Graded Rings

Section 9.1 Graded Rings

Subsection Graded Rings

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When we think of a polynomial ring \(R\text{,}\) we often think of \(R\) with its graded structure, even if we have never formalized what that means. Other rings we have seen also have a graded structure, and this structure is actually very powerful.

Definition 9.1. Graded Ring.

Let \(T\) be a monoid; in many examples we will take \(T = \N\text{,}\) which is a monoid since we follow the convention that \(0\) is a natural number. A ring \(R\) is \(T\)-graded if we can write a direct sum decomposition of \(R\) as an abelian group indexed by \(T\text{,}\)

\begin{equation*} R=\bigoplus_{a \geqslant 0} R_a, \end{equation*}

such that

\begin{equation*} R_a R_b \subseteq R_{a+b} \quad \textrm{for every } a,b \in T. \end{equation*}

This means that for all \(r \in R_a\) and \(s \in R_b\text{,}\) we have \(rs\in R_{a+b}\text{.}\)

Definition 9.2. Homogeneous Element.

An element in one of the summands of \(R_a\) is said to be homogeneous of degree \(a\text{;}\) we write \(|r|\) or \(\deg(r)\) to denote the degree of a homogeneous element \(r\text{.}\)

Definition 9.3. Homogeneous Component.

By definition, an element in a graded ring is a unique sum of homogeneous elements, which we call its homogeneous or graded components. One nice thing about graded rings is that many properties can usually be checked on homogeneous elements, and these are often easier to deal with.

Lemma 9.4. Graded Ring Properties.

Let \(R\) be a \(T\)-graded ring.

\(1\) is homogeneous of degree \(0 \in T\) (the identity of \(T\)).
\(R_0\) is a subring of \(R\text{.}\)
Each \(R_a\) is an \(R_0\)-module.

Proof.

Write \(1=\sum_a r_a\) with \(r_a\) homogeneous of degree \(a\text{.}\) Then \(r_0 = r_0 (\sum_a r_a) = \sum_a r_0 r_a\) implies \(r_0 r_a = 0\) for \(a\neq 0\text{.}\) Similarly, for any other \(a\) we have \(r_a = r_a (\sum_b r_b)\text{,}\) and thus \(r_a = r_a r_0\) (here is where we use the cancellative assumption). Thus \(r_a =0\) for \(a\neq 0\text{,}\) so \(1\in R_0.\)
We have shown that \(1 \in R_0\text{.}\) Moreover, \(R_0\) is a subgroup under addition, and \(r,s\in R_0\) implies \(rs\in R_0\text{.}\)
By assumption, \(R_a\) is a subgroup under addition. Given \(r\in R_0\) and \(s\in R_a\) we must have \(rs\in R_a\text{.}\)

Example 9.5. Graded Ring Examples.

Any ring \(R\) is trivially an \(\N\)-graded ring, by setting \(R_0 = R\) and \(R_n = 0\) for \(n \neq 0\text{.}\)
If \(k\) is a field and \(R=k[x_1,\dots,x_n]\) is a polynomial ring, there is an \(\N\)-grading on \(R\) called the standard grading where \(R_d\) is the \(k\)-vector space with basis given by the monomials of total degree \(d\text{,}\) meaning those of the form \(x_1^{\alpha_1}\cdots x_n^{\alpha_n}\) with \(\sum_i \alpha_i =d\text{.}\) For example, \(x_1^2+x_2x_3\) is homogeneous in the standard grading, while \(x_1^2+x_2\) is not.
If \(k\) is a field, and \(R=k[x_1,\dots,x_n]\) is a polynomial ring, we can give different \(\N\)-gradings on \(R\) by fixing some tuple \((\beta_1,\dots,\beta_n)\in \N^n\) and letting \(x_i\) be a homogeneous element of degree \(\beta_i\text{;}\) we call this a grading with weights \((\beta_1,\dots,\beta_n)\text{.}\) For example, in \(k[x_1,x_2]\text{,}\) \(x_1^2+x_2^3\) is not homogeneous in the standard grading, but it is homogeneous of degree \(6\) under the \(\mathbb{N}\)-grading with weights \((3,2)\text{.}\)
A polynomial ring \(R = k[x_1, \ldots, x_n]\) also admits a natural \(\N^n\)-grading: the grading with \(R_{(d_1,\dots,d_n)}= k \cdot x_1^{d_1}\cdots x_n^{d_n}\text{.}\) This is called the fine grading
Let \(\Gamma \subseteq \mathbb{N}^n\) be a subsemigroup of \(\mathbb{N}^n\text{.}\) Then

\begin{equation*} \bigoplus_{\gamma \in \Gamma} k \cdot \underline x ^ {\gamma} \subseteq k[\underline x] = k[x_1, \ldots, x_n] \end{equation*}

is an \(\mathbb{N}^n\)-graded subring of \(k[x_1,\dots,x_n]\) with the fine grading. Moreover, every \(\N^n\)-graded subring of \(k[x_1,\dots,x_n]\) is of this form.

Macaulay2.

Polynomial rings in Macaulay2 are graded with the standard grading by default. To define a different grading, we give Macaulay2 a list with the grading of each of the variables:

i1 : R = ZZ/101[a,b,c,Degrees=>{{1,2},{2,1},{1,0}}];

We can check whether an element of \(R\) isHomogeneous, and the function degree applied to an element of \(R\) returns the least upper bound of the degrees of its monomials:

i2 : degree (a+b)
o2 = {2, 2}
o2 : List

i3 : isHomogeneous(a+b)
o3 = false

Remark 9.6.

You may have seen the term homogeneous polynomial used to refer to a polynomial \(f(x_1, \ldots, x_n) \in k[x_1, \ldots, x_n]\) that satisfies

\begin{equation*} f(\lambda x_1, \ldots, \lambda x_n) = \lambda^d f(x_1, \ldots, x_n) \end{equation*}

for all \(\lambda \in k\) and some fixed \(d\text{.}\) This is equivalent to saying that all the terms in \(f\) have the same total degree \(d\text{,}\) or that \(f\) is homogeneous with respect to the standard grading.

Similarly, a polynomial is quasi-homogeneous, or weighted homogeneous if there exist integers \(w_1, \ldots, w_n\) such that the sum \(d = a_1 w_1 + \cdots + a_n w_n\) is the same for all monomials \(x_1^{a_1} \cdots x_n^{a_n}\) appearing in \(f\text{.}\) So \(f\) satisfies

\begin{equation*} f(\lambda^{w_1} x_1, \ldots, \lambda^{w_n} x_n) = \lambda^{d} f(x_1, \ldots, x_n), \end{equation*}

for all \(\lambda \in k\) and \(f(x_1^{w_1}, \ldots, x_n^{w_n})\) is homogeneous (in the previous sense, so with respect to the standard grading).

This condition is equivalent to asking that \(f\) be homogeneous with respect to some weighted grading on \(k[x_1, \ldots, x_n]\text{.}\)

Example 9.7. Quasi-Homogeneous.

In \(k[x,y,z]\text{,}\) the element \(x^2+y^3+z^5\) is not homogeneous in the standard grading, but it is homogeneous (of degree 30) if we set \(\deg(x) = 15\text{,}\) \(\deg(y)=10\text{,}\) and \(\deg(z)=6\text{.}\) This tells us that \(x^2+y^3+z^5\) is quasi-homogeneous; it is homogeneous with respect to the weights \((15,10,6)\text{.}\) Indeed,

\begin{equation*} (\lambda^{15}x)^2+(\lambda^{10}y)^3+(\lambda^6z)^5 = \lambda^{30}(x^2+y^3+z^5). \end{equation*}

Definition 9.8. Homogeneous Ideal.

An ideal \(I\) in a graded ring \(R\) is called homogeneous if it can be generated by homogeneous elements.

Lemma 9.9. Equivalent Homogeneous Ideal Definitions.

Let \(I\) be an ideal in a graded ring \(R\text{.}\) The following are equivalent:

\(I\) is homogeneous.
For any element \(f\in R\) we have \(f\in I\) if and only if every homogeneous component of \(f\) lies in \(I\text{.}\)
\(I = \bigoplus_{a \in T} I_a\text{,}\) where \(I_a = I \cap R_a\text{.}\)

Proof.

(1) \(\Rightarrow\) (2): Let \(f_1, \ldots, f_n\) be homogeneous generators for \(I\text{.}\) If \(f\in I\text{,}\) we can write \(f\) as

\begin{equation*} f = r_1 f_1 + \cdots + r_n f_n. \end{equation*}

We can also write each \(r_i\) as a sum of its components, say \(r_i = [r_i]_{d_{i,1}} + \cdots + [r_i]_{d_{i,m_i}}\text{.}\) Then, after substituting and collecting,

\begin{equation*} f= \sum_{d} ( [r_1]_{d-|f_1|} f_1 + \cdots + [r_n]_{d-|f_n|} f_n) \end{equation*}

expresses \(f\) as a sum of homogeneous elements of different degrees, so

\begin{equation*} f_d = [r_1]_{d-|f_1|} f_1 + \cdots + [r_n]_{d-|f_n|} f_n \in I. \end{equation*}
(2) \(\Rightarrow\) (1): Any element of \(I\) is a sum of its homogeneous components. Thus, in this case, the set of homogeneous elements in \(I\) is a generating set for \(I\text{.}\)
(1) \(\Rightarrow\) (3): As above, \(I\) is generated by the collection of additive subgroups \(\{I_a\}\) in this case; the sum is direct as there is no nontrivial \(\Z\)-linear combination of elements of different degrees.
(3) \(\Rightarrow\) (2): We can take generators for each abelian group \(I \cap R_a\text{,}\) and the collection of all of them is a generating set for \(I\text{.}\)

Example 9.10. .

Given an \(\N\)-graded ring \(R\text{,}\) then \(R_+=\bigoplus_{d>0} R_d\) is a homogeneous ideal.

We now observe the following:

Lemma 9.11. Quotients of Graded Rings.

Let \(R\) be an \(T\)-graded ring, and \(I\) be a homogeneous ideal. Then \(R/I\) has a natural \(T\)-graded structure induced by the \(T\)-graded structure on \(R\text{.}\)

Proof.

The ideal \(I\) decomposes as the direct sum of its graded components, so we can write

\begin{equation*} R/I = \frac{\oplus R_a}{\oplus I_a} \cong \oplus \frac{R_a}{I_a}. \end{equation*}

It’s elementary to check that this direct sum decomposition satisfies the desired properties.

Example 9.12. Homogeneous Ideals.

The ideal \(I = (w^2x+wyz+z^3,x^2+3xy+5xz+7yz+11z^2)\) in \(R=k[w,x,y,z]\) is homogeneous with respect to the standard grading on \(R\text{,}\) and thus the ring \(R/I\) admits an \(\N\)-grading with \(|w|=|x|=|y|=|z|=1\text{.}\)
The ring \(R=k[x,y,z]/(x^2+y^3+z^5)\) does not admit a grading with \(|x|=|y|=|z|=1\text{,}\) but by Example 9.7 it does admit a grading with \(|x|=15,|y|=10,|z|=6\text{.}\)

Subsection Graded Modules and Homomorphisms

Definition 9.13. Graded Module.

Let \(R\) be a \(T\)-graded ring and \(M\) an \(R\)-module. An \(R\)-module \(M\) is \(T\)-graded if there exists a direct sum decomposition of \(M\) as an abelian group indexed by \(T\text{:}\)

\begin{equation*} M=\bigoplus_{a\in T} M_a \textrm{ such that } R_a M_b \subseteq M_{a+b} \end{equation*}

for all \(a,b\in T\text{.}\)

The notions of homogeneous element of a module and degree of a homogeneous element of a module take the obvious meanings. A notable abuse of notation: we will often talk about \(\Z\)-graded modules over \(\N\)-graded rings, and likewise.

Definition 9.14. Graded Ring Homomorphism.

Let \(R\) and \(S\) be \(T\)-graded rings with the same grading monoid \(T\text{.}\) A ring homomorphism \(\varphi:R\to S\) is graded or degree-preserving if \(\varphi(R_a) \subseteq S_{a}\) for all \(a \in T\text{.}\)

Remark 9.15.

Note that our definition of ring homomorphism requires \(1_R \mapsto 1_S\text{,}\) and thus it does not make sense to talk about graded ring homomorphisms of degree \(d \neq 0\text{.}\) But we can have graded module homomorphisms of any degree.

Definition 9.16. Graded Module Homomorphism.

Let \(M\) and \(N\) be \(T\)-graded modules over the \(T\)-graded ring \(R\text{.}\) A homomorphism of \(R\)-modules] \(\varphi\!: M \to N\) is graded of degree \(d\) if \(\varphi(M_a) \subseteq N_{a+d}\) for all \(a \in T\text{.}\) A graded homomorphism of degree \(0\) is also called degree-preserving.

Example 9.17. Shift.

Consider the ring map \(k[x,y,z]\to k[s,t]\) given by \(x\mapsto s^2, y\mapsto st, z\mapsto t^2\text{.}\) If \(k[s,t]\) has the fine grading, meaning \(|s|=(1,0)\) and \(|t|=(0,1)\text{,}\) then the given map is degree preserving if and only if \(k[x,y,z]\) is graded by
\begin{equation*} |x|=(2,0), |y|=(1,1), |z|=(0,2). \end{equation*}
Let \(k\) be a field, and let \(R=k[x_1,\dots,x_n]\) be a polynomial ring with the standard grading. Given \(c\in k=R_0\text{,}\) the homomorphism of \(R\)-modules \(R\to R\) given by \(f\mapsto cf\) is degree preserving. However, if instead we take \(g\in R_d\) for some \(d>0\text{,}\) then the map
\begin{equation*} \begin{CD} R@>>> R\\\\ f@>>> gf \end{CD} \end{equation*}
is not degree preserving, although it is a graded map of degree \(d\text{.}\) We can make this a degree-preserving map if we shift the grading on \(R\) by defining \(R(-d)\) to be the \(R\)-module \(R\) but with the \(\Z\)-grading given by \(R(-d)_t=R_{t-d}\text{.}\) With this grading, the component of degree \(d\) of \(R(-d)\) is \(R(-d)_d=R_0=k\text{.}\) Now the map
\begin{equation*} \begin{CD} R(-d)@>>> R\\\\ f@>>> gf \end{CD} \end{equation*}
is degree preserving.

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