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\title{Regularity of Edge Ideals Via Suspension}
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%{A guide for authors preparing their accepted manuscript for AlCo}
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\author[\initial{A.} Banerjee]{\firstname{Arindam} \lastname{Banerjee}}
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%\address{Indian Institute of Technology\\
%Dept. of pure and applied mathematics\\
%2400 Clarksville st.\\
%Paris\\
%TX 75460 (USA)}
\address{Indian Institute of Technology, Kharagpur}
\email{123.arindam@gmail.com}
%\author{Eran Nevo}
%% Current address, if different from institutionnal address
%\curraddr{Coll\`ege Royal Henri-Le-Grand\\
%%Prytan\'ee National Militaire\\
%72000 La Fl\`eche, Sarthe (France)}
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%\email{m.mersenne@u-paris.tx.edu}
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\author[\initial{E.} Nevo]{\firstname{Eran} \lastname{Nevo}}
\address{Einstein Institute of Mathematics,
The Hebrew University of Jerusalem.}
\email{nevo@math.huji.ac.il}
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}
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%\author{\firstname{Joseph} \lastname{Fourier}}
%\address{Universit\'e de Grenoble\\
% Institut Moi-m\^eme\\
% BP74, 38402 SMH Cedex (France)}
%\email{fourier@fourier.edu.fr}
% Key words and phrases:
\keywords{Edge ideals, regularity, suspension}
%% Mathematical classification (2010)
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\subjclass{10X99, 14A12, 11L05}
\datepublished{2024-01-08}
\begin{document}
%% Abstracts must be placed before \maketitle
\begin{abstract}
We study the Castelnuovo--Mumford regularity of powers of edge ideals for arbitrary (finite simple) graphs. It has been repeatedly conjectured that for every graph $G$, $\mathrm{reg}(I(G)^s) \leq 2s+\mathrm{reg}\, I(G)-2$ for all $s\geq 2$, which is the best possible upper bound for any $s$. We prove this conjecture for every $s$ for all bipartite graphs, and for $s=2$ for all graphs. The $s=2$ case is crucial for our work and suspension plays a key role in its proof.
\end{abstract}
\maketitle
% First paragraph after a section is not indented. If there is text
% below the title before the first section, it should be unindented
% like this.
\section{Introduction}
Let $M$ be a finitely generated graded module over a polynomial ring $R = \K[x_1,\ldots,x_n]$, where
$\K$ is a field. The Castelnuovo--Mumford regularity (or simply, regularity) $\reg(M)$ of $M$
is defined as
$$
\reg(M)=\max \{j-i \mid \Tor_i^R(M, \K)_j \neq 0\}.
$$
$\text{ }$ $\text{ }$Regularity is an important invariant in commutative algebra and algebraic geometry that measures in some sense the complexity of ideals, modules, and sheaves. A question that has been studied by many is how the regularity behaves with respect to taking powers of homogeneous ideals. It is known that in the long-run $\reg(I^k)$ is linear in $k$, that is, there exist integers $a(I), b(I), c(I)$ such that $\reg(I^k) = a(I)k + b(I)$ for all $k\geq c(I)$ (see \cite{CHT,vijay}). For various classes of ideals people have studied these integers and also have looked for various upper and lower bounds for $\reg(I^k)$.
%For monomial ideals this study has a combinatorial angle to it
%as they are equipped with nice combinatorial structures
For monomial ideals these invariants, as well as bounds on them, reflect the underlying combinatorics (see e.g. \cite{BBH17,ha_adam2,Herzog'sBook,ms2005,MorVill,Nevo} for various works under this theme).
For monomial ideals $I$ generated in same degree $d$, Kodiyalam~\cite{vijay} showed that $a(I)=d$.
% Conca~\cite{C} showed that for any given integer $d > 1$ there exists an ideal $J$ generated by $d+5$ monomials of degree $d + 1$ in $4$ variables such that $\reg(J^k) = k(d + 1)$ for every $k 2$ is proved algebraically, via various uses of short exact sequences for related ideals. Part~\ref{it:2} improves the main result of ~\cite{jayanthan}, which proves that if $G$ is bipartite, then for all $s\geq 2$ there holds $\reg(I(G)^s) \leq 2s+\text{cochord(G)}-1$, where cochord(G) is the cochordal number of $G$ (see \cite{jayanthan} for definition).
\textbf{Outline}: Preliminaries are given in Section~\ref{pre}, Theorem~\ref{thm:main} is proved in Section~\ref{sec:main_results}, and concluding remarks are given in Section~\ref{sec:further}.
\section{Preliminaries}\label{pre}
In this section, we set up the basic definitions and notation needed for the main results. Let $M$ be a finitely generated graded $R=\K[x_1,\ldots,x_n]$-module.
Write the graded minimal free resolution of $M$ in the form:
\[
0 \longrightarrow \bigoplus_{j \in \mathbb{Z}} R(-j)^{\beta_{p,j}(M)}
\overset{\psi_{p}}{\longrightarrow} \cdots \overset{\psi_1}{\longrightarrow}
\bigoplus_{j \in \mathbb{Z}} R(-j)^{\beta_{0,j}(M)}
\overset{\psi_0}{\longrightarrow} M\longrightarrow 0,
\]
where $p \leq n$, $R(-j)$ indicates the ring $R$ with the
shifted grading such that, for all $a \in \mathbb{Z}$, $R(-j)_{a}=R_{a-j}$.
The nonnegative integers $\beta_{(i,j)}(M)$ are called
$i^{\text{th}}$-graded Betti numbers of $M$ in degree $j$.
The Castelnuovo-Mumford regularity (or regularity) of $M$ is defined to be
\[
\reg(M)=\max\{j-i \mid \beta_{i,j}(M)\neq 0\}.\label{def_reg}
\]
Let $I$ be a nonzero proper homogeneous ideal of $R$.
Then it follows from the definition
that $\reg(I) = \reg(R/I)+1.$
Let $I$ be any ideal of $R$ and $a\in R$ any element, the the \emph{colon ideal} $(I:a)$ is defined as the ideal $(I:a)\coloneqq (b \text{ } | b\in R, ab\in I)$.
\emph{Polarization} is a process that creates a squarefree monomial out of a monomial, possibly in a larger polynomial ring. If $f=x_1^{e_1}\cdots x_n^{e_n}$ is a monomial in $\mathbb{K}[x_1, \ldots , x_n]$ then the polarization of $f$ is defined as $\tilde{f}=x_{11}\cdots x_{1e_1} x_{21} \cdots x_{2e_2}\cdots x_{n1} \cdots x_{n e_n}$ in the ring $\mathbb{K}[x_{11},\ldots x_{1e_1}, x_{21},\ldots x_{2e_2}, \ldots, x_{n1}, \ldots, x_{ne_n}]$.
For convenience
we identify the variable $x_{i1}$ with $x_i$, so the new polynomial ring extends the old one.
For a monomial ideal $I$ with minimal monomial generators $\{m_1,\ldots, m_k\}$, we define the \emph{polarization} of $I$ as $\widetilde{I}\coloneqq(\widetilde{m_1}, \cdots, \widetilde{m_k})$ in a suitable ring, see e.g
\cite{Herzog'sBook} or \cite[Sec. 1.6]{kummini_thesis}. In the special case where degree of a variable $u=x_i$ is two in one or more generators then we call the unique new variable $x_{i2}$ a \emph{whisker variable} and denote it by $u'$ for short.
%If $u$ is a variable whose degree is two in the monomial that is being polarized then the new variable is denoted by $u'$.
In this paper we repeatedly use an important property of
polarization:
\begin{theorem}[e.g.~{\cite[Cor. 1.6.3(a)]{kummini_thesis}}] Let $I$ be a monomial
ideal in $R$.
%$\K[x_1, \ldots, x_n].$
Then
$$\reg(I)=\reg(\widetilde{I}).$$
\label{thm:2.1}
\end{theorem}
One of the main techniques that is used in this paper is that of short exact sequences.
In particular we shall use the following well known result~\cite[Lemma. 2.11]{banerjee}:
\begin{theorem}
\begin{enumerate}[label=(\roman*)]
\item Let $I$ be a homogeneous ideal in a polynomial ring $R$ and $m$ be an element of degree $d$ in $R$. Then the following is a short exact sequence: $$0 \longrightarrow \frac{R}{(I:m)} \overset{\cdot m} \longrightarrow \frac{R}{I} \longrightarrow \frac{R}{I+(m)} \longrightarrow 0.$$
Hence $$\reg(I) \leq \max \{\reg (I:m) +d, \reg (I+(m))\}.$$
\item In case $I$ is squarefree and $x$ a variable, then also $\reg(I,x) \leq \reg I$.
\end{enumerate}
\label{thm:2.2}
\end{theorem}
Let $G$ be a (finite simple) graph with vertex set $V(G)=\{x_1, \ldots ,x_n\}$ and the edge set $E(G)$. The \textit{edge ideal} $I(G)$ of $G$ is defined as the ideal in $R$: $$I(G)=(x_i x_j|x_ix_j \in E(G)).$$
For example, edge ideal of a $5$-cycle is $(x_1x_2,x_2x_3,x_3x_4,x_4x_5,x_5x_1)$.
The next couple of theorems allow for induction when increasing the power of an edge ideal.
\begin{theorem}[{\cite[Thm. 5.2]{banerjee}}]
\label{thm:2.3}
For any graph $G$ and any $s \geq 1,$ let the set of minimal monomial generators of $I(G)^s$ be $\{m_1, \dots, m_k\}.$ Then
$$ \reg (I(G)^{s+1}) \leq \max \{\reg (I(G)^s) , \reg ((I(G)^{s+1}): m_l)+2s , 1 \leq l \leq k\}.$$
\end{theorem}
By identifying the variables with the vertices of $G$, interpreting edges as squarefree quadratic monomials, defining neighborhood for any vertex $c$, $N(c)\coloneqq\{z\in V(G): cz \in E(G)\}$ and using \cite[Thm. 5.2]{banerjee} we get the following corollary:
\begin{coro}
\begin{enumerate}[label=(\roman*)]
\item The ideal $(I(G)^{s+1}: m_l)$ is a quadratic monomial ideal.
\item For the special case where $s=1$ and $m=ab$ is an edge we have $$(I(G)^2:ab)=I(G)+(xy| x\in N(a),y\in N(b)).$$
\end{enumerate}
\label{cor:I2:e}
\end{coro}
For bipartite graphs we further have:
\begin{theorem}[{\cite[Lem. 3.2, Prop. 3.4]{banerjee1}}]
Let $G$ be a bipartite graph and $s \geq 1$ an integer.
Then $(I(G)^{s+1}: e_1 \cdots e_s)$ is a quadratic squarefree monomial ideal where for all $i$ we have $e_i \in E(G)$. Moreover, the graph $G'$ associated to $(I(G)^{s+1} : e_1\cdots e_s)$ is bipartite on the same vertex set and same bipartition as G.
For any bipartite graph $G$ we have $$(I(G)^{s+1}:e_1\cdots e_s)=((I(G)^2:e_i)^s : \Pi_{j \neq i} e_j)$$ where for all $l \in \{1, \ldots,s\}$ we have $e_l \in E(G)$.
\label{thm:2.5}
\end{theorem}
\begin{rema} From Theorem~\ref{thm:2.5} one can show a slightly more general equality involving colons, under the same assumptions: for all $k$ and any $s \geq k+1$,
\begin{equation*}
(I(G)^{s+1}:e_1\cdots e_s)=((I(G)^{k+1}:e_1 \cdots e_k)^{s-k+1} : e_{k+1} \cdots e_s).
\end{equation*}
This follows from the following sequence of arguments:
\begin{enumerate}[align=parleft,left=0pt,label=(\alph*)]
\item From Theorem 2.5 for $i=1$ it follows that:
\begin{equation*}
(I(G)^{s+1}:e_1\cdots e_s)=((I(G)^2:e_1)^s : e_2 \cdots e_s ).
\end{equation*}
\item The first part of the Theorem~\ref{thm:2.5} tells us that $(I(G)^2:e_1)$ is a bipartite edge ideal.
\item By Corollary~\ref{cor:I2:e} above we get that each $e_i$ is an edge of the bipartite graph associated to $(I(G)^2:e_1)$.
\item Applying the Theorem~\ref{thm:2.5} again on the edge ideal $(I(G)^2:e_1)$ we get:
\begin{equation*}
((I(G)^2:e_1)^s : e_2 \cdots e_s )=(((I(G)^2:e_1)^2:e_2)^{s-1} : e_3 \cdots e_s).
\end{equation*}
\item By a similar application of the last result with $s=2$ we have
\begin{equation*}
(I(G)^3:e_1e_2)=((I(G)^2:e_1)^2:e_2).
\end{equation*}
\item Combining these we get
\begin{equation*}
(I(G)^{s+1}:e_1\cdots e_s)=((I(G)^3:e_1e_2)^{s-1} : e_3 \cdots e_s).
\end{equation*}
\item Inductively, assuming that for a fixed $k$ one has for any $s \geq k+1$
\begin{equation*}
(I(G)^{s+1}:e_1\cdots e_s)=((I(G)^{k+1}:e_1e_2 \cdots e_k)^{s-k+1} : e_{k+1} \cdots e_s).
\end{equation*}
\item Again applying Theorem~\ref{thm:2.5} on the bipartite edge ideal $(I(G)^{k+1}:e_1 \cdots e_k)$ we get
\begin{equation*}
((I(G)^{k+1} :e_1 \cdots e_k)^{s-k+1} :e_{k+1} \cdots e_s)=( ((I(G)^{k+1}: e_1 \cdots e_k )^2:e_{k+1})^{s-k} : e_{k+2} \cdots e_s).
\end{equation*}
\item But by induction $$(I(G)^{k+2}:e_1 \cdots e_{k+1})=((I(G)^{k+1}: e_1 \cdots e_k )^2:e_{k+1}).$$
So we get $$((I(G)^{k+1} :e_1 \cdots e_k)^{s-k+1} :e_{k+1} \cdots e_s)= ((I(G)^{k+2}:e_1 \cdots e_{k+1})^{s-k}:e_{k+2} \cdots e_s),$$ as claimed.
\end{enumerate}
\end{rema}
Now we recall some basic definitions about graphs and simplicial complexes that will be useful.
Let $G$ be a graph with vertex set $V(G)$
and edge set $E(G)$.
A subgraph $H \subseteq G$ is called \textit{induced} if $\{u,v\}$ is an edge of $H$ if and only if $u$ and $v$ are vertices of $H$ and $\{u,v\}$ is an edge of $G$.
A \textit{clique} in a graph is an induced subgraph that is a \emph{complete graph}.
For $u\in V(G)$, let $N_G(u) = \{v \in V (G)\mid \{u, v\} \in E(G)\}$ and $N_G[u]=N_G(u)\cup \{u\}$.
For $U \subseteq V(G)$, denote by $G \setminus U$
the induced subgraph of $G$ on the vertex set $V(G) \setminus U$.
Let $G$ be a graph. We denote the graph consisting of two disjoint edges
by $2K_2$.
% in $G$ if $G$ does not have an edge with one
%endpoint in $f_1$ and the other in $f_2$.
A graph without induced copy of $2K_2$ is called $2K_2$-free or \textit{gap-free} graph. The \textit{complement} of a graph $G$, denoted by $G^c$, is the graph on the same
vertex set in which $\{u,v\}$ is an edge of $G^c$ if and only if it is not an edge of $G$.
Then $G$ is gap-free if and only if $G^c$ contains no induced $4$-cycle.
%Thus, $G$ is gap-free if and only if it does not contain two vertex-disjoint edges as an induced subgraph.
%\par A \textit{matching} in a graph $G$ is a subgraph consisting of pairwise disjoint edges.
%If the
%subgraph is an induced subgraph, the matching is an \textit{induced matching}.
%T%he largest size of an induced matching in $G$ is called its induced
%matching number and denoted by $\nu(G)$.
%Let $H$ be a subgraph of $G$. The largest size of induced matching of $H$ as well as
%$G$ denoted by $\nu_{GH}$. A \textit{dominating induced matching} of $G$ is an induced matching which also
%forms a maximal matching of $G$
A graph $G$ is chordal (also called triangulated) if every induced
cycle in $G$ has length $3$, and is co-chordal if the complement graph $G^c$ is chordal.
The following important theorem(s) characterizes the edge ideals with regularity 2, and the regularity of their powers.
\begin{theorem}
\begin{enumerate}
\item \textup{(}Fr\"oberg~\cite[Thm.1]{froberg}\textup{)} For any graph $G$, we have $G^c$ is chordal if and only if $\reg(I(G))=2$.
\item \textup{(}Herzog--Hibi--Zheng~\cite[Thm.1.2]{HHZ}\textup{)} Further, in this case $\reg(I(G)^s)=2s$ for any $s$.
\end{enumerate}
\label{thm:chordal}
\end{theorem}
A \emph{simplicial complex} $\Delta$ on a vertex set $\{1,\ldots, n\}$ is a collection of subsets of $\{1,\ldots ,n\}$ such that if $\tau \in \Delta, \sigma \subseteq \tau$ then we have $\sigma \in \Delta$ and all singletons belong to that collection; if $\tau$ is such a subset belonging to $\Delta$ then $\tau$ is called a \emph{face} of $\Delta$.
The \emph{induced subcomplex} $\Delta [A]$ of $\Delta$ on vertex set $A\subset \{1,\ldots,n\}$ is the collection of faces $\tau'$ of $\Delta$ such that $\tau' \subset A$.
Clearly the induced subcomplex is a simplicial complex itself. We denote by $\Vr(\Delta)$ the set of vertices of $\Delta$.
For sets $A$ and $B$ we sometimes write $A\cup_C B$ for the set $A\cup B$ as a shorthand for denoting their intersection by $C=A\cap B$.
Likewise for simplicial complexes.
The \emph{link} of a vertex $v$ in $ \Delta$ is $$\text{link}_v (\Delta)= \{ \tau |\tau \cup \{v\} \in \Delta, \{v\} \cap \tau =\emptyset\}.$$
The (open) \emph{star} of a vertex $v$ in a simplicial complex $\Delta$ is the set of all faces that contain $v$, namely $\St_v(\Delta)=\{\tau| \tau \in \Delta, v \in \tau\}$. The \emph{closed star} $\bar{\St}_v (\Delta$ ) of $v$ is defined by the smallest subcomplex that contains $\St_v(\Delta)$. The \emph{antistar} of vertex $v$ is defined as the subcomplex $\text{ast}_v (\Delta) = \{\tau \in \Delta| \tau \cap \{v\} = \emptyset\}$.
The \emph{join} of two simplicial complexes $\Delta_1$ and $\Delta_2$ is defined by $\Delta_1 * \Delta_2=\{\sigma \cup \tau| \sigma \in \Delta_1, \tau \in \Delta_2\}$.
The \emph{suspension} of a simplicial complex $\Delta$, w.r.t. two points $a$ and $b$ not vertices of $\Delta$, is the join defined by $\Sigma_{a,b} \Delta=\Delta * \{\{a\},\{b\},\emptyset \}$; its geometric realization is homeomorphic to the topological suspension of the space $\Delta$.
For $H$ a graph
%and $\Delta=\cl(G^c)$, where
let $\cl(H)$ denote its clique complex, i.e. the simplicial complex whose faces are the cliques of $H$.
The following formulation of regularity follows from the so called Hochster's formula
%and Stanley-Reisner ideal formulation
(see {\cite{ms2005}} for further details):
\begin{theorem}[Hochster's formula]
For every graph $G$ whose edge set is nonempty and $\Delta=\cl(G^c)$ we have:
$$\reg(G)\coloneqq\reg(I(G))= \max\{l+2:\exists W \subseteq \Vr(\Delta), \widetilde{H}_l(\Delta[W];k)\neq 0\}.$$
\end{theorem}
(Sometimes Hochster's formula is phrased with the extra condition on $W$ that $|W|=l+2+i$ for some $i\ge 0$. However, this condition is superfluous: if $|W|\le l+1$ then either $\dim(\Delta[W]) 1$ there exists an ideal $J$ generated by $d+5$ monomials of degree $d + 1$ in $4$ variables such that $\reg(J^k) = k(d + 1)$ for every $k 4$ then this will be a counterexample. Unfortunately we could not verify the value of $\reg(I(G)^2)$ due to computational limitations. It will be great if this can be verified in the future.
\longthanks{We thank Aldo Conca for pointing us to~\cite{Raicu}, and the referees for very helpful remarks on the presentation.
Most of this work was done when first author was visiting the Institute of Mathematics of Hebrew University and he would like to thank faculty and staff of Hebrew University for their hospitality. Also, the first author was partially supported by DST INSPIRE (India) research grant (DST/INSPIRE/04/2017/000752) and he would like to acknowledge that.
The second author was partially supported by Israel Science Foundation grants ISF-1695/15 and ISF-2480/20, by grant 2528/16 of the ISF-NRF Singapore joint research program, and by ISF-BSF joint grant 2016288.}
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