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 Calgebra.

Definition 1. R is called a Ggraded 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.

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