% EJC papers *must* begin with the following two lines. \documentclass[12pt]{article} \usepackage{e-jc} % Please remove all other commands that change parameters such as % margins or pagesizes. % only use standard LaTeX packages % only include packages that you actually need % we recommend these ams packages \usepackage{amsthm,amsmath,amssymb} % we recommend the graphicx package for importing figures \usepackage{graphicx} % use this command to create hyperlinks (optional and recommended) \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} % use these commands for typesetting doi and arXiv references in the bibliography \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\texttt{doi:#1}}} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}} % all overfull boxes must be fixed; % i.e. there must be no text protruding into the margins % declare theorem-like environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{fact}[theorem]{Fact} \newtheorem{observation}[theorem]{Observation} \newtheorem{claim}[theorem]{Claim} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{open}[theorem]{Open Problem} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{note}[theorem]{Note} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % if needed include a line break (\\) at an appropriate place in the title \title{\bf Simplicial complexes of whisker type} % input author, affilliation, address and support information as follows; % the address should include the country, and does not have to include % the street address \author{Mina Bigdeli$^\ast$\\ \small Department of Mathematics\\[-0.8ex] \small Institute for Advanced Studies in Basic Sciences\\[-0.8ex] \small Zanjan, Iran\\ \small\tt m.bigdelie@iasbs.ac.ir\\ \and J\"urgen Herzog\\ \small Fachbereich Mathematik\\[-0.8ex] \small Universit\"at Duisburg-Essen\\[-0.8ex] \small Essen, Germany\\ \small\tt juergen.herzog@uni-essen.de \and Takayuki Hibi\\ \small Department\! of Pure and Applied Mathematics\\[-0.8ex] \small Osaka University\\[-0.8ex] \small Osaka, Japan\\ \small\tt hibi@math.sci.osaka-u.ac.jp \and Antonio Macchia\thanks{The paper was written while the first and the fourth author were visiting the Department of Mathematics of University Duisburg-Essen. They want to express their thanks for the hospitality.}\\ \small Fachbereich Mathematik und Informatik\\[-0.8ex] \small Philipps-Universit\"at Marburg\\[-0.8ex] \small Marburg, Germany\\ \small\tt macchia.antonello@gmail.com } % \date{\dateline{submission date}{acceptance date}\\ % \small Mathematics Subject Classifications: comma separated list of % MSC codes available from http://www.ams.org/mathscinet/freeTools.html} \date{\dateline{Dec 9, 2014}{}\\ \small Mathematics Subject Classifications: Primary 13C15, 05E40, 05E45; Secondary 13D02.} \begin{document} \maketitle % E-JC papers must include an abstract. The abstract should consist of a % succinct statement of background followed by a listing of the % principal new results that are to be found in the paper. The abstract % should be informative, clear, and as complete as possible. Phrases % like "we investigate..." or "we study..." should be kept to a minimum % in favor of "we prove that..." or "we show that...". Do not % include equation numbers, unexpanded citations (such as "[23]"), or % any other references to things in the paper that are not defined in % the abstract. The abstract will be distributed without the rest of the % paper so it must be entirely self-contained. \begin{abstract} Let $I \subset K[x_1,\dots,x_n]$ be a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that $\Delta(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $\Delta(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$. % keywords are optional \bigskip\noindent \textbf{Keywords:} depth function; linear quotients; vertex decomposable; whisker complexes; zero-dimensional ideals \end{abstract} \section{Introduction} Graphs with whiskers have first been considered by Villarreal in \cite{V}. They all share the nice property that they are Cohen-Macaulay. Various extensions of this concept and generalizations of his result have been considered in the literature, see \cite{CN}, \cite{VTV}, \cite{HHKO} and \cite{F}. The edge ideal of a whisker graph is obtained as the polarization of a monomial ideal $I\subset S$, where $S=K[x_1,\ldots,x_n]$ is the polynomial ring over a field $K$, $I$ is generated in degree $2$ and $\dim S/I=0$. In particular, $I$ contains the squares $x_1^2,\ldots,x_n^2$. More generally, given a simplicial complex $\Gamma$, the whisker complex $W(\Gamma)$ is studied in \cite{LL}. Its facet ideal is the polarization of a monomial ideal in $S$ which contains all the $x_i^2$. In \cite{LL}, Loiskekoski shows that the Stanley--Reisner ideal of the Alexander dual of the independence complex of $W(\Gamma)$ has a linear resolution, as well as its powers. %is called the face ideal of a simplicial complex. Such face ideals, under a different name, have also been studied in \cite{LL}. In \cite{HH1}, as well as in \cite{LL}, it is shown that face ideals have a linear resolution. In \cite{LL} Loiskekoski even shows that all powers of a face ideal have a linear resolution. In the present paper we generalize the above mentioned results by considering the polarization of any monomial ideal $I\subset S$ with $\dim S/I=0$. The simplicial complex $\Theta(I)$, whose facet ideal coincides with the polarization $I^\wp$ of $I$, is called of {\em whisker type} -- the whiskers being the simplices corresponding to the polarization of the pure powers contained in $I$. The independence complex of $\Theta(I)$, denoted $\Delta(I)$, is characterized by the property that the Stanley--Reisner ideal $I_{\Delta(I)}$ coincides with $I^\wp$. Note that $F\in \Delta(I)$ if and only if $F$ does not contain any facet of $\Theta(I)$. Given an arbitrary monomial ideal $I\subset S$, a multicomplex is associated with $I$, as defined by Popescu and the second author in \cite{HP}. Soleyman Jahan defines in \cite[Proposition 3.8]{S} a bijection between the facets of the multicomplex given by $I$ and the facets of the simplicial complex associated with $I^\wp$. In Theorem~\ref{min} we present a short proof of this bijection when $\dim S/I=0$, by using multiplicity theory. This result allows us to describe in Corollary~\ref{facets} the facets of $\Delta(I)$. By applying the Eagon--Reiner Theorem it is then shown in Corollary~\ref{linear} that the ideal $L(I)$ has a linear resolution, where $L(I)$ is generated by the monomials $x_{1,a_1+1} \cdots x_{n,a_n+1}$ for which $x_1^{a_1}\cdots x_n^{a_n}$ is a monomial in $S$ not belonging to $I$. In the case that $\dim S/I=0$, the case we consider here, the corresponding multicomplex is pretty clean, see \cite{HP}. Soleyman Jahan showed in \cite[Theorem 4.3]{S} that if $I$ defines a pretty clean multicomplex, then the simplicial complex associated with $I^\wp$ is clean, which, by a theorem of Dress \cite{D}, implies that the simplicial complex attached to $I^\wp$ is shellable. Applied to our situation it follows that $\Delta(I)$ is shellable. We give a direct proof of this fact by showing that $L(I)$ has linear quotients. This provides an explicit shelling of $\Delta(I)$, and as a side result we obtain a formula for the Betti numbers of $L(I)$ in terms of the $h$-vector of $S/I$, see Corollary~\ref{betti}. We conclude Section~2 with Corollary~\ref{cellular}, where it is shown that the minimal graded free resolution of $L(I)$ is cellular and supported on a regular CW-complex. The proof is based on a result of Dochtermann and Mohammadi\break \cite[Theorem~3.10]{DM}, who showed that the minimal graded free resolution of any ideal with regular decomposition function, as defined in \cite{HT}, have such nice cellular structure. In Section~3 we show that $\Delta(I)$ is not only shellable but even vertex decomposable. This was already known for whisker graphs (see \cite[Theorem~4.4]{DE}). Finally in Section~4 we prove that all powers of $L(I)$ have linear quotients, see Theorem~\ref{powers}. Analyzing the linear quotients, the depth function $f(k)=\mathrm{depth}\ S/L(I)^k$ can be computed. In Corollary~\ref{depth} a formula for the depth function is given and $\lim_{k\to \infty} \mathrm{depth}\ S/L(I)^k$ is determined. \section{The facets of the independence complex of a whisker type simplicial complex} Throughout this paper $S$ denotes the polynomial ring $K[x_1,\ldots,x_n]$ and $I\subset S$ a monomial ideal with $\dim S/I=0$, unless otherwise stated. The (finite) set of monomials in $S$ which belong to $S$ but not to $I$ will be denoted by $\mathrm{Mon}(S\setminus I)$. For an arbitrary monomial ideal $I$, we denote by $G(I)$ the unique minimal set of monomial generators of $I$. We will consider the polarization of $I$, denoted $I^\wp$. The polynomial ring in which $I^\wp$ is defined will be denoted by $S^\wp$. In the following theorem (cf. \cite[Proposition 3.8]{S}) we determine the set $\mathrm{Min}(I^\wp)$ of minimal prime ideals of $I^\wp$. \begin{theorem} \label{min} Let $I\subset S$ be a monomial ideal with $\dim S/I=0$. The map $\phi$ which assigns to each monomial $u=x_1^{a_1}\cdots x_n^{a_n}\in S\setminus I$ the monomial prime ideal $\phi(u)=(x_{1,a_1+1}, \ldots, x_{n,a_n+1})\subset S^\wp$, establishes a bijection between $\mathrm{Mon}(S\setminus I)$ and $\mathrm{Min}(I^\wp)$. \end{theorem} \begin{proof} We first observe that $\phi(\mathrm{Mon}(S\setminus I)) \subset S^\wp$. Indeed, since $\dim S/I=0$, there exists for each $1\leq i\leq n$ an integer $b_i>0$ such that $x_i^{b_i}\in I$ and $x_i^{b_i-1}\notin I$. It follows that $S^\wp$ is the polynomial ring in the variables $x_{i,1}, \ldots, x_{i,b_i}$ with $1\leq i\leq n$. Now let $u=x_1^{a_1}\cdots x_n^{a_n}\in \mathrm{Mon}(S\setminus I)$. Then $a_ia_i$. Therefore, $x_{i,a_i+1}$ divides $v^\wp$, as desired. Clearly, $\phi$ is injective. We will show that $|\mathrm{Mon}(S\setminus I)|=|\mathrm{Min}(I^\wp)|$. This will then imply that $\phi : \mathrm{Mon}(S\setminus I)\to \mathrm{Min}(I^\wp)$ is bijective. In order to see that these two sets have the same cardinality we observe that the multiplicity $e(S/I)$ of $S/I$ is equal to the length $\ell(S/I)$ of $S/I$, because $\dim S/I=0$, see \cite[Corollary 4.7.11(b)]{BH}. Since $\ell(S/I)=\dim_KS/I$ and since the elements of $\mathrm{Mon}(S\setminus I)$ form a $K$-basis of $S/I$, we see that $e(S/I)=\dim_KS/I=|\mathrm{Mon}(S \setminus I)|$. On the other hand, since $S/I$ is isomorphic to $S^\wp/I^\wp$ modulo a regular sequence of linear forms \cite[Proposition~1.6.2]{HHBook}, and since $S^\wp/I^\wp$ is reduced and equidimensional, \cite[Corollary 4.7.8]{BH} implies that $e(S/I)=e(S^\wp/I^\wp)=|\mathrm{Min}(I^\wp)|$. \end{proof} We denote by $\Delta(I)$ the simplicial complex whose Stanley-Reisner ideal is $I^\wp$. We view the variables $x_{i,j}\in S^\wp$ as the vertices of $\Delta(I)$. As an immediate consequence of Theorem~\ref{min} we obtain \begin{corollary} \label{facets} Let $\mathcal{S}$ be the set of variables of $S^\wp$. Then $F\subset \mathcal{S}$ is a facet of $\Delta(I)$ if and only if there exists $x_1^{a_1}\cdots x_n^{a_n}\in \mathrm{Mon}(S \setminus I)$ such that $$F=\mathcal{S}\setminus \{x_{1,a_1+1}, \ldots, x_{n,a_n+1}\}.$$ \end{corollary} Since $\Delta(I)$ is Cohen--Macaulay, the Eagon--Reiner Theorem \cite{EG} (see also \cite[Theorem~8.1.9]{HHBook}) implies that $I_{\Delta(I)^\vee}$ has a linear resolution. Here $\Delta(I)^\vee$ denotes the Alexander dual of $\Delta(I)$. Recall that, if $\Delta$ is an arbitrary simplicial complex on the vertex set $[n]=\{1,\ldots,n\}$ and $I_\Delta=\bigcap_{F} P_F$ where $P_F=(x_i\; :\; i\in F)$, then $I_{\Delta^{\vee}}$ is generated by the monomials $u_F$ where $u_F=\prod_{i\in F}x_i$. These facts applied to our case yield \begin{corollary} \label{linear} The ideal $L(I)$ generated by the monomials $x_{1,a_1+1} \cdots x_{n,a_n+1}$, with\break $x_1^{a_1}\cdots x_n^{a_n}\in \mathrm{Mon}(S\setminus I)$, has a linear resolution. \end{corollary} In the following we consider the special case that $x_i^2\in I$ for all $i$. In that case all other generators of $I$ are square-free. In simplified notation, the polarization $I^\wp$ of $I$ is generated by the square-free monomials in $I$ and by the monomials $x_iy_i$ for $i=1,\ldots,n$. Let $\Gamma$ be the simplicial complex with $I(\Gamma)=J$ and $W(\Gamma)$ be the simplicial complex with $I(W(\Gamma))= (J,x_1y_1,\ldots,x_ny_n)$. The edges of $W(\Gamma)$ corresponding to the $x_iy_i$ are called the {\em whiskers} of $W(\Gamma)$ and $W(\Gamma)$ is called the {\em whisker complex} of $\Gamma$. Given a simplicial complex $\Sigma$, the independence complex $\Lambda$ of $\Sigma$ is the simplicial complex such that $I_\Lambda=I(\Sigma)$. Notice that $F\in \Lambda$ if and only if no face of $\Sigma$ is contained in $F$. %As a special case of Theorem~\ref{min} we recover \cite[Theorem 1.1]{HH1}. \begin{corollary} \label{newproof} Let $\Gamma$ be a simplicial complex on the vertex set $[n]$, $I'=I(\Gamma)$ the facet ideal of $\Gamma$ and $W(\Gamma)$ its whisker complex. Let $I=(I', x_1^2,x_2^2,\ldots,x_n^2)$. Then $\Delta(I)$ is the independence complex of $W(\Gamma)$ and $L(I)$ is generated by the monomials $\prod_{i\in [n]\setminus F}x_i\prod_{i\in F}y_i$ with $F\in \Delta$, where $\Delta$ is the independence complex of $\Gamma$. \end{corollary} \section{Linear quotients} Let $I\subset S$ be a monomial ideal with $\dim S/I=0$. The main purpose of this section is to show that $L(I)$ not only has a linear resolution, but even has linear quotients. \begin{theorem} \label{quotients} The ideal $L(I)$ has linear quotients. \end{theorem} \begin{proof} Let $u,v\in G(L(I))$, $u=x_{1,a_1+1} \cdots x_{n,a_n+1}$ and $v=x_{1,b_1+1} \cdots x_{n,b_n+1}$. We set $u\leq v$ if $a_i\leq b_i$ for all $i$, and extend this partial order to a total order on $G(L(I))$. We claim that, with respect to this total order of the monomial generators of $L(I)$, the ideal $L(I)$ has linear quotients. Indeed, let $x_{1,a_1+1} \cdots x_{n,a_n+1}$ be the largest element in $G(L(I))$. Then $u=x_1^{a_1} \cdots x_n^{a_n}\in \mathrm{Mon}(S\setminus I)$ and $x_iu \in I$ for all $i$. Set $I' = I + (u)$. Then the polarization $(I')^\wp$ of $I'$ is equal to $I_{\Delta(I')}$. Notice that $L(I') \subset L(I)$ and $\ell(S/I') < \ell(S/I)$. In particular, $L(I) = (L(I'),x_{1,a_1+1} \cdots x_{n,a_n+1})$. Arguing by induction on the length, we may assume that $L(I')$ has linear quotients. Thus we just need to compute the colon ideal $Q=L(I') : x_{1,a_1+1} \cdots x_{n,a_n+1}$. We claim that \begin{eqnarray} \label{q} Q=(x_{1,1},x_{1,2},\ldots,x_{1,a_1},x_{2,1},\ldots,x_{2,a_2},\ldots, x_{n,1},\ldots, x_{n,a_n}). \end{eqnarray} Suppose that $j \in \{1,\dots,a_i\}$ for some $i$. Then $x_1^{a_1} \cdots x_i^{j-1} \cdots x_n^{a_n} \in \mathrm{Mon}(S \setminus I)$ and $$\phi(x_1^{a_1} \cdots x_i^{j-1} \cdots x_n^{a_n}) = x_{1,a_1+1} \cdots x_{i,j} \dots x_{n,a_n+1} \in L(I').$$ It follows that $x_{i,j} \in Q$. On the other hand, the elements $v/ \gcd(v, x_{1,a_1+1} \cdots x_{n,a_n+1})$ with $v\in G(L(I'))$ generate $Q$, see for example \cite[Proposition~1.2.2]{HHBook}. In fact, let $v\in G(L(I'))$. Then $v=x_{1,c_1+1}\cdots x_{n,c_n+1}$ and $x_1^{c_1}\cdots x_n^{c_n}\in \mathrm{Mon}(S\setminus I')$. There exists $i$ such that $c_i < a_i$ because $x_iu \in I$ for all $i$. Hence $x_{i,c_i+1}$ does not divide $x_{1,a_1+1} \cdots x_{n,a_n+1}$, and therefore $x_{i,c_i+1}$ divides $v/ \gcd(v, x_{1,a_1+1} \cdots x_{n,a_n+1})$. Since $c_i+1\leq a_i$, the desired conclusion follows. \end{proof} \begin{corollary} \label{betti} For every $i \geq 0$, \[ \beta_i (S^\wp/L(I)) = \sum_{j \geq 0} h_j \binom{j}{i-1}, \] where $h_j = h_j(S/I)$ is the $j$-th component of the $h$-vector of $S/I$. In particular,\break $\mathrm{proj\,dim}\ S^\wp/L(I)=\max\{\deg u : u\in \mathrm{Mon}(S\setminus I)\}+1$. \end{corollary} \begin{proof} As in the previous proof, let $u=x_1^{a_1} \cdots x_n^{a_n}\in \mathrm{Mon}(S \setminus I)$ with $x_iu \in I$ for all $i$. Set $I' = I + (u)$, and consider the short exact sequence $$ 0 \rightarrow L(I)/L(I') \rightarrow S^\wp/L(I') \rightarrow S^\wp/L(I) \rightarrow 0. $$ Notice that $L(I)/L(I') \cong S^{\wp}/Q (-n)$ with $Q$ as in (\ref{q}). Hence its minimal free resolution is the Koszul complex $\mathbb{K}$ on the variables $x_{i,j}$ with $x_{i,j}\in G(Q)$. Thus the minimal free resolution of $S^\wp/L(I)$ can be obtained as a mapping cone of $\mathbb{K}$ and the minimal free resolution of $S^\wp/L(I')$. Therefore $\beta_0 (S^\wp/L(I)) = \beta_0 (S^\wp/L(I'))$, and for $i \geq 1$ we obtain \begin{eqnarray*} \beta_i (S^\wp/L(I)) &=& \beta_i(S^\wp/L(I')) + \mathrm{rank}\ (K_{i-1}) = \beta_i(S^\wp/L(I')) + \binom{\deg u}{i-1} \\ &=& \sum_{u \in \mathrm{Mon}(S \setminus I)} \binom{\deg u}{i-1} = \sum_{j \geq 0} h_j \binom{j}{i-1}. \end{eqnarray*} \end{proof} It is easily seen that the geometric realization of $\Delta(I)$ is a sphere if $I$ is a complete intersection, and a ball otherwise. Both topological spaces admit shellable triangulations, though in general not all triangulations of these spaces are shellable, see \cite{Ru} and \cite{Li}. However, due to Theorem~\ref{quotients} we have \begin{corollary} \label{shelling} The simplicial complex $\Delta(I)$ is shellable. \end{corollary} As a further consequence of Theorem~\ref{quotients} we have \begin{corollary} \label{cellular} The graded minimal free resolution of $L(I)$ is cellular and supported on a regular CW-complex. \end{corollary} \begin{proof} Since $L(I)$ has linear quotients we may apply \cite[Theorem 3.10]{DM} and only need to show that $L(I)$ admits a regular decomposition function. In order to explain this, let $J=(u_1,\ldots,u_m)$ be an ideal with linear quotients with respect to the given order of the generators. The {\em decomposition function} of $J$ (with respect to the given order of the generators of $J$) is the map $b : \mathrm{Mon}(J) \to G(J)$ with $b(u) = u_j$, where $j$ is the smallest number such that $u\in (u_1,\ldots,u_j)$. For each $u_j\in G(J)$, let $\mathrm{set}(u_j)$ be the set of all $x_i$ such that $x_iu_j\in (u_1,\ldots,u_{j-1})$. According to \cite{HT}, the decomposition function $b$ is called {\em regular}, if $\mathrm{set}(b(x_iu_j))\subset \mathrm{set}(u_j)$ for all $u_j\in G(J)$ and all $x_i\in \mathrm{set}(u_j)$. Now let $u\in G(L(I))$, $u=x_{1,a_1+1} \cdots x_{n,a_n+1}$. By (\ref{q}) we have \[ \mathrm{set}(u)=\{x_{1,1},x_{1,2},\ldots,x_{1,a_1},x_{2,1},\ldots,x_{2,a_2},\ldots, x_{n,1},\ldots, x_{n,a_n}\}. \] Let $x_{i,j}\in \mathrm{set}(u)$. Then $b(x_{i,j}u)=x_{i,j}(u/x_{i,a_i+1})$, and so \[ \mathrm{set}(b(x_{i,j}u))=\mathrm{set}(u)\setminus\{x_{i,j+1},\ldots,x_{i,a_i}\}\subset \mathrm{set}(u), \] as desired. \end{proof} \section{Vertex decomposability} In \cite[Theorem~4.4]{DE} it was shown that for any graph, the independence complex of its whisker graph is vertex decomposable. Here we extend this result by showing that $\Delta(I)$ is vertex decomposable for any monomial ideal $I$ with $\dim S/I=0$. Recall that a simplicial complex $\Delta$ is called {\em vertex decomposable} if $\Delta$ is a simplex, or $\Delta$ contains a vertex $v$ such that \begin{itemize} \item[(i)] any facet of $\mathrm{del}_{\Delta}∆(v)$ is a facet of $\Delta$, and \item[(ii)] both $\mathrm{del}_{\Delta}(v)$ and $\mathrm{link}_{\Delta}(v)$ are vertex decomposable. \end{itemize} Here $\mathrm{link}_\Delta(v)=\{G\in \Delta : v\not\in G \text{ and } G \cup \{v\}\in \Delta\}$ is the {\em link} of $v$ in $\Delta$ and $\mathrm{del}_\Delta(v) =\{G\in \Delta : v\not\in G \}$ is the {\em deletion} of $v$ from $\Delta$. \medskip A vertex $v$ which satisfies condition (i) is called a {\em shedding vertex} of $\Delta$. \medskip For the proof of the next result we observe the following fact: let $\Delta$ be a simplicial complex, $\mathcal{F}(\Delta)$ the set of its facets and $v$ a vertex not belonging to $\Delta$. The {\em cone} of $v$ over $\Delta$, denoted by $v\ast \Delta$, is the simplicial complex whose set of facets is $\mathcal{F}(v\ast \Delta)=\{\{v\}\cup F : F\in \mathcal{F}(\Delta)\}$. If $\Delta$ is vertex decomposable, then $v\ast \Delta$ is again vertex decomposable (with respect to the same shedding vertex). \begin{theorem} \label{decomposable} Let $I$ be a monomial ideal in $S=K[x_1, \ldots, x_n]$ with $\dim S/I=0$. Then $\Delta(I)$ is vertex decomposable. \end{theorem} \begin{proof} By assumption, for each $1\leq i\leq n$ there exists $b_i \geq 1$ such that $x_i^{b_i}\in G(I)$. Then $\Delta(I)$ is a simplicial complex on $\mathcal{S}=\{x_{1,1},\dots, x_{1,b_1},\dots, x_{n,1},\dots,x_{n,b_n}\}$. We proceed by induction on $\sum_{i=1}^n b_i$. If $\sum_{i=1}^n b_i = n$, then $I=(x_1,\dots,x_n)$, which is a trivial case. Suppose that $\sum_{i=1}^n b_i > n$. Hence we may assume $b_n > 1$. We first show that the vertex $x_{n,1}$ is a shedding vertex of $\Delta(I)$. Clearly, $$\mathrm{del}_{\Delta(I)} (x_{n,1}) = \{ F : F \in \Delta(I), x_{n,1} \notin F \} \cup \{ F \setminus \{x_{n,1}\} : F \in \Delta(I), x_{n,1} \in F \}.$$ Obviously, any facet of $\mathrm{del}_{\Delta(I)} (x_{n,1})$ with $x_{n,1} \notin F$ is a facet of $\Delta(I)$. On the other hand, if we consider $F \setminus \{x_{n,1}\}$ with $F \in \mathcal{F}(\Delta(I))$ and $x_{n,1} \in F$, then $F \setminus \{x_{n,1}\}$ is not a facet of $\mathrm{del}_{\Delta(I)} (x_{n,1})$. Indeed, since $F \in \mathcal{F}(\Delta(I))$, there exists $u \in \mathrm{Mon}(S \setminus I)$ such that $\phi(u) = P_{\mathcal{S}\setminus F}$. Let $t$ be the largest integer such that $x_n^t$ divides $u$. Then $x_{n,t+1}\in P_{\mathcal{S}\setminus F}$ and so $x_{n,j}\in F$ for all $j\neq t+1$. Since $x_{n,1}\in F$, we have $t+1\neq 1$. Let $u'=u/x_n^t$. Then $u'\in \mathrm{Mon}(S\setminus I)$ and $\phi(u')= P_{((\mathcal{S}\setminus F)\setminus \{x_{n,t+1}\}) \cup \{x_{n,1}\}}$. Thus $G=(F\setminus \{x_{n,1}\})\cup \{x_{n,t+1}\} \in \mathcal{F}(\Delta(I))$. Since $G\in \mathrm{del}_{\Delta(I)} (x_{n,1})$, the claim follows. Consequently, $\mathcal{F}(\mathrm{del}_{\Delta(I)} (x_{n,1})) = \{ F : F \in \mathcal{F}(\Delta(I)), x_{n,1} \notin F \}$ which implies that $x_{n,1}$ is a shedding vertex of $\Delta(I)$. We now prove that $\mathrm{del}_{\Delta(I)} (x_{n,1})$ and $\mathrm{link}_{\Delta(I)}(x_{n,1})$ are vertex decomposable. First we consider $\mathrm{del}_{\Delta(I)} (x_{n,1})$. Let $J_1$ be the ideal in $S$ with $\mathrm{Mon}(S \setminus J_1)=\{u:\ u\in \mathrm{Mon}(S\setminus I), x_n \text{ does not divide } u\}$. Then $\Delta(J_1)$ is a simplicial complex on $\mathcal{S} \setminus \{x_{n,1},\dots,x_{n,b_n}\}$. By using Corollary~\ref{facets} we see that $$ \mathrm{del}_{\Delta(I)}(x_{n,1}) =x_{n,b_n}\ast(x_{n,b_n-1}\ast(\cdots\ast(x_{n,2}\ast\Delta(J_1)))). $$ Our induction hypothesis implies that $\Delta(J_1)$ is vertex decomposable, hence $\mathrm{del}_{\Delta(I)}(x_{n,1}) $ is vertex decomposable. As for $\mathrm{link}_{\Delta(I)}(x_{n,1})$, let $\Gamma$ be the simplicial complex whose faces are obtained from the faces of $\mathrm{link}_{\Delta(I)}(x_{n,1})$ as follows: for every $F \in \mathrm{link}_{\Delta(I)}(x_{n,1})$, we replace each $x_{n,j} \in F$ by $x_{n,j-1}$. Hence $\Gamma$ is a simplicial complex on $\mathcal{S}\setminus \{x_{n,b_n}\}$ and $\Gamma \cong \mathrm{link}_{\Delta(I)}(x_{n,1})$. Let $J_2$ be the monomial ideal in $S$ such that $\mathrm{Mon}(S \setminus J_2) = \{u/x_n : u \in \mathrm{Mon}(S \setminus I), x_n \text{ divides } u\}$. Then Corollary~\ref{facets} implies that $\Gamma=\Delta(J_2)$, which is vertex decomposable by induction hypothesis. It follows that $\mathrm{link}_{\Delta(I)}(x_{n,1})$ is vertex decomposable, as desired. \end{proof} \section{Powers} In this section we study the powers of $L(I)$. The main result is \begin{theorem} \label{powers} Let $I\subset S$ be a monomial ideal with $\dim S/I=0$. Then $L(I)^k$ has linear quotients for all $k$. In particular, all powers of $L(I)$ have a linear resolution. \end{theorem} \begin{proof} Any $u\in L(I)^k$ can be written in the form $u=u_1'u_2'\cdots u_n'$, where $u_i'=\break x_{i,j(i)_1}x_{i,j(i)_2}\cdots x_{i,j(i)_k}$ for $i=1,\ldots,n$ with $j(i)_1\leq j(i)_2\leq \cdots \leq j(i)_k$. We define a partial order on $G(L(I)^k)$ by setting $v\leq u$, if, with respect to the lexicographical order, $u_i'\leq v_i'$ for all $i$, and we extend this partial order to a total order on the set of monomial generators of $L(I)^k$. Now let $v,u\in L(I)^k$ with $v\mathrm{depth}\ S^\wp/L(I)^{k+1},$$ as long as $\mathrm{depth}\ S^\wp/L(I)^k>n-1$. \end{corollary} \begin{proof} In general, let $J\subset K[x_1,\ldots,x_n]$ be a graded ideal generated by a sequence $f_1, \ldots , f_s$ with linear quotients, and denote by $q_j(J)$ the minimal number of linear forms generating the ideal $(f_1,f_2, \ldots, f_{j-1}) : f_j$. Then $\mathrm{depth}\ K[x_1,\ldots,x_n]/J= n-q(J)-1$, where $q(J)=\max\{q_j(J) : 2\leq j\leq s\}$, see \cite[Formula (1)]{HH}. We apply this formula to $S^\wp/L(I)^k$. Since the Krull dimension of $S^\wp$ is equal to $\sum_{i=1}^nb_i$, it remains to be shown that \begin{eqnarray} \label{aa} q(L(I)^k)=\max\{\deg (\mathrm{lcm}(u_1,u_2,\ldots, u_k)) : u_1,u_2,\ldots,u_k\in \mathrm{Mon}(S\setminus I)\}. \end{eqnarray} To see this, let $u_1,u_2,\ldots,u_k\in \mathrm{Mon}(S\setminus I)$ where $u_j=x_{1}^{a_1(j)}\cdots x_{n}^{a_n(j)}$ for $j=1,\ldots,k$. Then $u=u_1'u_2'\cdots u_n'$ with $u_i' =\prod_{j=1}^kx_{i, a_i(j)+1}$ is a generator of $L(I)^k$. We may assume that $u$ is the $j$-th element in the given total order of the elements of $G(L(I)^k)$. As shown in the proof of Theorem~\ref{powers}, $q_j(L(I)^k)$ is the cardinality of the set \[ \{x_{1,1},\ldots, x_{1,c_1}, x_{2,1},\ldots,x_{2,c_2},\ldots,x_{n,1},\ldots,x_{n,c_n}\}, \] where $c_i =\max\{a_i(1),\ldots,a_i(k)\}$ for $i=1,\ldots,n$. If follows that $$q_j(L(I)^k)= \deg (\mathrm{lcm}(u_1,u_2,\ldots, u_k)), $$ and hence equation (\ref{aa}) follows. Suppose now that $k\geq n$. Then we may choose $u_i=x_i^{b_i-1}$ for $i=1,\ldots,n$ and $u_i\in \mathrm{Mon}(S\setminus I)$ arbitrary for $i>n$, and obtain $\deg(\mathrm{lcm}(u_1,u_2,\ldots u_k))=\sum_{i=1}^n(b_i-1)= \sum_{i=1}^nb_i-n$. Since this is the largest possible least common multiple of sequences of elements of $\mathrm{Mon}(S\setminus I)$, it follows that $\mathrm{depth}\ S^\wp/L(I)^k=n-1$ for all $k\geq n$. Finally, suppose that $\mathrm{depth}\ S^\wp/L(I)^k>n-1$. Then the formula for $\mathrm{depth}\ S^\wp/L(I)^k$ implies that $\max\{\deg (\mathrm{lcm}(u_1,\ldots, u_k)) : u_1,\ldots, u_k\in \mathrm{Mon}(S\setminus I)\}<\sum_{i=1}^n(b_i-1).$ Let $x_1^{a_1}\cdots x_n^{a_n}=\mathrm{lcm}(u_1,\ldots, u_k)$ attain this maximal degree. Since $\sum_{i=1}^na_i <\break \sum_{i=1}^n(b_i-1)$, there exists an index $i$ such that $a_i \deg (\mathrm{lcm}(u_1,\ldots, u_k))$. 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