Associated primes of monomial ideals

Primary decomposition

Let $R$ be a Noetherian commutative ring. An ideal $I$ of $R$ is called primary if it is a proper ideal and for each pair of elements $x$ and $y$ in $R$ such that $xy\in I$, either $x$ or some power of $y$ is in $I$; equivalently, every zero-divisor in the quotient $R/I$ is nilpotent. The radical of a primary ideal $Q$ is a prime ideal and $Q$ is said to be $\mathfrak{p}$-primary for $\mathfrak{p}=\sqrt{Q}$.

Any ideal $I$ in a Noetherian commutative ring has an irredundant primary decomposition into primary ideals (see Exercise 290.4):

$$I=Q_1\cap \cdots \cap Q_n.$$

By irredundant, we mean that removing any of the $Q_i$ changes the intersection, and all the prime ideals $\sqrt{Q_i}$ are distinct (note that the manner in which we constructed the primary decomposition here ensures this).

Associated primes

Definition 262.1(Associated primes).

Given a primary decomposition $I=Q_1\cap \cdots \cap Q_n$, the set ${\text{Ass}}_R(I):=\{\sqrt{Q_i}|i\}$ is uniquely determined by $I$, and is called the set of associated primes of $I$. Minimal elements in ${\text{Ass}}_R(I)$ are called isolated primes while the rest (those properly containing minimal primes) are called embedded primes.

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Here’s another way to think about associated primes: Let $R$ be a Noetherian ring, $M$ be a finitely generated $R$-module. Let $S\subseteq M$ be nonempty. Define the annihilator of $S$ to be

$${\text{Ann}}_R(S)=\{r\in R|rs=0\text{ for all }s\in S\}.$$

A nonzero $R$-module $N$ is called a prime module if ${\text{Ann}}_R(N)={\text{Ann}}_R(N^{\prime })$ for any nonzero submodule $N^{\prime }$ of $N$.

• For a prime module $N$, ${\text{Ann}}_R(N)$ is a prime ideal in $R$.
• An associated prime of $M$ is an ideal of the form ${\text{Ann}}_R(N)$ where $N$ is a prime submodule of $M$. Equivalently, when $R$ is commutative, an associated prime of $M$ is an ideal of the form ${\text{Ann}}_R(x)$, for $x\in M$.
• See that ${\text{Ass}}_R(R/I)$, interpreted using this definition, is the same thing as ${\text{Ass}}_R(I)$ interpreted using the previous definition.
• Easy to see that $\text{Ass}(Q)=\{\sqrt{Q}\}$ if $Q\subseteq R$ is primary.

Definition 262.2(Colon ideals).

$I:J:=\{r\in R:rJ\subseteq I\}$. Clearly, $I:J$ contains $I$.

Proposition 262.3(Proposition).

Let $R$ be a commutative ring. Let $I$ be an ideal of $R$. Let $P\in \text{Ass}(I)$. Then, there exists $a\in R$ such that $P=I:a$.

This is clear form the definition above: $P\in \text{Ass}(I)$ implies $P=\text{Ann}(a+I)$ for some $a+I\in R/I$, that is, $P=\{r\in R:ra\in I\}$. This can be equivalently written as $\{r\in R:r(a)\subseteq I\}$, so $P=I:(a)$.

Minimal primes

Definition 262.4(Minimal primes).

$\text{Min}(I)$ denotes the set of prime ideals $\mathfrak{p}$ that are minimal wrt inclusion among the prime ideals containing $I$.

Note that $\mathfrak{p}/I$ is a minimal prime ideal of $R/I$ iff $\mathfrak{p}\in \text{Min}(I)$.

We state the following without proof

Proposition 262.5(Proposition).

For an ideal $I$ of a Noetherian ring $R$, ==$\text{Ass}(I)$ is a finite set, and $\text{Min}(I)\subseteq \text{Ass}(I)$==.

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Lemma 262.6(Lemma).

Let $I$ be any ideal. Then $\text{Min}(I^n)=\text{Min}(I)$ for any $n$.

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From Proposition 5 and Lemma 6, it follows that

Corollary 262.7(Corollary).

$\text{Min}(I)=\text{Min}(I^n)\subseteq \text{Ass}(I^n)$.

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Stabilization of associated primes

Theorem 262.8(Brodmann (1979)).

Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ a finitely generated $R$-module. Let $A$ be the map from $\mathbb{N}$ to subsets of $\text{spec}(R)$ defined by $A(n)={\text{Ass}}_R(M/I^{n}M)$. Then, there exists $n_0$ such that for all $n\ge n_0$, $A(n)=A(n_0)$.

In particular, for $M=R$, we have $A(n)={\text{Ass}}_R(I^n)$. The smallest $n_0$ for which $A(n)$ stabilizes is denoted by $\text{astab}(I)$.

Monomial ideals ¹

Definition 262.9(Monomial ideals).

Let $K$ be a field, and $S=K[x_1,\dots ,x_n]$. An ideal $I\subseteq S$ is called a monomial ideal if it is generated by monomials.

The set $\text{Mon}(S)$ of monomials of $S$ is a $K$-basis of $S$: any polynomial $f\in S$ is a unique $K$-linear combination of monomials.

For $f\in S$, we define the support of $f$ to be the set of all monomials in $f$.

Proposition 262.10(Proposition).

The set $\mathcal{N}$ of monomials belonging to $I$ is a $K$-basis of $I$.

Corollary 262.11(Corollary).

Let $I\subseteq S$ be an ideal. The following are equivalent:

1. $I$ is a monomial ideal;
2. for all $f\in S$ one has $f\in I$ iff $\text{supp}(f)\subseteq I$.

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Proposition 262.12(Proposition).

Let $\{u_1,\dots ,u_m\}$ be a monomial system of generators of the monomial ideal $I$. Then the monomial $v$ belongs to $I$ iff there exists a monomial $w$ such that $v=w{u}_i$ for some $i$.

Proposition 262.13(Proposition).

Each monomial ideal has a unique minimal monomial set of generators. More precisely, let $G$ denote the set of monomials in $I$ which are minimal with respect to divisibility. Then $G$ is the unique minimal set of monomial generators.

We denote the unique minimal set of monomial generators of the monomial ideal $I$ by $G(I)$.

Computational aides

Theorem 262.14(Theorem).

Let $I$ and $J$ be monomial ideals. Then $I\cap J$ is a monomial ideal, and $\{\text{lcm}(u,v):u\in G(I),v\in G(J)\}$ is a set of generators of $I\cap J$.

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Theorem 262.15(Theorem).

Let $I$ and $J$ be monomial ideals. Then $I:J$ is a monomial ideal, and

$$I:J=\bigcap _{v\in G(J)}I:(v).$$

Moreover, $\{u/\text{gcd}(u,v):u\in G(I)\}$ is a set of generators of $I:(v)$.

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Easier to remember: $(I:(f_1,\dots ,f_n))=\bigcap (I:(f_i))$.

Primary decomposition and associated primes in the context of monomial ideals

Definition 262.16(Irreducible monomial ideals).

A monomial ideal is called irreducible if it cannot be written as a proper intersection of two other monomial ideals.

Proposition 262.17(Proposition).

1. A monomial ideal is irreducible iff it is generated by pure powers of variables.
2. Every monomial ideal has a unique presentation as an irredundant intersection of irreducible monomial ideals.

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Note that a monomial ideal is prime iff it is of the form $(x_{i_1},x_{i_2},\dots ,x_{i_k})$, that is, monomial prime are generated by single powers of variables. If $I$ is a squarefree monomial ideal, its presentation as an irredundant intersection of irreducible monomial ideals will only feature prime ideals.

An ideal $I$ in a Noetherian ring $R$ is $P$-primary, if $\text{Ass}(I)=\{P\}$. In other words, $I$ is primary, and $\sqrt{I}=P$.

Associated primes of monomial ideals

Lemma 262.18(Lemma).

The irreducible ideal $(x_{i_1}^{a_1},\dots ,x_{i_k}^{a_k})$ is $(x_{i_1},\dots ,x_{i_k})$-primary.

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It follows from Proposition 17 and Lemma 18 that

Theorem 262.19(Theorem).

The associated prime ideals of a monomial ideal are monomial prime ideals.

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Theorem 262.20(Herzog & Hibi (2011) Corollary 1.3.10).

Let $I\subseteq S$ be a monomial ideal, and let $P\in \text{Ass}(I)$. Then there exists a monomial $v\in S$ such that $P=I:v$.

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Squarefree monomial ideals

Corollary 262.21(Corollary).

A squarefree monomial ideal is an intersection of monomial prime ideals.

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The following Lemma holds in general.

Lemma 262.22(Lemma).

Suppose $I$ has irredundant presentation $I=P_1\cap \cdots \cap P_m$ as an intersection of prime ideals. Then $\text{Min}(I)=\{P_1,\dots ,P_m\}$.

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Theorem 262.23(Theorem).

Combining Lemma 22 and Corollary 21, if $I\subseteq S$ is a squarefree monomial ideal, then

$$I=\bigcap _{P\in \text{Min}(I)}P.$$

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Footnotes

1. See Herzog & Hibi (2011) Chapter 1 for proofs. ↩

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References

Brodmann, M. (1979). Asymptotic Stability of Ass(M/I^nM).

Herzog, J., & Hibi, T. (2011). Monomial Ideals. Springer London. https://doi.org/10.1007/978-0-85729-106-6