$k$-clean monomial ideals

TL;DR

This paper introduces $k$-clean monomial ideals, extending the concept of clean ideals, and explores their properties, showing their relation to $k$-decomposability of simplicial complexes and identifying classes of ideals that are $k$-clean.

Contribution

It defines $k$-clean monomial ideals, establishes their properties, and connects them to $k$-decomposability and various classes of monomial ideals, broadening the understanding of their structure.

Findings

$k$-clean ideals generalize clean ideals with hierarchical properties.

A $(d-1)$-dimensional simplicial complex is $k$-decomposable iff its Stanley-Reisner ideal is $k$-clean.

Classes like monomial complete intersections and Cohen-Macaulay ideals of codimension 2 are $k$-clean for all $k",

Abstract

In this paper, we introduce the concept of $k$-clean monomial ideals as an extension of clean monomial ideals and present some homological and combinatorial properties of them. Using the hierarchal structure of $k$-clean ideals, we show that a $(d-1)$-dimensional simplicial complex is $k$-decomposable if and only if its Stanley-Reisner ideal is $k$-clean, where $k\leq d-1$. We prove that the classes of monomial ideals like monomial complete intersection ideals, Cohen-Macaulay monomial ideals of codimension 2 and symbolic powers of Stanley-Reisner ideals of matroid complexes are $k$-clean for all $k\geq 0$.

Equations

$$\mathcal{F}:0=M_0\subset M_1\subset \dots \subset M_{r-1}\subset M_r=M$$

$$\begin{array}[]{l}\mathrm{link}_{\Delta}(\sigma)=\{F\in\Delta:\sigma\cap F=\emptyset,\sigma\cup F\in\Delta\},\\ {\mathrm{star}}_{\Delta}(\sigma)=\{F\in\Delta:\sigma\cup F\in\Delta\}\ \mbox{and}\\ \Delta\backslash\sigma=\{F\in\Delta:\sigma\nsubseteq F\}.\end{array}$$

$$\mathcal{F}:(0)=M_0\subset M_1\subset \dots \subset M_{r-1}\subset M_r=S/I$$

$$I$$

$$l(I)=\min \{l(\mathcal{T}):\mathcal{T}\text{mbox}isanidealtreeofI\}.$$

$${\mathcal{F}}_1:I+Su=J_0\subset J_1\subset \dots \subset J_r=S$$

$${\mathcal{F}}_2:0=\frac{L_0}{I:u}\subset \frac{L_1}{I:u}\subset \dots \subset \frac{L_s}{I:u}=\frac{S}{I:u}.$$

$$\frac{H_i}{H_{i-1}}\cong \frac{H_i/I}{H_{i-1}/I}\cong \frac{(L_i/I:u)(-\mathbf{a})}{(L_{i-1}/I:u)(-\mathbf{a})}\cong \frac{S}{Q_i}(-\mathbf{a}-{\mathbf{a}}_i).$$

$${\mathcal{F}}_3:I=H_0\subset H_1\subset \dots \subset H_s=I+Su.$$

$$\mathcal{F}:I=H_0\subset H_1\subset \dots \subset H_s=I+Su\subset J_1\subset \dots \subset J_r=S.$$

$$\mathcal{F}:(0)=M_0\subset M_1\subset \dots \subset M_{r-1}\subset M_r=S/I$$

$$(I:u)+(\frac{v}{\gcd (u,v)})=(I+(v)):u\text{mbox}and(I:u):\frac{v}{\gcd (u,v)}=(I:v):\frac{u}{\gcd (u,v)}.$$

$$u^p=x_{i_{1}1}\dots x_{i_1{a}_1}\dots x_{i_{t}1}\dots x_{i_t{a}_t}.$$

$$\begin{array}[]{rl}\mathrm{pd}(S/I)=\mathrm{pd}(S^{p}/I_{\Delta})=&\max\{\mathrm{pd}(S^{p}/I_{\Delta\backslash\sigma}),\mathrm{pd}(S^{p}/J_{\mathrm{link}_{\Delta}\sigma})\}\\ =&\max\{\mathrm{pd}(S^{p}/(I+(u))^{p}),\mathrm{pd}(S^{p}/(I:u)^{p})\}\\ =&\max\{\mathrm{pd}(S/I+(u)),\mathrm{pd}(S/I:u)\}\end{array}$$

$$\mathcal{F}:0=M_0\subset M_1\subset \dots \subset M_r=S/I$$

$$\dim (M_1/M_0)\le \dim (M_2/M_1)\le \dots \le \dim (M_r/M_{r-1}.$$

$$\min (I_{\Delta })=\{P_{F^c}:F\in \mathcal{F}(\Delta )\}$$

$$\min (I_{\Delta }+({\mathbf{x}}^{\sigma }))=\{P_{F^c}:F\in \mathcal{F}(\Delta \backslash \sigma )\}.$$

$$\Delta =⟨124,125,126,135,136,145,236,245,256,345,346⟩$$

$$\min (I+(x_1^a))=\{P+(x_1):P\in \min (J)\}$$

$$\min (I)=\{P+(x_i):P\in \min (J)\text{mbox}and{x}_i|M_1\}.$$

$$G(J)=\{x_{i_1,j_1}{x}_{i_2,j_2}\dots x_{i_r,j_r}:\{i_1,\dots ,i_r\}\text{mbox}isafacetof{\Delta }^c\}$$

$$x_{i_r^{\prime },j_r^{\prime }}\dots x_{i_{l-1}^{\prime },j_{l-1}^{\prime }}{x}_{i_{l+1}^{\prime },j_{l+1}^{\prime }}\dots x_{i_t,j_t}{x}_{i_{t-1}^{\prime },j_{t-1}^{\prime }}\dots x_{i_1^{\prime },j_1^{\prime }}\in G(J).$$

Full text

Introduction

Let $R$ be a Noetherian ring and $M$ be a finitely generated $R$-module. It is well known that there exists a so called prime filtration

[TABLE]

that is such that $M_{i}/M_{i-1}\cong R/P_{i}$ for some $P_{i}\in{\mathrm{Supp}}(M)$. We call any such filtration of M a prime filtration. Set ${\mathrm{Supp}}(\mathcal{F})=\{P_{1},\ldots,P_{r}\}$. Let $\mathrm{Min}(M)$ denote the set of minimal prime ideals in ${\mathrm{Supp}}(M)$. If $I$ is an ideal of $R$ then we set $\min(I)=\mathrm{Min}(R/I)$. Dress [7] calls a prime filtration $\mathcal{F}$ of $M$ clean if ${\mathrm{Supp}}(\mathcal{F})=\mathrm{Min}(M)$. The module $M$ is called clean, if $M$ admits a clean filtration and $R$ is clean if it is a clean module over itself.

Let $S=K[x_{1},\ldots,x_{n}]$ be the polynomial ring in $n$ indeterminate over a field $K$. Let $\Delta$ be a simplicial complex on the vertex set $[n]=\{1,2,\ldots,n\}$. Dress [7] showed that $\Delta$ is (non-pure) shellable in the sense of Björner and Wachs [3], if and only if the Stanley-Reisner ring $S/I_{\Delta}$ is clean. The result of Dress is, in fact, the algebraic counterpart of shellability for simplicial complexes. Some subclasses of shellable complexes are $k$-decomposable simplicial complexes which were introduced by Billera and Provan [2] on pure simplicial complexes and then by Woodroofe [31] on not necessarily pure ones. Simon in [25] introduced “completed clean ideal trees” as an algebraic counterpart of pure $k$-decomposable complexes. Actually, in the sense of Simon, the Stanley-Reisner ideal of a $k$-decomposable complex is completed clean ideal tree.

Let $I\subset S$ be a monomial ideal. We call $I$ Cohen-Macaulay (clean) if the quotient ring $S/I$ has this property. In this paper, we define the concept of $k$-clean monomial ideals. The class of $k$-clean monomial ideals are, actually, subclass of clean monomial ideals. It is the aim of this paper to study the properties of $k$-clean monomial ideals and describe relations between these ideals and $k$-decomposable simplicial complexes. Moreover, some classes of $k$-clean monomial ideals are introduced. Also, some results of [1, 16] are extended.

In Section 2, we introduce $k$-clean monomial ideals. We show that $k$-clean monomial ideals are clean and, also, every clean monomial ideal is $k$-clean for some $k\geq 0$ (see Theorem 2.4). In Section 3, we discuss some of basic properties of $k$-clean ideals. Some homological invariants of $k$-clean monomial ideals like depth and Castelnuovo-Mumford regularity are described in this section. In the fourth section, we show that a $(d-1)$-dimensional simplicial complex $\Delta$ is $k$-decomposable if and only if its associated Stanley-Reisner ideal is $k$-clean, where $k\leq d$ (see Theorem 4.1). The last section is devoted to presenting some examples of $k$-clean monomial ideals. We show that irreducible monomial ideals and monomial complete intersection ideals are $k$-clean, for all $k\geq 0$ (see Theorems 5.1 and 5.2). Then by showing that Cohen-Macaulay monomial ideals of codimension 2 (see Theorem 5.4) are $k$-clean, we improve Proposition 1.4. of [16]. Finally, in Theorem 5.5, we show that symbolic powers of Stanley-Reisner ideals of matroid complexes are $k$-clean for all $k\geq 0$. In this way, we improve Theorem 2.1 of [1].

1. Preliminaries

Let $\Delta$ be a simplicial complex of dimension $d-1$ with the vertex set $[n]:=\{1,2,\ldots,n\}$. Let $K$ be a field. The Stanley-Reisner monomial ideal of $\Delta$ is denoted by $I_{\Delta}$ and it is a squarefree monomial ideal in the polynomial ring $S=K[x_{1},\ldots,x_{n}]$ generated by the monomials $\mathbf{x}^{F}=\underset{i\in F}{\prod}x_{i}$ which $F$ is a non-face in $\Delta$. The quotient ring $S/I_{\Delta}$ is called the face ring or Stanley-Reiner ring of $\Delta$. If $\mathcal{F}(\Delta)=\{F_{1},\ldots,F_{r}\}$ is the set of maximal faces (facets) of $\Delta$ then we set $\Delta=\langle F_{1},\ldots,F_{r}\rangle$.

For all undefined terms or notions on simplicial complexes we refer the reader to the books [13] or [27].

Given a simplicial complex $\Delta$ on $[n]$, the link, star and deletion of $\sigma$ in $\Delta$ are defined, respectively, by

[TABLE]

Moreover, the Alexander dual of $\Delta$ is defined as $\Delta^{\vee}=\{F\in\Delta:[n]\backslash F\not\in\Delta\}$.

Let $I\subset S$ be a squarefree monomial ideal generated by monomials of degree at least $2$. Then there exists a simplicial complex $\Delta$ on $[n]$ such that $I=I_{\Delta}$. The Alexander dual of $I$ is defined $I^{\vee}=I_{\Delta^{\vee}}$.

Definition 1.1**.**

[31] Let $\Delta$ be a simplicial complex on vertex set $[n]$. Then a face $\sigma\in\Delta$ is called a shedding face if it satisfies the following property:

no facet of $({\mathrm{star}}_{\Delta}\sigma)\backslash\sigma$ is a facet of $\Delta\backslash\sigma$.

Definition 1.2**.**

[31] A $(d-1)$-dimensional simplicial complex $\Delta$ is recursively defined to be $k$-decomposable if either $\Delta$ is a simplex or else has a shedding face $\sigma$ with $\dim(\sigma)\leq k$ such that both $\mathrm{link}_{\Delta}\sigma$ and $\Delta\backslash\sigma$ are $k$-decomposable.

We consider the complexes $\{\}$ and $\{\emptyset\}$ to be $k$-decomposable for $k\geq-1$. Also $k$-decomposability implies to $k^{\prime}$-decomposability for $k^{\prime}\geq k$.

A [math]-decomposable simplicial complex is called vertex-decomposable.

We say that the simplicial complex $\Delta$ is (non-pure) shellable if its facets can be ordered $F_{1},F_{2},\ldots,F_{r}$ such that, for all $r\geq 2$, the subcomplex $\langle F_{1},\ldots,F_{j-1}\rangle\cap\langle F_{j}\rangle$ is pure of dimension $\dim(F_{j})-1$ [3]. It was shown in [31] or [18] that a $(d-1)$-dimensional (not necessarily pure) simplicial complex $\Delta$ is shellable if and only if it is $(d-1)$-decomposable.

Let $I$ be a monomial ideal of $S$. We denote by $G(I)$ the set of minimal monomial generators of $I$. Let $\min(I)$ be the set of minimal (under inclusion) prime ideals of $S$ containing $I$.

For $\mathbf{a}\in\mathbb{N}^{n}$, set $\mathbf{x}^{\mathbf{a}}=\underset{\mathbf{a}(i)>0}{\prod}x^{\mathbf{a}(i)}_{i}$ and define the support of $\mathbf{a}$ by ${\mathrm{supp}}(\mathbf{a})=\{i:\mathbf{a}(i)>0\}$. We set ${\mathrm{supp}}(\mathbf{x}^{\mathbf{a}}):={\mathrm{supp}}(\mathbf{a})$. Also, we define $\bar{\mathbf{a}}$ an $n$-tuple in $\{0,1\}^{n}$ with $\bar{\mathbf{a}}(i)=1$ if $\mathbf{a}(i)\neq 0$ and $\bar{\mathbf{a}}(i)=0$, otherwise. Set $\nu_{i}(\mathbf{x}^{\mathbf{a}}):=\mathbf{a}(i)$.

Let $u,v\in S$ be two monomials. We set $[u,v]=1$ if for all $i\in{\mathrm{supp}}(u)$, $x^{a_{i}}_{i}\nmid v$ and $[u,v]\neq 1$, otherwise.

For the monomial $u\in S$ and the monomial ideal $I\subset S$ set

$I^{u}=\langle v\in G(I):[u,v]\neq 1\rangle$ and $I_{u}=\langle v\in G(I):[u,v]=1\rangle.$

Definition 1.3**.**

[23] Let $I$ be a monomial ideal with the minimal system of generators $\{u_{1},\ldots,u_{r}\}$. The monomial $v=x^{a_{1}}_{1}\ldots x^{a_{n}}_{n}$ is called shedding if $I_{v}\neq 0$ and for each $u_{i}\in G(I_{v})$ and each $l\in{\mathrm{supp}}(u)$ there exists $u_{j}\in G(I^{v})$ such that $u_{j}:u_{i}=x_{l}$.

Definition 1.4**.**

[23] Let $I$ be a monomial ideal minimally generated with set $\{u_{1},\ldots,u_{r}\}$. We say $I$ is a $k$-decomposable ideal if $r=1$ or else has a shedding monomial $v$ with $|{\mathrm{supp}}(v)|\leq k+1$ such that the ideals $I^{v}$ and $I_{v}$ are $k$-decomposable. (Note that since the number of minimal generators of $I$ is finite, the recursion procedure will stop.)

A [math]-decomposable monomial ideal is called variable-decomposable.

Theorem 1.5**.**

[23, Theorem 2.10.]** Let $\Delta$ be a (not necessarily pure) $(d-1)$-dimensional simplicial complex on vertex set $[n]$. Then $\Delta$ is $k$-decomposable if and only if $I_{\Delta^{\vee}}$ is $k$-decomposable, where $k\leq d-1$.

Definition 1.6**.**

[20] A monomial ideal $I$ is called weakly polymatroidal if for every two monomials $u=x^{a_{1}}_{1}\ldots x^{a_{n}}_{n}>_{lex}v=x^{b_{1}}_{1}\ldots x^{b_{n}}_{n}$ in $G(I)$ such that $a_{1}=b_{1},\ldots,a_{t-1}=b_{t-1}$ and $a_{t}>b_{t}$, there exists $j>t$ such that $x_{t}(v/x_{j})\in I$.

Theorem 1.7**.**

[24, Theorem 4.33.]** Every weakly polymatroidal ideal $I$ is variable-decomposable.

2. $k$-clean monomial ideals

In this section we extend the concept of cleanness introduced by Dress [7]. Let $I\subset S$ be a monomial ideal. A prime filtration

[TABLE]

of $S/I$ is called multigraded, if all $M_{i}$ are multigraded submodules of $S/I$, and if there are multigraded isomorphisms $M_{i}/M_{i-1}\cong S/P_{i}(-\mathbf{a}_{i})$ with some $\mathbf{a}_{i}\in\mathbb{Z}^{n}$ and some multigraded prime ideals $P_{i}$.

A multigraded prime filtration $\mathcal{F}$ of $S/I$ is called clean if ${\mathrm{Supp}}(\mathcal{F})\subseteq\min(I)$.

Definition 2.1**.**

Let $I\subset S$ be a monomial ideal. A non unit monomial $u\not\in I$ is called a cleaner monomial of $I$ if $\min(I+Su)\subseteq\min(I)$.

Definition 2.2**.**

Let $I\subset S$ be a monomial ideal. We say that $I$ is $k$-clean whenever $I$ is a prime ideal or $I$ has no embedded prime ideals and there exists a cleaner monomial $u\not\in I$ with $|{\mathrm{supp}}(u)|\leq k+1$ such that both $I:u$ and $I+Su$ are $k$-clean.

We recall the concept of ideal tree from [25]:

Let $I\subset S$ be a $k$-clean monomial ideal. By the definition, there are cleaner monomials $u_{1},u_{2},\ldots$ with $|{\mathrm{supp}}(u_{i})|\leq k+1$ decomposing $I$. Therefore we obtain the rooted, finite, directed and binary tree $\mathcal{T}$:

[TABLE]

$\mathcal{T}$ is called the ideal tree of $I$ and the number of all cleaner monomials appeared in $\mathcal{T}$ is called the length of $\mathcal{T}$. We denote the length of $\mathcal{T}$ by $l(\mathcal{T})$.

We define the $k$-cleanness length of the $k$-clean monomial ideal $I$ by

[TABLE]

Example 2.3*.*

Consider the monomial ideal

$I=(x_{1}x_{2}x_{4},x_{1}x_{2}x_{5},x_{1}x_{2}x_{6},x_{1}x_{3}x_{5},x_{1}x_{3}x_{6},x_{1}x_{4}x_{5},x_{2}x_{3}x_{6},$

$x_{2}x_{4}x_{5},x_{2}x_{5}x_{6},x_{3}x_{4}x_{5},x_{3}x_{4}x_{6})$

and

$J=(x_{1}x_{2},x_{1}x_{3},x_{1}x_{4})$

of the polynomial ring $S=K[x_{1},\ldots,x_{6}]$. $I$ and $J$ are, respectively, $1$-clean and [math]-clean and have ideal trees $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ such that the cleaner monomials appeared in $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ are, respectively, $x_{2}x_{3},x_{1}x_{4},x_{1}x_{5},x_{2}x_{4},x_{2}x_{5},x_{2},x_{1},x_{3}x_{6},x_{3}$ and $x_{1}$.

Theorem 2.4**.**

Every $k$-clean monomial ideal $I$ is clean. Also, every clean monomial ideal is $k$-clean for some $k\geq 0$.

Proof.

Let $I$ be a $k$-clean monomial ideal. We use induction on the $k$-cleanness length of $I$. Let $I$ be not prime and there exists a cleaner monomial $u\not\in I$ of multidegree $\mathbf{a}$ with $|{\mathrm{supp}}(u)|\leq k+1$ such that both $I:u$ and $I+Su$ are $k$-clean. By induction, $I:u$ and $I+Su$ are clean. Let

[TABLE]

and

[TABLE]

be clean prime filtrations and let $(L_{i}/I:u)/(L_{i-1}/I:u)\cong S/Q_{i}(-\mathbf{a}_{i})$ where $Q_{i}$ are prime ideals. It is known that the multiplication map $\varphi:S/I:u(-\mathbf{a})\overset{.u}{\longrightarrow}I+Su/I$ is an isomorphism. Restricting $\varphi$ to $L_{i}/I:u$ yields a monomorphism $\varphi_{i}:L_{i}/I:u\overset{.u}{\longrightarrow}I+Su/I$. Set $H_{i}/I:=\varphi_{i}(L_{i}/I:u)$. Hence $H_{i}/I\cong(L_{i}/I:u)(-\mathbf{a})$. It follows that

[TABLE]

Therefore we obtain the following prime filtration induced from $\mathcal{F}_{2}$:

[TABLE]

By adding $\mathcal{F}_{1}$ to $\mathcal{F}_{3}$ we obtain the following prime filtration

[TABLE]

Finally, ${\mathrm{Supp}}(\mathcal{F})={\mathrm{Supp}}(\mathcal{F}_{1})\cup{\mathrm{Supp}}(\mathcal{F}_{2})\subset\min(I+Su)\cup\min(I:u)\subseteq\min(I)$ and therefore $I$ is clean.

To prove the second assertion, suppose that $I$ is a clean monomial ideal. If $I$ is prime then we are done. Suppose that $I$ is not prime and let

[TABLE]

be a clean prime filtration of $S/I$ with $M_{i}/M_{i-1}\cong S/P_{i}(-\mathbf{a}_{i})$. We use induction on the length of the prime filtration $\mathcal{F}$. Since that $\mathrm{Ass}(S/I)\subseteq{\mathrm{Supp}}(\mathcal{F})\subseteq\min(I)$, we have $\mathrm{Ass}(S/I)=\min(I)$. Hence $I$ has no embedded prime ideal. It follows from Proposition 10.1. of [15] that there is a chain of monomial ideals $I=I_{0}\subset I_{1}\subset\ldots\subset I_{r}=S$ and monomials $u_{i}$ of multidegree $\mathbf{a}_{i}$ such that $I_{i}=I_{i-1}+Su_{i}$ and $I_{i-1}:u_{i}=P_{i}$. Since that $I+Su_{1}$ has a clean filtration, it is $k$-clean, by induction hypothesis, where $|{\mathrm{supp}}(u_{1})|\leq k+1$. On the other hand, $I+Su_{1}/I\cong S/P_{1}$. Therefore $\min(I+Su_{1})=\{P_{1}\}\subset\min(I)$. This means that $I$ is $k$-clean. ∎

3. Some properties of $k$-clean monomial ideals

Theorem 3.1**.**

Let $I\subset S$ be $k$-clean. Then for all monomial $u\in S$, $I:u$ is $k$-clean.

Proof.

We use induction on the $k$-cleanness length of $I$. If $I$ is prime then $I:u$ is prime, too and we have nothing to prove. Assume that $I$ is not prime. Suppose $v$ is a cleaner monomial of $I$ with $|{\mathrm{supp}}(v)|\leq k+1$ and $I:v$ and $I+(v)$ are $k$-clean. We consider two cases:

Case 1. Let $v|u$. Then $I:u=(I:v):u/v$ and by induction hypothesis $I:u$ is $k$-clean.

Case 2. Let $v\nmid u$. We show that $v/\gcd(u,v)$ is a cleaner monomial of $I:u$. We have

[TABLE]

By induction, $(I:u)+(\frac{v}{\gcd(u,v)})$ and $(I:u):\frac{v}{\gcd(u,v)}$ are $k$-clean. Since $\min(I+(v))\subset\min(I)$, by some elementary computations, we obtain that $\min((I+(v)):u)\subset\min(I:u)$. Therefore $v/\gcd(u,v)$ is a cleaner monomial of $I:u$. ∎

Theorem 3.2**.**

The radical of each $k$-clean monomial ideal is $k$-clean.

Proof.

Let $I=(\mathbf{x}^{\mathbf{a}_{1}},\ldots,\mathbf{x}^{\mathbf{a}_{r}})$ be a $k$-clean monomial ideal with cleaner monomial $u=\mathbf{x}^{\mathbf{b}}$ with $|{\mathrm{supp}}(u)|\leq k+1$. We use induction on the $k$-cleanness length of $I$. Denote the radical of $I$ by $\sqrt{I}$. By induction hypothesis, $\sqrt{I+Su}$ and $\sqrt{I:u}$ are $k$-clean. Let $v=\mathbf{x}^{{\mathrm{supp}}(u)}$ and let $w$ be the product of variables $x_{i}$ with $i\in{\mathrm{supp}}(u)$ and $\mathbf{a}_{j}(i)>\mathbf{b}(i)>0$ for some $1\leq j\leq r$. $\sqrt{I}+Sv$ is $k$-clean, because $\sqrt{I}+Sv=\sqrt{I+Su}$. Also, $\sqrt{I}:v=(\sqrt{I:u}):w$ and so $\sqrt{I}:v$ is $k$-clean, by Theorem 3.1. On the other hand, $\min(\sqrt{I}+Sv)\subset\min(\sqrt{I+Su})=\min(I+Su)\subset\min(I)=\min(\sqrt{I})$ and so $v$ is a cleaner monomial of $\sqrt{I}$. ∎

Let $u=x^{a_{1}}_{i_{1}}\ldots x^{a_{t}}_{i_{t}}\in S$. The polarization of $u$ is defined by

[TABLE]

If $I\subset S$ is a monomial ideal. The polarization of $I$ is a monomial ideal of $S^{p}=K[x_{ij}:x_{ij}|u^{p}\ \mbox{for some}\ u\in G(I)]$ given by $I^{p}=(u^{p}:u\in G(I))$.

Define the $K$-algebra homomorphism $\pi:S^{p}\rightarrow S$ by $\pi(x_{ij})=x_{i}$.

Theorem 3.3**.**

Let $I$ be a monomial ideal with no embedded prime ideal. If $I^{p}$ is $k$-clean then $I$ is $k$-clean, too.

Proof.

We use induction on the $k$-cleanness length of $I^{p}$. If $I$ is a prime ideal then we have nothing to prove. Suppose that $I$ is not prime. Let $u$ be a cleaner monomial of $I^{p}$ with $|{\mathrm{supp}}(u)|\leq k+1$ and let $I^{p}:u$ and $I^{p}+(u)$ be $k$-clean. We claim that $\pi(u)$ is a cleaner monomial of $I$. Note that

$I:\pi(u)=\pi(I^{p}:u)$ and $I+(\pi(u))=\pi(I^{p}+(u))$.

By induction hypothesis, $I:\pi(u)$ and $I+(\pi(u))$ are $k$-clean. Since $|{\mathrm{supp}}(\pi(u))|\leq|{\mathrm{supp}}(u)|\leq k+1$, it remains to show that $\pi(u)$ is a cleaner monomial of $I$. Let $P\in\min(I+(\pi(u)))$. Hence there exists $Q\in\min(I^{p}+(u))$ such that $P=\pi(Q)$. Since $Q\in\min(I^{p})$, it follow that $P\in\min(I)$, as desired. ∎

Lemma 3.4**.**

Let $I\subset S$ be a $k$-clean monomial ideal with cleaner monomial $u$. Then $u^{p}$ is a cleaner monomial of $I^{p}$.

Proof.

Let $Q\in\min(I^{p}+(u^{p}))$. Then $Q\in\mathrm{Ass}(S^{p}/I^{p}+(u^{p}))$. By Corollary 2.6 of [9], $\pi(Q)\in\mathrm{Ass}(S/I+(u))=\min(I+(u))\subset\min(I)$. Again, by Proposition 2.3 of [9], $Q\in\min(I^{p})$, as desired. ∎

The following theorem describes projective dimension and Castelnuovo-Mumford regularity of $k$-clean monomial ideals

Theorem 3.5**.**

Let $I\subset S$ be a $k$-clean monomial ideal with the cleaner monomial $u$. Then

1. (i)

$\mathrm{pd}(S/I)=\max\{\mathrm{pd}(S/I+(u)),\mathrm{pd}(S/I:u)\}$; 2. (ii)

$\mathrm{reg}(S/I)=\max\{\mathrm{reg}(S/I+(u)),\mathrm{reg}(S/I:u)+\deg(u)\}$.

Proof.

(i) Without loss of generality we may assume that $I\subset\mathfrak{m}^{2}$. By Corollary 1.6.3. of [13], $\mathrm{pd}(S/I)=\mathrm{pd}(S^{p}/I^{p})$ and $\mathrm{reg}(S/I)=\mathrm{reg}(S^{p}/I^{p})$. Let $\Delta$ be a simplicial complex with $I_{\Delta}=I^{p}$. By Lemma 3.4, $u^{p}$ is a cleaner monomial of $I^{p}$. Let $u^{p}=\mathbf{x}^{\sigma}$ for some $\sigma\in\Delta$. Therefore $\Delta$ is a $k$-decomposable simplicial complex with shedding monomial $\sigma$, by Theorem 4.1. Now it follows from Theorem 2.8 of [21] that

[TABLE]

where $J_{\mathrm{link}_{\Delta}\sigma}$ is the Stanley-Reisner ideal of $\mathrm{link}_{\Delta}\sigma$ considered as a complex on $V(\Delta)\backslash\sigma$.

(ii) follows by a similar argument from Theorem 2.8 of [21] and Theorem 4.1. ∎

Remark 3.6*.*

The concept of sequentially Cohen-Macaulayness was introduced in [27] for finitely generated (graded) modules. We specially recall this concept for the quotient rings. Let $I\subset S$ be a monomial ideal. We say that $I$ is sequentially Cohen-Macaulay if there exists a finite filtration

[TABLE]

of submodules of $S/I$ with these properties that $M_{i}/M_{i-1}$ is Cohen-Macaulay and

[TABLE]

It was proven in [15] that cleanness implies sequentially Cohen-Macaulayness. Therefore the class of $k$-clean monomial ideals is contained in the class of sequentially Cohen-Macaulay monomial ideals. In particular, since that every unmixed sequentially Cohen-Macaulay monomial ideal is Cohen-Macaulay, we conclude that the unmixed $k$-clean monomial ideals are Cohen-Macaulay.

4. A view toward $k$-decomposable simplicial complexes

In this section, we prove the main result of this paper. In fact, we show that a squarefree $k$-clean monomial ideal is Stanley-Reisner ideal of a $k$-decomposable simplicial complex, and vice versa.

Theorem 4.1**.**

Let $\Delta$ be a ($d-1$)-dimensional simplicial complex. Then $\sigma\in\Delta$ is a shedding face of $\Delta$ if and only if $\mathbf{x}^{\sigma}$ is a cleaner monomial of $I_{\Delta}$.

In particular, $\Delta$ is $k$-decomposable if and only if $I_{\Delta}$ is $k$-clean, where $0\leq k\leq d-1$.

Proof.

We first show that $\sigma$ is a shedding face of $\Delta$ if and only if $\min(I_{\Delta}+(\mathbf{x}^{\sigma}))\subseteq\min(I_{\Delta})$. Since that Stanley-Reisner rings are reduced, it follows that

[TABLE]

and

[TABLE]

Let $\sigma$ be the shedding face of $\Delta$. To show that $\mathbf{x}^{\sigma}$ is a cleaner monomial of $I_{\Delta}$, it suffices to prove $\mathcal{F}(\Delta\backslash\sigma)\subseteq\mathcal{F}(\Delta)$. Suppose, on the contrary, that $F\in\mathcal{F}(\Delta\backslash\sigma)$ and $F\subsetneqq G$ with $G\in\mathcal{F}(\Delta)$. This implies that $\sigma\subset G$ and so $G\in{\mathrm{star}}_{\Delta}\sigma$. On the other hand, since $F$ is a facet of $\Delta\backslash\sigma$, it follows that there is $t\in\sigma$ such that $\sigma\backslash\{t\}\subset F$. We claim that $G=F\dot{\cup}\{t\}$. The inclusion “$\supseteq$” is clear. For the converse inclusion, if $s\in G\backslash(F\cup\{t\})$ for some $s$, then $\sigma\nsubseteq F\cup\{s\}$ and so $F\cup\{s\}\in\mathcal{F}(\Delta\backslash\sigma)$, a contradiction. Therefore $G=F\dot{\cup}\{t\}$ and it follows that $F\in\mathcal{F}(({\mathrm{star}}_{\Delta}\sigma)\backslash\sigma)$. But this contradicts the assumption that $\sigma$ is a shedding face of $\Delta$. Hence $\mathbf{x}^{\sigma}$ is a cleaner monomial.

Let $\Delta$ be $k$-decomposable with the shedding face $\sigma\in\Delta$. By the first part, $\mathbf{x}^{\sigma}$ is a cleaner monomial of $I_{\Delta}$. To showing that $I_{\Delta}$ is $k$-clean, we use induction on the number of the facets of $\Delta$. If $\Delta$ is a simplex then the assertion is trivial. So assume that $|\mathcal{F}(\Delta)|>1$. It is easy to check that $J_{\mathrm{link}_{\Delta}\sigma}=I_{\Delta}:\mathbf{x}^{\sigma}$ and $I_{\Delta\backslash\sigma}=I_{\Delta}+(\mathbf{x}^{\sigma})$. By induction hypothesis, $\mathrm{link}_{\Delta}\sigma$ and $\Delta\backslash\sigma$ are $k$-decomposable if and only if $I_{\Delta}:\mathbf{x}^{\sigma}$ and $I_{\Delta}+(\mathbf{x}^{\sigma})$ are $k$-clean. Therefore $I_{\Delta}$ is $k$-clean.

The reverse directions of both parts follow easily in similar arguments. ∎

Remark 4.2*.*

Note that a $k$-clean monomial ideal need not be $k^{\prime}$-clean for $k^{\prime}<k$. Consider the monomial ideal $I\subset K[x_{1},\ldots,x_{6}]$ with the minimal generator set

$G(I)=\{x_{1}x_{2}x_{4},x_{1}x_{2}x_{5},x_{1}x_{2}x_{6},x_{1}x_{3}x_{5},x_{1}x_{3}x_{6},x_{1}x_{4}x_{5},$

$x_{2}x_{3}x_{6},x_{2}x_{4}x_{5},x_{2}x_{5}x_{6},x_{3}x_{4}x_{5},x_{3}x_{4}x_{6}\}$.

$I$ is the Stanley-Reisner ideal of the simplicial complex

[TABLE]

on $[6]$. It was shown in [25] that $\Delta$ is shellable but not vertex-decomposable. It follows from Theorem 4.1 that $I$ is clean but not [math]-clean. To see more examples of clean ideals which are not [math]-clean we refer the reader to [11, 22].

Remark 4.3*.*

Let $I$ be a clean monomial ideal and $\dim(S/I)=d$. By Theorem 2.4, $I$ is $k$-clean for some $k\geq 0$ with cleaner monomial $u$. It follows from Theorem 3.2 that $\sqrt{I}$ is $k$-clean with cleaner monomial $v=\mathbf{x}^{{\mathrm{supp}}(u)}$. Let $I_{\Delta}=\sqrt{I}$ for some simplicial complex $\Delta$ on $[n]$. By Theorem 4.1, we have $|{\mathrm{supp}}(u)|=|{\mathrm{supp}}(v)|\leq\dim(\Delta)+1=d$. Therefore $I$ is $(d-1)$-clean.

On the other hand, every $k$-clean monomial ideal is also $(k+1)$-clean. This means that the $k$-cleanness is a hierarchical structure. Therefore we have the following implications:

[math]-clean $\Rightarrow$ $1$-clean $\Rightarrow$ …$\Rightarrow$ ($d-1$)-clean $\Leftrightarrow$ clean.

In Remark 4.2 we implied that above implications are strict.

Corollary 4.4**.**

Let $I\subset S$ be a squarefree monomial ideal generated by monomials of degree at least $2$. Then $I$ is $k$-clean if and only if $I^{\vee}$ is $k$-decomposable.

Proof.

Let $\Delta$ be a simplicial complex on $[n]$ such that $I=I_{\Delta}$. The assertion follows from Theorems 4.1 and 1.5. ∎

5. Some classes of $k$-clean ideals

In this section, we introduce some classes of $k$-clean monomial ideals.

5.1. Irreducible monomial ideals

Theorem 5.1**.**

Every irreducible monomial ideal is [math]-clean.

Proof.

Let $I$ be a irreducible monomial ideal. We want to show that $I$ is [math]-clean. By Theorem 1.3.1. of [13], $I$ is generated by pure powers of the variables. Without loss of generality we may assume that $I=(x^{a_{1}}_{1},\ldots,x^{a_{m}}_{m})$ with $a_{i}\neq 0$ for all $i$. We use induction on $\sum^{m}_{i=1}a_{i}$. If $\sum^{m}_{i=1}a_{i}=m$, then $I$ is prime and we are done. Suppose that $\sum^{m}_{i=1}a_{i}>m$. So we can assume that $a_{1}>1$. We have

$I:x_{1}=(x^{a_{1}-1}_{1},x^{a_{2}}_{2},\ldots,x^{a_{m}}_{m})$ and $I+(x_{1})=(x_{1},x^{a_{2}}_{2},\ldots,x^{a_{m}}_{m})$.

By induction hypothesis, $I:x_{1}$ and $I+(x_{1})$ are [math]-clean. Clearly, $x_{1}$ is a cleaner monomial and so the proof is completed. ∎

5.2. Monomial complete intersection ideals

Theorem 5.2**.**

Let $I\subset S$ be a monomial complete intersection ideal. Then $S/I$ is [math]-clean.

Proof.

Let $G(I)=\{M_{1},\ldots,M_{r}\}$. By the assumption $M_{1},\ldots,M_{r}$ is a regular sequence. Hence $\gcd(M_{i},M_{j})=1$ for all $i\neq j$. If $I$ is a primary ideal then we are done, by Theorem 5.1. Suppose that $I$ is not primary. We use induction on $n$ the number of variables. Let $|{\mathrm{supp}}(M_{1})|>1$ and let $\nu_{1}(M_{1})=a$. Then

$I:x^{a}_{1}=(M_{1}/x^{a}_{1},M_{2},\ldots,M_{r})$ and $I+(x^{a}_{1})=(x^{a}_{1},M_{2},\ldots,M_{r})$.

Since that $(M_{1}/x^{a}_{1},M_{2},\ldots,M_{r})$ and $(x^{a}_{1},M_{2},\ldots,M_{r})$ are complete intersection monomial ideals with the number of variables less that $n$, we deduce that $I:x^{a}_{1}$ and $I+(x^{a}_{1})$ are [math]-clean, by induction hypothesis. Set $J:=(M_{2},\ldots,M_{r})$. Since that

[TABLE]

and

[TABLE]

we conclude that $\min(I+(x^{a}_{1}))\subset\min(I)$ and so $x^{a}_{1}$ is a cleaner monomial. ∎

5.3. Cohen-Macaulay monomial ideals of codimension $2$

Proposition 2.3 from [14] says that if $I\subset S$ is a squarefree monomial ideal with 2-linear resolution, then after suitable renumbering of the variables, one has the following property:

if $x_{i}x_{j}\in I$ with $i\neq j$, $k>i$ and $k>j$, then either $x_{i}x_{k}$ or $x_{j}x_{k}$ belongs to $I$.

Let $I$ has a 2-linear resolution and the monomials in $G(I)$ be ordered by the lexicographical order induced by $x_{n}>x_{n-1}>\ldots>x_{1}$. Let $u=x_{s}x_{t}>v=x_{i}x_{j}$ be squarefree monomials in $G(I)$ with $s<t$ and $i<j$. We have $t\geq j$. If $t=j$, then $x_{s}(v/x_{i})=u\in G(I)$. If $t>j$ then by the above property either $x_{i}x_{t}\in G(I)$ or $x_{j}x_{t}\in G(I)$. This immediately implies the following lemma.

Lemma 5.3**.**

If $I$ is a squarefree monomial ideal generated in degree 2 which has a linear resolution, then after suitable renumbering of the variables, $I$ is weakly polymatroidal.

Theorem 5.4**.**

Let $I\subset S$ be a monomial ideal which is Cohen-Macaulay and of codimension $2$. Then $S/I$ is [math]-clean.

Proof.

Since $I$ has no embedded prime ideals, if we show that $I^{p}$ is [math]-clean then it follows from Theorem 3.3 that $I$ is [math]-clean. Let $\Delta$ be a simplicial complex with $I_{\Delta}=I^{p}$. Since $I$ is Cohen-Macaulay, by Corollary 1.6.3. of [13], $I_{\Delta}$ is Cohen-Macaulay, too. In particular, $I_{\Delta}^{\vee}$ has linear resolution, by the Eagon-Reiner theorem [8]. It follows from Lemma 5.3 and Theorems 1.7 and 4.1 that $I_{\Delta}=I^{p}$ is [math]-clean, as desired. ∎

5.4. Symbolic powers of Stanley-Reisner ideals of matroid complexes

Let $\Delta$ be a simplicial complex and let $I^{(m)}_{\Delta}$ denote the $m$th symbolic power of $I_{\Delta}$. Minh and Trung [19] and Varbaro [30] independently proved that $\Delta$ is a matroid if and only if $I^{(m)}_{\Delta}$ is Cohen-Macaulay for all $m\in\mathbb{N}$. Later, in [28], Terai and Trung showed that $\Delta$ is a matroid if and only if $I^{(m)}_{\Delta}$ is Cohen-Macaulay for some integer $m\geq 3$. Recently, Bandari and Soleyman Jahan [1] proved that if $\Delta$ is a matroid, then $I^{(m)}_{\Delta}$ is clean for all $m\in\mathbb{N}$. In this section, we improve this result by showing that if $\Delta$ is a matroid, then $I^{(m)}_{\Delta}$ is [math]-clean for all $m\in\mathbb{N}$.

Theorem 5.5**.**

Let $\Delta$ be a matroid complex with $I=I_{\Delta}$. Then for all $m\geq 1$, $I^{(m)}$ is [math]-clean.

Proof.

Let $\Delta=\langle F_{1},\ldots,F_{t}\rangle$. Then $I=I_{\Delta}=\bigcap^{t}_{i=1}P_{F^{c}_{i}}$ and $(I_{\Delta})^{(m)}=\bigcap^{t}_{i=1}(P_{F^{c}_{i}})^{(m)}$. Since $\Delta$ is a matroid and $I$ is Cohen-Macaulay, it follows that $I^{(m)}$ has no embedded prime ideal. Therefore if we show that $(I^{(m)})^{p}$ is [math]-clean then the proof is completed, by Theorem 3.3.

In [1] the authors introduced an ordering on the variables of $S^{p}$ and showed that $((I^{(m)})^{p})^{\vee}$ has linear quotients with respect to this ordering. We improve this result by considering the same ordering to show that $((I^{(m)})^{p})^{\vee}$ is weakly polymatroidal. Then by Theorem 1.7 and Corollary 4.4, $(I^{(m)})^{p}$ is [math]-clean. We use some notations of the proof of [1, Theorem 2.1.]. It is known that $\Delta^{c}$ is a matroid. Let $\dim(\Delta^{c})=r-1$. We set $J=((I^{(m)})^{p})^{\vee}$. Then

[TABLE]

where $1\leq j_{l}\leq m$ and $\sum^{r}_{l=1}j_{l}\leq m+r-1$.

Consider the order $<$ on the variables of $S^{\alpha}$ by setting $x_{i,j}>x_{i^{\prime},j^{\prime}}$ if either $j<j^{\prime}$, or $j=j^{\prime}$ and $i<i^{\prime}$. Let $u,v\in G(J)$ with $u=x_{i_{r},j_{r}}\ldots x_{i_{2},j_{2}}x_{i_{1},j_{1}}>v=x_{i_{r}^{\prime},j_{r}^{\prime}}\ldots x_{i_{2}^{\prime},j_{2}^{\prime}}x_{i_{1}^{\prime},j_{1}^{\prime}}$ such that $x_{i_{l},j_{l}}=x_{i_{l}^{\prime},j_{l}^{\prime}}$ for all $l>t$ and $x_{i_{t},j_{t}}>x_{i_{t}^{\prime},j_{t}^{\prime}}$. We have two cases:

Case 1. $x_{i_{t}}|x_{i_{r}^{\prime}}\ldots x_{i_{t+1}^{\prime}}x_{i_{t}^{\prime}}$. Let $i_{l}^{\prime}=i_{t}$. It is clear that $j_{t}<j_{l}^{\prime}$. In particular, $x_{i_{t},j_{t}}(v/x_{i_{l}^{\prime},j_{l}^{\prime}})\in G(J)$.

Case 2. $x_{i_{t}}\nmid x_{i_{r}^{\prime}}\ldots x_{i_{t+1}^{\prime}}x_{i_{t}^{\prime}}$. Since $I_{\Delta^{\vee}}$ is matroidal, it follows from [12, Lemma 3.1.] that there exists $i^{\prime}_{l}\not\in\{i_{1},\ldots,i_{r}\}$ such that $x_{i_{t}}(x_{i_{r}^{\prime}}\ldots x_{i_{1}^{\prime}}/x_{i_{l}^{\prime}})\in I_{\Delta^{\vee}}$. Therefore

[TABLE]

Therefore $J$ is weakly polymatroidal, as desired. ∎

It follows from Theorem 5.5 that we can add the condition “[math]-cleanness of $I^{(m)}_{\Delta}$ for all $m>0$” to [1, Corollary 2.3.]:

Corollary 5.6**.**

Let $\Delta$ be a pure simplicial complex and $I=I_{\Delta}\subset S$. Then the following conditions are equivalent:

1. (i)

$\Delta$* is a matroid;* 2. (ii)

$S/I^{(m)}$* is [math]-clean for all integer $m>0$;* 3. (iii)

$S/I^{(m)}$* is clean for some integer $m>0$;* 4. (iv)

$S/I^{(m)}$* is clean for some integer $m\geq 3$;* 5. (v)

$S/I^{(m)}$* is Cohen-Macaulay for some integer $m\geq 3$;* 6. (vi)

$S/I^{(m)}$* is Cohen-Macaulay for all integer $m>0$.*

Cowsik and Nori in [6] proved that for any homogeneous radical ideal $I$ in the polynomial ring $S$, all the powers of $I$ are Cohen-Macaulay if and only if $I$ is a complete intersection. We call the simplicial complex $\Delta$ complete intersection if $I_{\Delta}$ is a complete intersection ideal. Therefore the simplicial complex $\Delta$ is a complete intersection if and only if $I^{m}_{\Delta}$ is Cohen-Macaulay for any $m\in\mathbb{N}$ ([29, Theorem 3]). We improve this result in the following. By the fact that if $I^{m}_{\Delta}$ is Cohen-Macaulay then $I^{m}_{\Delta}$ is equal to the $m$th symbolic power $I^{(m)}_{\Delta}$ of $I_{\Delta}$ we have

Corollary 5.7**.**

Let $\Delta$ be a pure simplicial complex and $I=I_{\Delta}\subset S$. Then the following conditions are equivalent:

1. (i)

$\Delta$* is a complete intersection;* 2. (ii)

$S/I^{m}$* is [math]-clean for all integer $m>0$;* 3. (iii)

$S/I^{m}$* is clean for some integer $m>0$;* 4. (iv)

$S/I^{m}$* is clean for some integer $m\geq 3$;* 5. (v)

$S/I^{m}$* is Cohen-Macaulay for some integer $m\geq 3$;* 6. (vi)

$S/I^{m}$* is Cohen-Macaulay for all integer $m>0$.*

Acknowledgments: The author would like to express his sincere gratitude to the referee for his/her helpful comments that helped to improve the quality of the manuscript. The work was supported by the research council of the University of Maragheh.