Archive for the ‘Graded Algebras & Modules’ Category

For basic definitions and examples see here. We continue our discussion with a few more examples.

Example 1. Let R=M_m(A) be the algebra of m \times m matrices with entries in an algebra A. Let e_{ij} be the m \times m matrix with 1 in the i-th row and j-th column and 0 anywhere else. Let R_n = \sum_{i \geq 1} Ae_{i, i+n} for all integers |n| < m. For |n| \geq m we define R_n=0. So, for example, if R=M_2(A), then R_{-1}=Ae_{21}, \ R_0=Ae_{11}+Ae_{22} and R_1=Ae_{12} and R_n = 0 for all |n| \geq 2. The reason that R = \bigoplus_{|n| < m} R_n holds is that for every 1 \leq i,j \leq m we may let n=j-i. Then obviously |n| < m and e_{ij}=e_{i,i+n} \in R_n.  Besides, every e_{ij} appears in R_n for the unique n = j-i. Finally R_nR_k = \sum_{i,j \geq 1} Ae_{i,i+n}e_{j,j+k} and e_{i,i+n}e_{j,j+k} is non-zero if and only if i+n=j. Thus

R_nR_k \subseteq \sum_{i \geq 1} Ae_{i,i+n+k} = R_{n+k}.

Example 2. Suppose G_1, G_2 are two groups and f: G_1 \longrightarrow G_2 is an onto group homomorphism. If R is G_1-graded, then R is also G_2-graded if we define R_h = \bigoplus_{ \{g \in G_1: \ f(g)=h \} } R_g for all h \in G_2. Clearly

\bigoplus_{h \in G_2} R_h = \bigoplus_{g \in G_1} R_g = R.

It is also easy to see that R_h R_u \subseteq R_{hu} for all h,u \in G_2.

Example 3. This example is an application of Example 2. Let G_1 = \mathbb{Z} and G_2= \mathbb{Z}/2\mathbb{Z}. Let f: G_1 \longrightarrow G_2 be the natural group homomorphism. Let R=M_m(A) with the grading introduced in Example 1. So, by Example 2, we have a G_2-grading for R=R_0 \oplus R_1, where

R_0 = \bigoplus_{ \{n \in G_1: \ f(n)=0 \}} R_n = \bigoplus_{2 \mid n} R_n = \bigoplus Ae_{i,i+2n} = \bigoplus_{2 \mid i+j} Ae_{ij}

and, similarly, R_1 is the direct sum of all Ae_{ij} such that i+j is odd.

We now move on to define an important concept, i.e. a graded ideal of a graded algebra.

Definition. Let  R=\bigoplus_{g \in G}R_g be a G-graded algebra. A graded or homogenous ideal of R is an ideal I such that I= \bigoplus_{g \in G} (I \cap R_g). Graded left or right ideals and graded subalgebras of R are defined analogously.

Theorem. Let  R=\bigoplus_{g \in G}R_g be a G-graded algebra and let I be an ideal of R. Then I is graded if and only if, as an ideal, I is generated by a subset of \bigcup_{g \in G} R_g. In other words, I is graded if and only if I can be generated by a set of homogeneous elements of R.

Proof. If I is generated by S \subseteq \bigcup_{g \in G} R_g, then I = \sum_{s \in S} Rs. Now, if s \in S, then s \in R_u, for some u \in G. Let r \in I. Then r = r_{g_1} + \ldots + r_{g_n}, where r_{g_i} \in R_{g_i}. Clearly r_{g_i}s \in I because s \in I and I is an ideal of R. Also, r_{g_i}s \in R_{g_iu} because s \in R_u and r_{g_i} \in R_{g_i}. Thus r_{g_i}s \in I \cap R_{g_iu} and hence r \in \bigoplus_{i=1}^n (I \cap R_{g_iu}) \subseteq \bigoplus_{g \in G} (I \cap R_g). So we have proved that I \subseteq \bigoplus_{g \in G}(I \cap R_g) and, since the other side of the inclusion is trivial, we get that I is graded. Conversely, suppose that I = \bigoplus_{g \in G}(I \cap R_g) and let S = \bigcup_{g \in G} (I \cap R_g). We want to prove that I=\sum_{s \in S} Rs.  Obviously \sum_{s \in S} Rs \subseteq I because S \subseteq I and I is an ideal.  Also every element of I is a finite sum of elements of S because I = \bigoplus_{g \in G}(I \cap R_g). Thus I \subseteq \sum_{s \in S} Rs and we are done. \Box

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 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 general, the standard grading of the polynomial algebra R=A[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.