## Graded algebras; basic definitions and examples

Posted: December 18, 2010 in Graded Algebras & Modules, Noncommutative Ring Theory Notes
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Throughout $G$ is a group, $C$ is a commutative ring with 1 and $R$ is a $C$algebra.

Definition 1. $R$ is called a $G$graded $C$-algebra if for every $g \in G$ there exists a $C$-module $R_g \subseteq R$ such that

1) $R_gR_h \subseteq R_{gh},$ for all $g,h \in G,$

2) $R = \bigoplus_{g \in G} R_g,$ as $C$-modules.

A homogeneous element of $R$ is any element of $R_g, \ g \in G.$ If $0 \neq r_g \in R_g,$ then $r_g$ is called homogeneous of degree $g.$ If $r \in R,$ then $r$ is written uniquely as $r = \sum_{g \in G} r_g,$ where $r_g \in R_g$ and all but finitely many of $r_g$ are zero. Each $r_g$ is called a homogeneous component of $r.$

Definition 2. We say that $R$ is positively graded if $R$ is $\mathbb{Z}$-graded and $R_n = 0$ for all $n < 0.$

The concept of graded algebras is just a very natural generalization of polynomial algebras.

Example 1. Let $A$ be a $C$-algebra and consider the polynomial algebra $R=A[x].$ Then $R$ is positively graded because clearly $R=\bigoplus_{n=0}^{\infty}R_n,$ where $R_n=Ax^n, \ n \geq 0.$ Note that $R_nR_m=Ax^{n+m}=R_{n+m}.$ This grading for $R$ is called the standard grading of $R.$

Example 2. Let $R=A[x,y],$ the polynomial algebra in the indeterminates $x$ and $y.$ Let $R_n, \ n \geq 0,$ be the set of all polynomials of total degree $n.$ For example, $R_0=A, \ R_1=Ax+Ay, \ R_2=Ax^2+Axy+Ay^2,$ etc. Then $R=\bigoplus_{n=0}^{\infty}R_n.$ Note that $R_nR_m=R_{n+m}.$ So $R$ is positively graded and we call this the standard grading of $R.$ In genral, the standard grading of the polynomial algebra $R=A[x_1, \ldots , x_m]$ in the indeterminates $x_1, \ldots , x_m$ is defined by $R = \bigoplus_{n=0}^{\infty}R_n,$ where $R_n$ is the set of polynomials of total degree $n.$

Example 3. The standard grading of the Laurent polynomial algebra $R=A[x,x^{-1}]$ is $R=\bigoplus_{n \in \mathbb{Z}} Ax^n.$

Remark. Let $R=\bigoplus_{n=0}^{\infty} R_n$ be a (positively) graded algebra and let $R_{+}=\bigoplus_{n=1}^{\infty} R_n.$ Then

1) $R_0$ is a subring of $R$ and $R_n$ is an $R_0$-module for all $n \geq 0.$

2) $1_R \in R_0.$

3) $R_{+}$ is a two-sided ideal of $R.$

Proof. 1) By the property 1) in Definition 1, we have $R_0R_n \subseteq R_n$ and $R_nR_0 \subseteq R_n.$ That means $R_n$ is both left and right $R_0$-module. Also, $R_0R_0 \subseteq R_0$ and so $R_0$ is a subring of $R.$ To prove 2), let $1 = \sum_{n=0}^{\infty} r_n,$ where $r_n \in R_n$ and only a finitely many of $r_n$ are non-zero. We need to show that $r_n = 0$ for all $n > 0.$ To prove this, let $m \geq 0.$ Then $r_m = \sum_{n \geq 0} r_mr_n.$ But $r_mr_n \in R_{m+n}$ and if $n > 0,$ then $m + n \neq m.$ Thus $r_m r_n = 0,$ for all $m \geq 0$ and $n > 0.$ Thus if $t > 0,$ then $r_t = \sum_{n \geq 0}r_nr_t = 0.$ Part 3) of the remark is trivial. $\Box$

Definition 3. The ideal $R_{+}$ in the above remark is called the augmentation ideal of $R.$

Example 4. Let $R$ be the polynomial algebra in Example 1. Then $R_{+}=\bigoplus_{n=1}^{\infty} Ax^n = \langle x \rangle,$ the ideal generated by $x.$ Similarly, the augmentation ideal of the polynomial algebra $R=A[x_1, \ldots , x_m]$ (see Example 2) is $\langle x_1, \ldots , x_m \rangle.$