every graded module $M$ can be uniquely decomposed as follows. $$ \left( \bigoplus_{i=1}^{n} \sum^{\alpha_{i}} D \right) \oplus \left( \bigoplus_{j=1}^{m} \sum^{\gamma_{j}} D / d_{j} D \right) $$ Here, $d_{j} \in D$ satisfy $d_{j} \mid d_{j+1}$, where $\alpha_{i} , \gamma_{j} \in \mathbb{Z}$, and $\sum^{\alpha}$ denotes the increase in grading by $\alpha$. The left side is called the Free part, and the right side the Torsional part.