# Minimal Cellular Resolutions of Path Ideals

Trung Chau  Department of Mathematics, University of Utah, 155 South 1400
East, Salt Lake City, UT 84112, USA<EMAIL_ADDRESS>,  Selvi Kara
Department of Mathematics, Bryn Mawr College, 906 New Gulph Rd, Bryn Mawr, PA
19010, USA<EMAIL_ADDRESS>and  Kyle Wang  Department of Mathematics &
Statistics, Dalhousie University 6316 Coburg Rd., PO BOX 15000, Halifax, NS,
Canada<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove that the path ideals of both paths and cycles have
minimal cellular resolutions. Specifically, these minimal free resolutions
coincide with the Barile-Macchia resolutions for paths, and their generalized
counterparts for cycles. Our result on cycles marks the first instance in the
literature where the minimal free resolution of edge ideals of cycles has been
established. Furthermore, edge ideals of cycles represent the first class of
ideals that lack a minimal Barile-Macchia resolution, yet have a minimal
generalized Barile-Macchia resolution.

##  1\. Introduction

It has been a powerful approach to associate a combinatorial object with an
algebraic one and study its algebraic properties via combinatorics [11, 12,
14, 15, 16]. In the spirit of this approach, the algebraic objects of interest
in this paper are path ideals, while the combinatorial counterparts are
graphs. Specifically, our focus is on studying the path ideals of graphs and
their minimal free resolutions by leveraging the underlying structure of the
graphs. Central to this work is the use of (generalized) Barile-Macchia
resolutions from [8]. Such resolutions fall under the umbrella of Morse
resolutions, a class of cellular resolutions first introduced by Batzies and
Welker in [4]. These resolutions are obtained using homogeneous acyclic
matchings, a concept from discrete Morse theory.

Path ideals, first introduced by Conca and De Negri in [10], have been studied
for their algebraic properties [1, 2, 3, 6, 7]. Path ideals can be seen as a
generalization of edge ideals, which have been of significant interest in
recent years. Let $G=(V,E)$ be a finite, simple graph with the vertex set
$V=\\{x_{1},\ldots,x_{n}\\}$. Associating the vertices of $G$ with the
variables in the polynomial ring $R=\Bbbk[x_{1},\ldots,x_{n}]$, where $\Bbbk$
is any field, the edge ideal of $G$ is then generated by monomials of the form
$x_{i}x_{j}$ for every $\\{x_{i},x_{j}\\}\in E$. In essence, edge ideals arise
from monomials corresponding to edges in $G$, which are inherently paths of
length 1. Extending this idea, ideals generated by monomials corresponding to
paths of a specified length in $G$ are called the path ideals of $G$.

In this paper, our goal is to construct minimal resolutions for path ideals of
paths and cycles. While there is literature discussing and providing explicit
formulas for the (graded) Betti numbers of these ideals [2], no construction
has yet been provided for their minimal resolution. We achieve this by working
with (generalized) Barile-Macchia resolutions from [8], thereby expanding the
class of ideals for which these resolutions are minimal.

Our two main results are:

  1. (1)

We establish that path ideals of paths have the _bridge-friendly_ property
(Theorem 3.8). This property ensures the minimality of Barile-Macchia
resolutions as described in [8]. Thus, we determine that the path ideals of
paths admit a minimal Barile-Macchia resolution. From this, we derive formulas
for their projective dimension and graded Betti numbers, recovering results
from [2] and [6] (Corollary 3.15, Theorem 3.16).

  2. (2)

We shift our focus to path ideals of cycles. In this context, Barile-Macchia
resolutions are not always minimal. One instance is the edge ideal of a
9-cycle, $I_{2}(C_{9})$, which, as indicated in [8], does not have any minimal
Barile-Macchia resolution. Nonetheless, we prove that path ideals of cycles
have a minimal generalized Barile-Macchia resolution (Theorem 4.16). To the
best of our knowledge, this is the first instance where a minimal free
resolution for the edge ideal of a cycle has been provided in the literature.

Both Barile-Macchia and generalized Barile-Macchia resolutions are induced by
homogeneous acyclic matchings, called Barile-Macchia and generalized Barile-
Macchia matchings, respectively. At the heart of the Barile-Macchia matching
construction is the comparison of least common multiples of subsets of the
minimal generating set ${\mathcal{G}}(I)$ of a monomial ideal $I$, with
respect to a total order on ${\mathcal{G}}(I)$. An algorithm for producing
Barile-Macchia matchings was introduced by the first two authors in [8]. In
addition, MorseResolutions Macaulay2 package dedicated to Barile-Macchia
resolutions were introduced by the first two authors and O’Keefe in [9]. While
the generalized version adopts the same foundational principle, it extends to
multiple total orders on ${\mathcal{G}}(I)$ as discussed in Section 4. For
further details, refer to [8].

An important concept in relating homogeneous acyclic matchings and free
resolutions is _critical cells_ : These are subsets of ${\mathcal{G}}(I)$ that
remain untouched by a homogeneous acyclic matching of $I$. In [4], it was
shown that these cells are in one-to-one correspondence with the ranks of free
modules from (generalized) Barile-Macchia resolutions. Thus, in this paper, we
focus on characterizing critical cells of path ideals using _bridges, gaps_
and _true gaps_ – simple yet powerful concepts rooted in the graph’s structure
as introduced in [8]. We use these notions to produce minimal free resolutions
of path ideals of paths and cycles.

A key observation concerning the critical cells of path ideals of paths is
that two distinct critical cells have different least common multiples. This
observation proves useful in constructing a critical cell of maximum size,
which in turn allows us to deduce a formula for the projective dimension of
path ideals of paths. Additionally, this insight is helpful in identifying all
critical cells of path ideals of cycles. Such cells consists of critical cells
of path ideals of induced paths as well as critical cells whose least common
multiple has the largest multidegree – for a cycle on $n$ vertices, it would
be $x_{1}\dots x_{n}$. Consequently, in the paper’s final section, our study
primarily focuses on identifying the critical cells of the latter type for
cycles.

The structure of this paper is as follows. In Section 2, we revisit essential
concepts and results relevant to Morse resolutions, as well as Barile-Macchia
matchings and resolutions. In Section 3, we explore path ideals of paths. We
start this section by delving into the characterizations of bridges, gaps, and
true gaps specific to paths. With these characterizations in hand, we obtain
the bridge-friendliness and affirm the minimality of the associated Barile-
Macchia resolution. This paves the way for us to introduce formulas for the
projective dimension and to provide a recursive formula for graded Betti
numbers. Lastly, in Section 4, we turn our attention to path ideals of cycles,
offering both a review of and insights into the application of generalized
Barile-Macchia resolutions. After characterizing the bridges, gaps, and true
gaps, we verify the minimality of their generalized Barile-Macchia resolutions
by drawing upon our earlier findings on path ideals.

##  2\. Preliminaries

###  2.1. Morse resolutions

Let $I$ be a monomial ideal in the polynomial ring
$R=\Bbbk[x_{1},\ldots,x_{N}]$ with a minimal generating set denoted by
$\mathcal{G}(I)=\\{m_{1},\ldots,m_{n}\\}$. We associate to $I$ the Taylor
simplex $X$. The vertices of $X$ correspond to the generators of $I$, whereas
its cells are labeled by the least common multiple of the labels of their
incident vertices. We also associate a directed graph $G_{X}=(V,E)$ with this
structure, where $V$ denotes the cells of $X$, or equivalently, the subsets of
$\mathcal{G}(I)$. The edge set $E$ consists of directed edges of the form
$(\sigma,\sigma^{\prime})$ where $\sigma^{\prime}\subseteq\sigma$ and
$|\sigma^{\prime}|=|\sigma|-1$. For any subset $A$ of $E$, we define
$G_{X}^{A}$ as the directed graph having vertex set $V$ and edge set

$E(G_{X}^{A})=(E\setminus
A)\cup\\{(\sigma^{\prime},\sigma)\mid(\sigma,\sigma^{\prime})\in A\\}.$

Essentially, $G_{X}^{A}$ is derived from $G_{X}$ by reversing the direction of
edges belonging to $A$.

Central to our discussion on a Morse resolution of $I$ is the notion of
homogeneous acyclic matchings from discrete Morse theory, a concept we revisit
below.

######  Definition 2.1.

A subset $A\subseteq E$ is called a homogeneous acyclic matching of $I$ if the
following conditions hold:

  1. (1)

(_matching_) Each cell appears in, at most, one edge of $A$.

  2. (2)

(_acyclicity_) The graph $G^{A}_{X}$ does not contain a directed cycle.

  3. (3)

(_homogeneity_) For any edge $(\sigma,\sigma^{\prime})$ in $A$, we have
$\operatorname{lcm}(\sigma)=\operatorname{lcm}(\sigma^{\prime})$.

A cell $\sigma$ that does not appear in any edge of $A$ is called an
$A$-critical cell of $I$. In contexts where there is no ambiguity, we simply
refer to it as critical.

Recall that $X$ is a $\mathbb{Z}^{N}$-graded complex that induces a free
resolution $\mathcal{F}$ where $\mathcal{F}_{r}$ is the free $R$-module with a
basis indexed by all cells of cardinality $r$. The differentials
$\partial_{r}\colon\mathcal{F}_{r}\to\mathcal{F}_{r-1},$ are defined as

$\partial_{r}(\sigma)=\sum_{\begin{subarray}{c}\sigma^{\prime}\subseteq\sigma,\\\
|\sigma^{\prime}|=r-1\end{subarray}}[\sigma:\sigma^{\prime}]\frac{\operatorname{lcm}(\sigma)}{\operatorname{lcm}(\sigma^{\prime})}\sigma^{\prime}.$

where $[\sigma:\sigma^{\prime}]$ denotes the coefficient of $\sigma^{\prime}$
in the boundary of $\sigma$ and is either $1$ or $-1$. The complex
$\mathcal{F}$ is called the Taylor resolution of $R/I$.

Morse resolutions are refinements of the Taylor resolution. Each homogeneous
acyclic matching yields a Morse resolution, and these resolutions may
coincide. To precisely define the differentials of a Morse resolution, we need
to introduce some additional terminology.

Given a directed edge $(\sigma,\sigma^{\prime})\in E(G^{A})$, set

$m(\sigma,\sigma^{\prime})=\begin{cases}-[\sigma^{\prime}:\sigma]&\text{if
}(\sigma^{\prime},\sigma)\in A,\\\
[\sigma:\sigma^{\prime}]&\text{otherwise}.\end{cases}$

A gradient path $\mathcal{P}$ from $\sigma_{1}$ to $\sigma_{t}$ is a directed
path

$\mathcal{P}\colon\sigma_{1}\to\sigma_{2}\to\cdots\to\sigma_{t}$

in $G^{A}_{X}$. Similarly, set $m(\mathcal{P})=m(\sigma_{1},\sigma_{2})\cdot
m(\sigma_{2},\sigma_{3})\cdots m(\sigma_{t-1},\sigma_{t}).$

We are now ready to recall Morse resolutions of monomial ideals.

######  Theorem 2.2.

[4, Proposition 2.2, Proposition 3.1, Lemma 7.7] Let $A$ be a homogeneous
acyclic matching of $I$. Then $A$ induces a cellular resolution
$\mathcal{F}_{A}$, where:

  * •

$(\mathcal{F}_{A})_{r}$ is the free $R$-module with a basis indexed by all
critical cells of cardinality $r$.

  * •

The differentials
$\partial_{r}^{A}:(\mathcal{F}_{A})_{r}\to(\mathcal{F}_{A})_{r-1}$ are defined
by:

$\partial_{r}^{A}(\sigma)=\sum_{\begin{subarray}{c}\sigma^{\prime}\subseteq\sigma,\\\
|\sigma^{\prime}|=r-1\end{subarray}}[\sigma:\sigma^{\prime}]\sum_{\begin{subarray}{c}\sigma^{\prime\prime}\text{
is critical},\\\
|\sigma^{\prime\prime}|=r-1\end{subarray}}\sum_{\mathcal{P}\text{ is a
gradient path }\\\ \text{from }\sigma^{\prime}\text{ to
}\sigma^{\prime\prime}}m(\mathcal{P})\frac{\operatorname{lcm}(\sigma)}{\operatorname{lcm}(\sigma^{\prime\prime})}\sigma^{\prime\prime}.$

The resulting (cellular) free resolution $\mathcal{F}_{A}$ is called the
_Morse resolution_ of $R/I$ associated to $A$. Furthermore, $\mathcal{F}_{A}$
is minimal if for any two $A$-critical cells $\sigma$ and $\sigma^{\prime}$
with $|\sigma^{\prime}|=|\sigma|-1$, we have
$\operatorname{lcm}(\sigma)\neq\operatorname{lcm}(\sigma^{\prime})$.

###  2.2. Barile-Macchia matchings and Barile-Macchia resolutions.

In this subsection, we recall Barile-Macchia matchings and resolutions. To
ensure that this paper is self-contained and accessible, we present relevant
definitions and results from [8] that are instrumental to our discussions.

Given Theorem 2.2, the key to producing a Morse resolution of $R/I$ is finding
a homogeneous acyclic matching of $I$. However, systematically crafting such a
matching for any monomial ideal is a challenging task. A recent development in
this direction is the Barile-Macchia algorithm, as presented in [8, Theorem
2.8], which produces a homogeneous acyclic matching. In the context of [8], a
matching arising from this algorithm is called a Barile-Macchia matching, and
the resulting resolution is called a Barile-Macchia resolution.

Below, we recall some of the terminology useful for discussing Barile-Macchia
resolutions. Throughout this section, we fix a total order $(\succ_{I})$ on
${\mathcal{G}}(I)$. For simplicity, we write $S\setminus s$ (resp, $S\cup s$)
instead of $S\setminus\\{s\\}$ (resp, $S\cup\\{s\\}$) where $S$ is a set and
$s\in S$ (resp, $s\notin S$).

Additionally, throughout our discussion, we use the terms “subsets of
${\mathcal{G}}(I)$” and “cells of $I$” interchangeably. By cells of $I$, we
refer to cells of the corresponding Taylor simplex $X$.

######  Definition 2.3.

Let $\sigma$ be a subset of ${\mathcal{G}}(I)$. A monomial $m$ is called a
bridge of $\sigma$ if $m\in\sigma$ and removing $m$ from $\sigma$ does not
change the $\operatorname{lcm}$, i.e., $\operatorname{lcm}(\sigma\setminus
m)=\operatorname{lcm}(\sigma)$.

######  Notation 2.4.

If $\sigma$ has a bridge, the notation $\operatorname{sb}(\sigma)$ denotes the
smallest bridge of $\sigma$ with respect to $(\succ_{I})$. We set
$\operatorname{sb}(\sigma)=\emptyset$ if $\sigma$ has no bridges.

To fully understand the terms and context we discuss, in Algorithm 1 we recall
the Barile-Macchia algorithm as outlined in [8, Algorithm 2.9]. The Barile-
Macchia Algorithm systematically constructs a matching in $G_{X}$ based on the
concept of bridges within subsets on ${\mathcal{G}}(I)$ and a fixed total
order $(\succ_{I})$ of ${\mathcal{G}}(I)$. As it was shown in [8, Theorem
2.11], this process always produces a homogeneous acyclic matching.

Algorithm 1 Barile-Macchia Algorithm

Input: A total order $(\succ_{I})$ on ${\mathcal{G}}(I)$.
Output: Set of directed edges $A$ in $G_{X}$.

1:$A\leftarrow\emptyset$

2:$\Omega\leftarrow\\{\text{all subsets of }{\mathcal{G}}(I)\text{ with
cardinality at least }3\\}$

3:while $\Omega\neq\emptyset$ do

4: Pick $\sigma\in\Omega$ with maximal cardinality

5: Remove $\\{\sigma,\sigma\setminus\operatorname{sb}(\sigma)\\}$ from
$\Omega$

6: if $\operatorname{sb}(\sigma)\neq\emptyset$ then

7: Add edge $(\sigma,\sigma\setminus\operatorname{sb}(\sigma))$ to $A$

8:for all distinct edges $(\sigma,\tau)$ and $(\sigma^{\prime},\tau^{\prime})$
in $A$ with $\tau=\tau^{\prime}$ do

9: if $\operatorname{sb}(\sigma^{\prime})\succ_{I}\operatorname{sb}(\sigma)$
then

10: Remove $(\sigma^{\prime},\tau^{\prime})$ from $A$

11: else

12: Remove $(\sigma,\tau)$ from $A$ return $A$

######  Definition 2.5.

Given a Barile-Macchia matching $A$ of $I$:

  1. (1)

For any edge $(\sigma,\tau)$ in the final $A$ from Algorithm 1 , the cell
$\sigma$ is called type-2 while the cell $\tau$ is called type-1.

  2. (2)

During the execution of Algorithm 1, if a directed edge $(\sigma,\tau)$ is
added to $A$, the cell $\sigma$ is called potentially-type-2. It is important
to note that this edge may not persist in the final $A$ produced by Algorithm
1.

######  Remark 2.6.

If a subset of ${\mathcal{G}}(I)$ has a bridge, then it is either type-1 or
potentially-type-2.

In earlier discussions, we emphasized the one-to-one correspondence between
the ranks of the free $R$-modules in a Barile-Macchia resolution of $R/I$ and
the $A$-critical cells of $I$. Here, $A$ represents the Barile-Macchia
matching of $I$ with respect to $(\succ_{I})$. Thus, delving into the nature
of $A$-critical cells, along with the remaining cells, is crucial for a deeper
comprehension of the Barile-Macchia resolution of $R/I$. It is important to
point out that critical cells of $I$ consists of those cells either left out
of $A$ during the Barile-Macchia Algorithm or added initially but later
excluded in the final refinement of the algorithm. We name these critical
cells as follows:

######  Definition 2.7.

Let $\sigma$ be a subset of ${\mathcal{G}}(I)$. If $\sigma$ is never added to
$A$ in any of the steps throughout Algorithm 1, it is called absolutely
critical. If $\sigma$ is potentially-type-2 but not type-2 (initially added to
$A$ but removed in the final $A$ produced by Algorithm 1), we call it
fortunately critical.

The following concepts from [8] are useful in characterizing critical and non-
critical cells of $I$.

######  Definition 2.8.

[8, Definition 2.19] Let $m,m^{\prime}\in{\mathcal{G}}(I)$ and $\sigma$ be a
subset of ${\mathcal{G}}(I)$.

  1. (1)

We say that $m$ dominates $m^{\prime}$ if and only if $m\succ_{I}m^{\prime}$.

  2. (2)

The monomial $m$ is called a gap of $\sigma$ if $m\notin\sigma$, and
$\operatorname{lcm}(\sigma\cup m)=\operatorname{lcm}(\sigma)$.

  3. (3)

The monomial $m$ is called a true gap of $\sigma$ if it is a gap of $\sigma$,
and any bridge $m^{\prime}$ of $\sigma\cup m$ where $m\succ_{I}m^{\prime}$ is
also a bridge of $\sigma$.

In [8, Theorem 2.24], the first two authors characterized type-1, potentially-
type-2, and type-2 cells utilizing the concepts of bridges and true gaps. We
focus primarily on the characterization of potentially-type-2 cells.
Therefore, we present a slightly adjusted version of [8, Theorem 2.24 (b)] in
the following.

######  Remark 2.9.

In a Barile-Macchia matching of $I$, a cell $\sigma$ is potentially-type-2 if
and only if $m\succ_{I}\operatorname{sb}(\sigma)$ for each true gap $m$ of
$\sigma$.

Based on the characterizations of type-1 and potentially-type-2 cells in [8,
Theorem 2.24], we provide the following characterization of absolutely
critical cells:

######  Corollary 2.10.

A cell is absolutely critical if and only if it has neither bridges nor true
gaps.

In [8], the first two authors introduced a pivotal class of ideals called
”bridge-friendly” for analyzing the minimality of Barile-Macchia resolutions.
As established in [8, Theorem 2.29], bridge-friendliness is a sufficient
condition for an ideal to have a minimal Barile-Macchia resolution. We revisit
the definition of this class of ideals using the concept of absolutely
critical cells.

######  Definition 2.11.

[8, Definition 2.27] A monomial ideal $I$ is called bridge-friendly if, for
some total order $\succ_{I}$ on ${\mathcal{G}}(I)$, every potentially-type-2
cell is of type-2 with respect to ($\succ_{I}$). Equivalently, all
$A$-critical cells of $I$ are absolutely critical. Here, $A$ represents the
Barile-Macchia matching of $I$ with respect to ($\succ_{I}$).

In the next chapter, we study a class of ideals with the bridge-friendly
property.

##  3\. Minimal free resolutions of path ideals of paths

In this section, our primary goal is to investigate the bridge-friendliness
and, consequently, the minimal Barile-Macchia resolutions of path ideals of
paths.

Consider a path $L$ on the vertices $\\{x_{1},\ldots,x_{n+p-1}\\}$. Let
$R=\Bbbk[x_{1},\ldots,x_{N}]$ with $N=n+p-1$. The $p$-path ideal of $L$,
denoted as $I_{p}(L_{N})$, is generated by monomials in $R$ corresponding to
paths on $p$ vertices along $L$. Explicitly, we have:

$I_{p}(L_{N})=(x_{1}x_{2}\cdots x_{p},\ x_{2}x_{3}\cdots x_{p+1},\ \cdots,\
x_{n}x_{n+1}\cdots x_{n+p-1}).$

######  Remark 3.1.

Path ideals can be viewed as an extension of edge ideals of graphs.
Specifically, the $2$-path ideal of a graph coincides with its edge ideal.

The set of minimal generators of the $p$-path ideal of $L$ is

${\mathcal{G}}(I_{p}(L_{N}))=\\{m_{1},m_{2},\ldots,m_{n}\\}$

where $m_{i}:=x_{i}x_{i+1}\cdots x_{i+p-1}$ for each $1\leq i\leq n$. Fix a
total order $(\succ)$ on ${\mathcal{G}}(I_{p}(L_{N}))$ such that

$m_{1}\succ m_{2}\succ\cdots\succ m_{n}.$

Throughout the rest of this chapter, our focus is on the monomial ideal
$I_{p}(L_{N})$. In particular, we examine its Barile-Macchia matching and
resolution with respect to the aforementioned total order. For ease of
readability, we introduce the following notation, which is consistently
employed throughout the paper.

######  Notation 3.2.

For each $1\leq i\leq n$, let $M_{i}$ denote the set of all monomials in
${\mathcal{G}}(I_{p}(L_{N}))$ that are divisible by $x_{i}$. In other words,

$M_{i}=\\{m_{i-p+1},\ldots,m_{i-1},m_{i}\\}.$

Note that there are less than $p$ monomials in $M_{i}$ when $i\leq p-1$.

We begin our analysis with the following lemma, which serves as a foundational
tool for the classification of bridges, gaps, and true gaps of a subset
$\sigma$ of ${\mathcal{G}}(I_{p}(L_{N}))$.

######  Lemma 3.3.

Let $\sigma$ be a cell of $I_{p}(L_{N})$ and consider a monomial
$m_{i}\in{\mathcal{G}}(I_{p}(L_{N}))$ such that $m_{i}\notin\sigma$. Assume
that $\operatorname{lcm}(\sigma)$ is divisible by $m_{i}$. Then there exist
monomials $m_{j}\in\sigma\cap M_{i}$ and $m_{k}\in\sigma\cap M_{i+p-1}$ such
the distance between these monomials along $m_{i}$ is at most $p$, i.e.,
$k-j\leq p$.

###### Proof.

Since $\operatorname{lcm}(\sigma)$ is divisible by $m_{i}$, there exist
monomials $m_{j}\in\sigma\cap M_{i}$ and $m_{k}\in\sigma\cap M_{i+p-1}$. Pick
the largest such $j$ and smallest such $k$. Then any factor of $m_{i}$ divides
either $m_{j}$ or $m_{k}$. In other words, $m_{i}$ divides
$\operatorname{lcm}(m_{j},m_{k})$. For the sake of contradiction, assume that
$k-j>p$. Consider the variable $x_{j+p}$. Note that $x_{j+p}$ does not divide
neither $m_{j}$ nor $m_{k}$, where the latter comes from the assumption that
$j+p<k$. On the other hand, since $m_{j}\in M_{i}$ and $m_{k}\in M_{i+p-1}$,
we have $i+1\leq j+p<k\leq i+p-1$, which implies that $x_{j+p}$ divides
$m_{i}$, a contradiction. Thus, we conclude that $k-j\leq p$. ∎

From Lemma 3.3, we can directly derive the characterization of bridges and
gaps. It is important to note that in any cell $\sigma$ of $I_{p}(L_{N})$,
neither $m_{1}$ nor $m_{n}$ can be a gap or a bridge.

######  Proposition 3.4.

Let $\sigma$ be a cell of $I_{p}(L_{N})$. For a monomial
$m_{i}\in{\mathcal{G}}(I_{p}(L_{N}))$ with $1<i<n$, the following statements
hold:

  1. (1)

$m_{i}$ is a bridge of $\sigma$ if and only if $m_{i}\in\sigma$ and there
exist monomials $m_{j}\in\sigma\cap M_{i}$ and $m_{k}\in\sigma\cap M_{i+p-1}$
such that $j<i<k$ and $k-j\leq p$.

  2. (2)

$m_{i}$ is a gap of $\sigma$ if and only if $m_{i}\notin\sigma$ and there
exist monomials $m_{j}\in\sigma\cap M_{i}$ and $m_{k}\in\sigma\cap M_{i+p-1}$
such that $k-j\leq p$. In particular, we have $j<i<k$.

######  Remark 3.5.

The monomials $m_{j}$ and $m_{k}$, which are highlighted in the proof of Lemma
3.3 and mentioned in the statement of Proposition 3.4, can be chosen to be
those closest to $m_{i}$ among the monomials in $\sigma\setminus m_{i}$ in
terms of their distance to $m_{i}$.

We present a characterization of true gaps below, which proves to be more
intricate than the characterization of bridges and gaps.

######  Proposition 3.6.

Let $\sigma$ be a cell of $I_{p}(L_{N})$ and consider a monomial
$m_{i}\in{\mathcal{G}}(I_{p}(L_{N}))$ where $1<i<n$. Assume $m_{i}$ does not
dominate any bridges of $\sigma$. Then the monomial $m_{i}$ is a true gap of
$\sigma$ if and only if the following statements hold:

  1. (a)

$m_{i}\notin\sigma$ and there exist monomials $m_{j}\in\sigma\cap M_{i}$ and
$m_{k}\in\sigma\cap M_{i+p-1}$ such that $k-j\leq p$.

  2. (b)

$\sigma\cap M_{i+p-1}=\\{m_{k}\\}$ where $m_{k}$ is the element from part (a).

  3. (c)

If $i+p\leq n$, then $m_{i+p}\notin\sigma$.

###### Proof.

For the forward direction, suppose $m_{i}$ is a true gap of $\sigma$. In order
to prove (a), (b), and (c), it is useful to recall that $m_{i}$ is a true gap
$\sigma$ that does not dominate any bridges of $\sigma$ if and only if $m_{i}$
is a gap of $\sigma$ and $\operatorname{sb}(\sigma\cup m_{i})=m_{i}$ by [8,
Proposition 2.21].

  1. (a)

By the definition of a true gap, $m_{i}$ is a gap. Hence, (a) holds by
Proposition 3.4 (2).

  2. (b)

It follows from (a) that $m_{k}\in\sigma\cap M_{i+p-1}$ for $i<k\leq n$.
Suppose, for the sake of contradiction, that there exists another monomial
$m_{t}$ in $\sigma\cap M_{i+p-1}$. Referring to Remark 3.5, without loss of
generality, we may assume $k$ is the smallest index of a monomial in
$\sigma\cap M_{i+p-1}$. So, $k<t$. Observe that $m_{k}$ divides
$\operatorname{lcm}(m_{i},m_{t})$, implying that $m_{k}$ is a bridge of
$\sigma\cup m_{i}$. This contradicts the condition
$\operatorname{sb}(\sigma\cup m_{i})=m_{i}$. Thus, $m_{k}$ is the only
monomial in $\sigma\cap M_{i+p-1}$.

  3. (c)

Assume $i+p\leq n$. Since $\operatorname{sb}(\sigma\cup m_{i})=m_{i}$, the
monomial $m_{k}$ is not a bridge of $\sigma\cup m_{i}$. In particular, this
implies that $m_{i+p}\notin\sigma$.

For the other direction, assume the conditions (a), (b), and (c) hold. For
contradiction, suppose that $m_{i}$ is not a true gap of $\sigma$. Since
$m_{i}$ is a gap of $\sigma$ by (a), $m_{i}$ is not a true gap of $\sigma$ if
and only if $m_{i}\succ m_{t}\coloneqq\operatorname{sb}(\sigma\cup m_{i})$ by
[8, Proposition 2.21]. In particular, this implies that $i<t$. Since $m_{i}$
does not dominate any bridges of $\sigma$, $m_{t}$ is not a bridge of
$\sigma$. According to Proposition 3.4 (1), we have $m_{t}\in\sigma$, and
there are monomials $m_{j_{t}},m_{k_{t}}\in\sigma\cup m_{i}$ such that
$j_{t}<t<k_{t}$ and $k_{t}-j_{t}\leq p$. Since $m_{t}$ is not a bridge of
$\sigma$, either $j_{t}$ or $k_{t}$ has to be $i$. Given that $i<t$, the
feasible scenario is $j_{t}=i$. This situation means that $m_{t}$ is an
element of $\sigma\cap M_{i+p-1}$ as $i<t<k_{t}\leq i+p$. The condition (b)
implies that $m_{k_{t}}$ is in $\sigma$ but is not contained in $M_{i+p-1}$.
Hence, $k_{t}=i+p$, which leads to $m_{k_{t}}=m_{i+p}\in\sigma$, a direct
contradiction to condition (c). As a result, $m_{i}$ must indeed be a true gap
of $\sigma$. ∎

######  Example 3.7.

Consider the $3$-path ideal of an $8$-path,

$I=(x_{1}x_{2}x_{3},x_{2}x_{3}x_{4},x_{3}x_{4}x_{5},x_{4}x_{5}x_{6},x_{5}x_{6}x_{7},x_{6}x_{7}x_{8}).$

Consider the subset $\sigma=\\{m_{1},m_{4},m_{6}\\}$. It has no bridges, and
its gaps are $m_{2},m_{3}$, and $m_{5}$. Using Proposition 3.6, we identify
which among these gaps are true gaps.

  * •

The monomial $m_{2}$ is a true gap of $\sigma$. This is confirmed by: (a) the
observation that $m_{2}\notin\sigma$ and $m_{1},m_{4}\in\sigma$, with the
difference in their indices satisfying $4-1\leq 3$; (b) within the set
$M_{4}=\\{m_{2},m_{3},m_{4}\\}$, only $m_{4}$ belongs to $\sigma$; and (c) the
monomial $m_{2+3}=m_{5}$ is not in $\sigma$.

  * •

The monomial $m_{3}$ is not a true gap of $\sigma$. This is due to the failure
of part (c) of Proposition 3.6 (3), given that $m_{6}\in\sigma$.

  * •

The monomial $m_{5}$ is a true gap of $\sigma$ based on: (a)
$m_{5}\notin\sigma$ and both $m_{4}$ and $m_{6}$ are in $\sigma$, satisfying
$6-4\leq 3$; (b) from the set $M_{7}=\\{m_{5},m_{6}\\}$, only $m_{6}$ is in
$\sigma$. Furthermore, (c) is not applicable since $8>6$, and thus,
$m_{5+3}=m_{8}$ does not exist.

In what follows, we show that path ideals of paths are bridge-friendly.

######  Theorem 3.8.

The path ideal $I_{p}(L_{N})$ is bridge-friendly, and its Barile-Macchia
resolution is minimal.

###### Proof.

To demonstrate the bridge-friendliness of $I_{p}(L_{N})$, we use [8, Lemma
2.33]. Specifically, our goal is to establish that, for any potentially-type-2
cell $\sigma$ (should it exist), there does not exist an
$m\in{\mathcal{G}}(I_{p}(L_{N}))$ such that $m$ is a true gap of
$\sigma\setminus\operatorname{sb}(\sigma)$ and that
$\operatorname{sb}(\sigma)\succ m$.

If $I_{p}(L_{N})$ lacks potentially-type-2 cells, then its Taylor resolution
is inherently minimal, making the path ideal bridge-friendly. So, we may
assume $\sigma$ has potentially-type-2 cells. Now, consider a potentially-
type-2 cell $\sigma$. This means $\operatorname{sb}(\sigma)$ exists. Note that
$\operatorname{sb}(\sigma)\neq m_{n}$ since $m_{n}$ cannot be a bridge of
$\sigma$. Then, there exists a monomial $m_{i}\in{\mathcal{G}}(I_{p}(L_{N}))$
satisfying $\operatorname{sb}(\sigma)\succ m_{i}$. We claim that $m_{i}$ is
not a true gap of $\sigma\setminus\operatorname{sb}(\sigma)$. Suppose, for the
sake of contradiction, that $m_{i}$ is a true gap. We first note that $m_{i}$
does not dominate any bridges of $\sigma$ due to the fact that
$\operatorname{sb}(\sigma)\succ m_{i}$. Hence, by Proposition 3.6, we deduce
that $m_{i}$ is a true gap of $\sigma$. The detailed justification is omitted
as it directly arises from the application of Proposition 3.6 to our current
hypothesis - that $m_{i}$ is a true gap of
$\sigma\setminus\operatorname{sb}(\sigma)$, combined with our other hypothesis
that $\operatorname{sb}(\sigma)\succ m_{i}$. This scenario implies that every
bridge of $\sigma$ dominates a true gap, contradicting Remark 2.9 since
$\sigma$ is potentially-type-2. Thus, no such $m_{i}$ exists, and by [8, Lemma
2.33], the path ideal $I_{p}(L_{N})$ is bridge-friendly.

The minimality of the Barile-Macchia resolution is a direct consequence of [8,
Theorem 2.26]. ∎

In the subsequent discussion, our goal is to demonstrate that for every
multidegree $m$, i.e., a monomial in $R$, there exists at most one critical
cell $\sigma$ such that $\operatorname{lcm}(\sigma)=m$. Establishing the
uniqueness of this critical cell for each multidegree allows us to compute
both the projective dimension and Betti numbers of the path ideal via its
Barile-Macchia resolution. To pave the way for this claim, we first introduce
several auxiliary lemmas addressing the necessary technical details.

Before delving into these lemmas, it is worth recalling [8, Corollary 2.28]
which characterizes the critical cells of $I_{p}(L_{N})$ by their lack of
bridges and true gaps, a consequence of $I_{p}(L_{N})$ being bridge-friendly.

######  Lemma 3.9.

Let $\sigma$ be a critical cell of $I_{p}(L_{N})$, and let $x_{i}$ be a
variable that divides $\operatorname{lcm}(\sigma)$. If none of the monomials
$m_{k}$ with $k\in\\{i-p+2,\ldots,i\\}$ are in $\sigma$, then
$m_{i-p+1}\in\sigma$.

###### Proof.

Note that $\sigma\cap M_{i}\neq\emptyset$ since $x_{i}$ divides
$\operatorname{lcm}(\sigma)$. Given the conditions of the lemma, it is
immediate that $m_{i-p+1}\in\sigma$. ∎

######  Lemma 3.10.

Let $\sigma$ be a critical cell of $I_{p}(L_{N})$, and
$b\in\mathbb{N}\cup\\{\infty\\}$. Assume that there exists an integer
$b^{\prime}\leq b-p-1$ such that among the set of monomials
$\\{m_{b^{\prime}-p+1},m_{b^{\prime}-p+2},\ldots,m_{b-p-1}\\}$, only
$m_{b^{\prime}-p+1}$ belongs to $\sigma$. Then the following statements hold:

  1. (1)

No monomial in the set $\\{m_{b^{\prime}-2p+1},\ldots,m_{b^{\prime}-p-1}\\}$
belongs to $\sigma$.

  2. (2)

$\operatorname{lcm}(\sigma)$ is divisible by $x_{b^{\prime}-p}$ if and only if
$m_{b^{\prime}-p}$ belongs to $\sigma$.

###### Proof.

Suppose, for the sake of contradiction, that (1) does not hold, i.e., there
exists a monomial $m_{k}$ in
$\sigma\cap\\{m_{b^{\prime}-2p+1},\ldots,m_{b^{\prime}-p-1}\\}$. Under this
assumption, we show that $m_{b^{\prime}-p}$ is either a bridge or a true gap
of $\sigma$ which leads to a contradiction because $I_{p}(L_{N})$ is bridge-
friendly and $\sigma$ has no bridges or true gaps by [8, Corollary 2.28].

Observe that $m_{b^{\prime}-p}$ divides $\operatorname{lcm}(\sigma\setminus
m_{b^{\prime}-p})$. This is because both $m_{k}$ and $m_{b^{\prime}-p+1}$ are
in $\sigma$ and the difference $(b^{\prime}-p+1)-k$ is at most $p$. Hence,
$m_{b^{\prime}-p}$ is either a bridge or a gap of $\sigma$ by Proposition 3.4.
If $m_{b^{\prime}-p}$ is a bridge of $\sigma$, we are done as discussed above.
So, assume it is a gap of $\sigma$. Note that $m_{b^{\prime}-p}$ does not
dominate any bridges of $\sigma$. Under these assumptions, $m_{b^{\prime}-p}$
is a true gap of $\sigma$ by Proposition 3.6 as its conditions (a), (b) and
(c) are met:

  1. (a)

$m_{b^{\prime}-p}$ is a gap of $\sigma$.

  2. (b)

It follows from the hypothesis of the lemma that the only monomial in
$\sigma\cap M_{b^{\prime}}$ is $m_{b^{\prime}-p+1}$.

  3. (c)

Following the previous point, it is clear that $m_{b^{\prime}}$ does not
belong to $\sigma$ (assuming it exists).

This completes the proof of (1). In order to prove (2), observe that
$m_{b^{\prime}-p}\in\sigma$ implies $x_{b^{\prime}-p}$ divides
$\operatorname{lcm}(\sigma)$. On the other hand, if
$\operatorname{lcm}(\sigma)$ is divisible by $x_{b^{\prime}-p}$, then
$\sigma\cap M_{b^{\prime}-p}$ is nonempty. By (1), the only such monomial in
$\sigma\cap M_{b^{\prime}-p}$ is $m_{b^{\prime}-p}$. ∎

######  Lemma 3.11.

Let $\sigma$ be a critical cell of $I_{p}(L_{N})$ and
$b\in\mathbb{N}\cup\\{\infty\\}$. Define the number $b^{\prime}$ as:

$b^{\prime}\coloneqq\begin{cases}\sup\\{i\colon
x_{i}\mid\operatorname{lcm}(\sigma)\text{ and }i\leq b-p-1\\}&\text{if such
}i\text{ exists},\\\ -\infty&\text{otherwise}.\end{cases}$

Assume none of the monomials in the set $\\{m_{b-2p+1},\ldots,m_{b-p-1}\\}$
belong to $\sigma$ when $b$ is finite. Then among the monomials from the set
$\\{m_{b^{\prime}-p+1},m_{b^{\prime}-p+2},\ldots,m_{b-p-1}\\}$, only
$m_{b^{\prime}-p+1}$ belongs to $\sigma$.

###### Proof.

We analyze the two possible cases for $b$ separately.

Case 1: $b=\infty$. By the definition of $b^{\prime}$, it is clear that
$b^{\prime}$ is finite. Given this, for any integer $k\geq b^{\prime}+1$,
$\operatorname{lcm}(\sigma)$ is not divisible by $x_{k}$. This immediately
implies that $\sigma\cap\\{m_{b^{\prime}-p+2},\ldots,m_{k}\\}$ is empty.
Moreover, as $x_{b^{\prime}}$ divides $\operatorname{lcm}(\sigma)$, Lemma 3.9
guarantees that $m_{b^{\prime}-p+1}$ belongs to $\sigma$.

Case 2: $b\in\mathbb{N}$. From the definition of $b^{\prime}$, it is
straightforward to see that $b^{\prime}\leq b-p-1$. If $b^{\prime}<b-p-1$, it
can be deduced that $\operatorname{lcm}(\sigma)$ is not divisible by any
variable from the set $\\{x_{b^{\prime}+1},\ldots,x_{b-p-1}\\}$. Consequently,
$\sigma\cap\\{m_{b^{\prime}-p+2},\ldots,m_{b-p-1}\\}$ is empty. With the
divisibility of $\operatorname{lcm}(\sigma)$ by $x_{b^{\prime}}$, Lemma 3.9
indicates that $m_{b^{\prime}-p+1}$ is indeed in $\sigma$. On the other hand,
if $b^{\prime}=b-p-1$, then by the definition of $b^{\prime}$, we know that
$x_{b-p-1}$ divides $\operatorname{lcm}(\sigma)$. Since
$\sigma\cap\\{m_{b-2p+1},\ldots,m_{b-p-1}\\}$ is empty by the assumption of
this lemma, the monomial $m_{b^{\prime}-p+1}=m_{b-2p}$ belongs to $\sigma$ by
Lemma 3.9. ∎

Building upon the preceding three lemmas, we introduce a sequence that is key
to understanding the content of a given critical subset $\sigma$. Here, by
“content of $\sigma$”, we refer to the collection of monomials from
${\mathcal{G}}(I_{p}(L_{N}))$ that are in $\sigma$.

######  Proposition 3.12.

For a critical cell $\sigma$ of $I_{p}(L_{N})$, define the sequence
$\\{b_{i}\\}_{i\in\mathbb{N}\cup\\{0\\}}$ by setting $b_{0}=\infty$ and, for
$i\geq 1$,

$b_{i}=\sup\\{j:x_{j}\mid\operatorname{lcm}(\sigma)\text{ and }j\leq
b_{i-1}-p-1\\},$

with $b_{i}=-\infty$ if no such $j$ exists. Then:

  1. (1)

There exists $l\geq 1$ such that $b_{l}\neq-\infty$ and $b_{l+1}=-\infty$.

  2. (2)

When $1\leq i\leq l$, only $m_{b_{i}-p+1}$ from the set
$\\{m_{b_{i}-p+1},\ldots,m_{b_{i-1}-p-1}\\}$ is in $\sigma$.

  3. (3)

When $i=l+1$, none of the monomials $m_{k}$ with $k\leq b_{l}-p-1$ are in
$\sigma$.

###### Proof.

Given that $\sigma$ is a critical cell of $I_{p}(L_{N})$, it is non-empty,
ensuring $b_{1}$ is finite. As the sequence $\\{b_{i}\\}$ decreases strictly
after $b_{1}$ until it reaches $-\infty$, there must be an $l\geq 1$ for which
$\\{i:x_{i}\mid\operatorname{lcm}(\sigma)\text{ and }i\leq
b_{l-1}-p-1\\}\neq\emptyset$ but
$\\{i:x_{i}\mid\operatorname{lcm}(\sigma)\text{ and }i\leq
b_{l}-p-1\\}=\emptyset$. This confirms that $b_{l}\neq-\infty$ and
$b_{l+1}=-\infty$.

For $1\leq i\leq l$, we use induction on $i$. Lemma 3.11 covers our base case.
For a fixed $i<l$, suppose
$\sigma\cap\\{m_{b_{j}-p+2},\ldots,m_{b_{j-1}-p-1}\\}$ is empty and
$m_{b_{j}-p+1}\in\sigma$ for each $j\leq i$. The induction hypothesis implies
that only $m_{b_{i}-p+1}$ from the set
$\\{m_{b_{i}-p+1},\ldots,m_{b_{i-1}-p-1}\\}$ belongs to $\sigma$. By Lemma
3.10, $\sigma\cap\\{m_{b_{i}-2p+1},\ldots,m_{b_{i}-p-1}\\}$ is empty. Given
the constraints on $b_{i}$ and $b_{i+1}$, Lemma 3.11 implies that only
$m_{b_{i+1}-p+1}$ from the set $\\{m_{b_{i+1}-p+1},\ldots,m_{b_{i}-p-1}\\}$ is
in $\sigma$.

For the case $i=l+1$, note that no $x_{i}$ divides
$\operatorname{lcm}(\sigma)$ for $i\leq b_{l}-p-1$. Thus,
$\operatorname{lcm}(\sigma)$ is not divisible by any variable $x_{k}$ with
$k\leq b_{l}-p-1$. This implies that $\sigma\cap\\{m_{k-p+1},\ldots,m_{k}\\}$
is empty for any $k$. Hence, $m_{k}$ is not in $\sigma$ for any $k\leq
b_{l}-p-1$. ∎

######  Proposition 3.13.

For two critical cells $\sigma$ and $\sigma^{\prime}$ of $I_{p}(L_{N})$, if
$\operatorname{lcm}(\sigma)=\operatorname{lcm}(\sigma^{\prime})$, then
$\sigma=\sigma^{\prime}$.

###### Proof.

Assume that $\operatorname{lcm}(\sigma)=\operatorname{lcm}(\sigma^{\prime})$.
Then, the decreasing sequences defined for $\sigma$ and $\sigma^{\prime}$ in
Proposition 3.12 are identical. This is because the values of $b_{i}$ of a
cell are determined solely by the least common multiple of the cell.

Based on Proposition 3.12, the monomial $m_{{b_{i}}-p+1}$ is present in both
$\sigma$ and $\sigma^{\prime}$ for all $1\leq i\leq l$. The only remaining
monomials in ${\mathcal{G}}(I_{p}(L_{N}))$ that could potentially belong to
either $\sigma$ or $\sigma^{\prime}$ are those denoted by
$m_{{b_{1}}-p},\ldots,m_{{b_{l}}-p}$. We can justify this claim with both
Proposition 3.12 and the following decomposition of the real line:

$\displaystyle\mathbb{R}=$
$\displaystyle\,(b_{l+1},b_{l}]\cup\left(\bigcup_{i=1}^{l-1}(b_{i+1},b_{i}]\right)\cup(b_{1},b_{0})$
$\displaystyle=$
$\displaystyle\,(b_{l+1},b_{l}-p+1]\cup\left(\bigcup_{i=1}^{l-1}(b_{i+1}-p+1,b_{i}-p+1]\right)\cup(b_{1}-p+1,b_{0}).$

Considering Lemma 3.10, which states that $m_{b_{i}-p}$ is contained in the
critical cell $\tau$ if and only if $x_{b_{i}-p}$ divides
$\operatorname{lcm}(\tau)$ for $\tau\in\\{\sigma,\sigma^{\prime}\\}$, we
conclude that $\sigma=\sigma^{\prime}$. ∎

From the preceding proposition, we deduce that the least common multiples of
distinct critical cells of $I_{p}(L_{N})$ are different. This particularly
implies the following information on its multi-graded Betti numbers.

######  Corollary 3.14.

For any monomial $m$ and any integer $i$, we have

$\beta_{i,m}(R/I_{p}(L_{N}))=\begin{cases}1&\text{ if there exists a critical
subset of cardinality }i\text{ whose lcm equals }m,\\\ 0&\text{
otherwise.}\end{cases}$

From the characterization of critical cells in Proposition 3.12 and the
insights from the proof of Proposition 3.13, we can deduce the projective
dimension of $I_{p}(L_{n+p-1})$. Specifically, among every collection of $p+1$
consecutive monomials, at most two can be in a critical cell. Consequently, we
can derive the maximal cardinality of a critical cell, thus obtaining the
projective dimension.

######  Corollary 3.15.

Let $n$ be expressed as $n=(p+1)q+s$ where $0\leq s\leq p$. The projective
dimension of $R/I_{p}(L_{n+p-1})$ is given by:

$\operatorname{pdim}(R/I_{p}(L_{n+p-1}))=\begin{cases}2q&\text{if }s=0,\\\
2q+1&\text{if }s=1,\\\ 2q+2&\text{otherwise}.\end{cases}$

A formula for the projective dimension of the path ideal of a path was also
given in [2]. The formula from [2] matches ours. However, while we express the
projective dimension based on the number of minimal generators of the $p$-path
ideal, [2] does so using the length of the path.

We also recover the recursive formula for graded Betti numbers from [6]. This
formula was utilized in [2] to provide explicit calculations for the
projective dimension and regularity of path ideals of paths and cycles.

######  Theorem 3.16.

For all indices $r,d$, we have

$\beta_{r,d}(R/I_{p}(L_{n}))=\beta_{r,d}(R/I_{p}(L_{n-1}))+\beta_{r-1,d-{p}}(R/I_{p}(L_{n-(p+1)}))+\beta_{r-2,d-(p+1)}(R/I_{p}(L_{n-(p+1)})).$

###### Proof.

By Theorem 3.14, $\beta_{r,d}(R/I_{p}(L_{n}))$ counts the critical cells of
cardinality $r$ and degree $d$. To derive our desired expression, it suffices
to partition the set of critical subsets of cardinality $r$ and degree $d$ in
an appropriate way. Consider a critical cell $\sigma$ of $I_{p}(L_{n})$. The
following three scenarios for $\sigma$ completes the proof:

  1. (1)

If $m_{n-p+1}\notin\sigma$, then $\sigma$ is a critical cell of $I_{p}(L_{n})$
if and only if it is a critical cell of $I_{p}(L_{n-1})$.

  2. (2)

If $m_{n-p+1}\in\sigma$ but $m_{n-p}\notin\sigma$, then $\sigma$ is a critical
cell of $I_{p}(L_{n})$ if and only if $\sigma\backslash\\{m_{n-p+1}\\}$ is a
critical cell of $I_{p}(L_{n-(p+1)})$ by Corollary 3.4 and Proposition 3.6.

  3. (3)

If both $m_{n-p+1}$ and $m_{n-p}$ are in $\sigma$, then $\sigma$ is a critical
cell of $I_{p}(L_{n})$ if and only if
$\sigma\backslash\\{m_{n-p+1},m_{n-p}\\}$ is a critical cell of
$I_{p}(L_{n-(p+1)})$ by Corollary 3.4 and Proposition 3.6. ∎

Barile-Macchia resolutions are cellular, i.e., they are supported on CW
complexes. We first remark that the dimension of the CW complex that supports
the minimal resolution of a monomial ideal equals its projective dimension. In
general, the minimal resolution of any monomial ideal of projective dimension
$1$ is supported on a tree [13, Theorem 1]. In fact, in the cases where the
path ideals of paths have projective dimension $1$, their minimal resolutions
are supported on paths, which can be shown using the techniques that will be
employed in the next example. In what follows, we provide an example where the
path ideal of a path has projective dimension 2.

######  Example 3.17.

Consider the path ideal $I=I_{p}(L_{2p+1})$ under the total order

$m_{1}\succ m_{2}\succ\cdots\succ m_{p+2}.$

For any subset $\sigma=\\{m_{i_{1}},\ldots,m_{i_{k}}\\}$, each element of
$\sigma$ (except $m_{i_{1}}$ and $m_{i_{k}}$) is a bridge of $\sigma$ due to
Proposition 3.4 (1). Consequently, $m_{i_{k-1}}$ emerges as the smallest
bridge of $\sigma$. Therefore, the Barile-Macchia matching of $I$ with respect
to $(\succ)$, denoted by $A$, is achieved by removing the penultimate element
at every iteration. Note that there is only one cell (of cardinality of at
least 3) with no bridges: $\\{m_{1},m_{p+1},m_{p+2}\\}$. This results in the
following list of all critical cells of $I$:

$\emptyset,\\{m_{1}\\},\dots,\\{m_{p+2}\\},\\{m_{1},m_{2}\\},\\{m_{2},m_{3}\\},\dots,\\{m_{p+1},m_{p+2}\\},\\{m_{p+2},m_{1}\\},\\{m_{1},m_{p+1},m_{p+2}\\}.$

Given two distinct critical cells, $\sigma$ and $\sigma^{\prime}$, their least
common multiples are different by Proposition 3.13. This uniqueness ensures
that the Barile-Macchia resolution of $I$ is minimal by Theorem 2.2.

By discrete Morse theory, there exists a cellular complex that supports this
minimal free resolution, and it has $(p+2)$ many $0$-cells, $(p+2)$ many
$1$-cells, and one $2$-cell. One obvious object that fits this description is
the following $(p+2)$-gon:

$\\{m_{1},m_{2}\\}$$\\{m_{2},m_{3}\\}$$\\{m_{3},m_{4}\\}$$\\{m_{4},m_{5}\\}$$\\{m_{1},m_{p+2}\\}$$\\{m_{1}\\}$$\\{m_{2}\\}$$\\{m_{3}\\}$$\\{m_{4}\\}$$\\{m_{5}\\}$$\\{m_{p+2}\\}$$\\{m_{1},m_{p+1}m,m_{p+2}\\}$

One can show that this $(p+2)$-gon, denoted by $\Delta$, supports a free
resolution of $R/I$. By [5, Lemma 2.2], it suffices to demonstrate that for
each monomial $m$, the subcomplex
$\Delta[m]\coloneqq\\{\sigma\in\Delta\colon\operatorname{lcm}(\sigma)\text{
divides }m\\}$ of $\Delta$ is either empty or contractable. By exhausting all
possible cases, one can confirm that the subcomplex $\Delta[m]$ is either
empty, a line segment, or the $(p+2)$-gon $\Delta$ itself for each squarefree
monomial $m$. Thus, it is well-known that the latter two objects are both
contractable.

##  4\. Minimal free resolutions of path ideals of cycles

In this section, we turn our attention to path ideals of cycles, demonstrating
that these ideals have minimal cellular resolutions. While in the preceding
section we derived this outcome for path ideals of paths through identifying a
minimal Barile-Macchia resolution of $R/I$ with respect to a specific total
order on ${\mathcal{G}}(I)$, this approach falls short for path ideals of
cycles. For instance, when we consider the edge ideal of a 9-cycle, it has no
minimal Barile-Macchia resolutions as pointed out in [8, Remark 4.23].

In our investigation of path ideals of cycles, we transition our focus towards
the generalized Barile-Macchia resolutions. These are Morse resolutions and
can be considered as an extension of the Barile-Macchia resolutions,
introduced in [8]. The crux of these resolutions lies in utilizing a
collection of total orders on ${\mathcal{G}}(I)$, instead of one. For the
reader’s convenience, we restate the construction of generalized Barile-
Macchia resolutions from [8] along with a theorem stating that they induce
cellular free resolutions. First, recall that $G_{X}$ is the directed graph
obtained from the Taylor complex of $I$.

######  Theorem 4.1.

[8, Theorem 5.19] For a monomial $u\in R$, let $G_{u}$ be the induced subgraph
of $G_{X}$ on the vertices $\sigma\subseteq{\mathcal{G}}(I)$ where
$\operatorname{lcm}(\sigma)=u$. Consider a total order $(\succ_{u})$ on
${\mathcal{G}}(I)$ for each monomial $u\in R$.

Let $A$ be the union of all $A_{u}$, where $A_{u}$ is the collection of
directed edges obtained by applying the Barile-Macchia Algorithm to $G_{u}$
with respect to $(\succ_{u})$ for each monomial $u\in R$. Then, $A$ is a
homogeneous acyclic matching of $I$. The Morse resolution induced by $A$ is
called the generalized Barile-Macchia resolution of $R/I$ with respect to
$(\succ_{u})_{u\in R}$.

Consider a cycle $C_{n}$ on $n$ vertices $\\{x_{1},\ldots,x_{n}\\}$. Let
$R=\Bbbk[x_{1},\ldots,x_{n}]$. The $p$-path ideal of $C_{n}$, denoted as
$I_{p}(C_{n})$, is generated by monomials in $R$ corresponding to paths on $p$
vertices along $C_{n}$. Then, we have $I_{p}(C_{n})=(x_{1}\cdots
x_{p},\ldots,x_{n}\cdots x_{p-1})$. The minimal generating set of $I$ is
$G(I)=(m_{1},\dots,m_{n})$ where $m_{i}=x_{i}\cdots x_{i+p-1}$ for each $1\leq
i\leq n$. We restrict our attention to $2\leq p\leq n$ and consider the
indices modulo $n$.

We begin our discussion with an observation on the construction of the set
$A$, as outlined in Theorem 4.1, leveraging our previous findings on path
ideals of paths.

######  Observation 4.2.

Let $A$ be a homogeneous acyclic matching constructed in accordance with
Theorem 4.1 for the ideal $I_{p}(C_{n})$. To obtain a generalized Barile-
Macchia resolution of $R/I_{p}(C_{n})$, it is necessary to determine its
$A$-critical cells.

  1. (1)

The following is immediate from the definition of $A$:

$\\{A\text{-critical cells}\\}=\bigcup_{u:\text{ monomial in
}R}\\{A_{u}\text{-critical cells}\\}.$

  2. (2)

Assuming $u\neq x_{1}\cdots x_{n}$, remember that each vertex of $G_{X}$ is a
subset of ${\mathcal{G}}(I_{p}(C_{n}))$. Note that $V(G_{u})$, the vertex set
of $G_{u}$, is empty or every vertex of $G_{u}$ is a subset of
${\mathcal{G}}(J)$ where $J$ is the path ideal of some path. When $V(G_{u})$
is non-empty, Proposition 3.13 assures the existence of a total order
$(\succ)$ on ${\mathcal{G}}(J)$ that allows for precisely one $A$-critical
cell.

In light of the above observation, our attention is on the $A_{u}$-critical
cells where $u=x_{1}\cdots x_{n}$ to derive a minimal generalized Barile-
Macchia resolution of $R/I_{p}(C_{n})$.

For the remainder of this section, let $u=x_{1}\cdots x_{n}$ and adopt the
following total order $(\succ)$ for $u$:

$m_{1}\succ m_{2}\succ\cdots\succ m_{n}.$

Our primary objective is to demonstrate that all $A_{u}$-critical cells have
the same cardinality. Consequently, the resulting generalized Barile-Macchia
resolution is minimal by Theorem 2.2 and 4.2.

As a preliminary step, we examine the structure of $\sigma\in V(G_{u})$.
Recall that $\operatorname{lcm}(\sigma)=u$ for each vertex $\sigma$ in
$G_{u}$.

######  Proposition 4.3.

Let $\sigma\in V(G_{u})$. Then $\sigma$ is of the form
$\\{m_{i_{1}},\dots,m_{i_{t}}\\}\subseteq{\mathcal{G}}(I_{p}(C_{n}))$ with the
property that the distance between consecutive elements of $\sigma$ is at most
$p$, i.e., $i_{j+1}-i_{j}\leq p$ for all $j\in\\{1,\ldots,t-1\\}$ and
$i_{1}+n-i_{t}\leq p$.

###### Proof.

Suppose that either $i_{j+1}-i_{j}>p$ for some $j\in\\{1,\ldots,t-1\\}$ or
$i_{1}+n-i_{t}>p$. In both cases, this implies that $u=x_{1}\cdots x_{n}$
cannot be divisible by $x_{i_{j}+p}$ or $x_{i_{t}+p}$, respectively, leading
to a contradiction. ∎

Building upon our earlier observations, a key point we aim to elucidate is
that every critical cell contains $m_{n}$.

######  Lemma 4.4.

If $\sigma$ is an $A_{u}$-critical cell, then $m_{n}$ belongs to $\sigma$.

###### Proof.

Suppose for contradiction that $\sigma$ is an $A_{u}$-critical cell that does
not contain $m_{n}$. By Proposition 4.3, $m_{n}$ is a gap of $\sigma$. In
particular, this implies that $m_{n}=\operatorname{sb}(\sigma\cup m_{n})$.
Thus, $\sigma\cup m_{n}$ is potentially-type-2 by Remark 2.9. Moreover, it is
not possible to find a cell $\tau$ such that
$\tau\setminus\operatorname{sb}(\tau)=\sigma$ where
$m_{n}\succ\operatorname{sb}(\tau)$. Hence, we can infer that the directed
edge $(\sigma\cup m_{n},\sigma)$ belongs to $A_{u}$. This contradicts the
assumption that $\sigma$ is an $A_{u}$-critical cell. Thus, $m_{n}\in\sigma$.
∎

######  Notation 4.5.

Similar to Notation 3.2 for paths, define $M_{i}$ to be the collection of all
monomials in ${\mathcal{G}}(I_{p}(C_{n}))$ that are divisible by $x_{i}$. For
any $i\geq p$, we have $M_{i}=\\{m_{i-p+1},\ldots,m_{i}\\}$. For any given $i$
such that $1\leq i\leq p-1$, we have

$M_{i}=\\{m_{(n+i)-p+1},\ldots,m_{n},m_{1},\ldots,m_{i}\\}.$

Clearly, $|M_{i}|=p$ for each value of $i$.

In the subsequent discussion, we analyze the vertices of $G_{u}$ and describe
their bridges, gaps, and true gaps. Analogous to the path case, we can
classify the bridges in a similar manner. The proof is omitted since it
directly follows from Lemma 3.3, keeping in mind that indices are now
considered modulo $n$. First, we consider gaps as their classification is
immediate.

######  Proposition 4.6.

Let $\sigma\in V(G_{u})$. For $i\in\\{1,\ldots,n\\}$, a monomial $m_{i}$ is a
gap of $\sigma$ if and only if $m_{i}\notin\sigma$.

###### Proof.

Note that adding $m_{i}$ to $\sigma$ does not change the least common
multiple. Specifically,
$\operatorname{lcm}(\sigma)=\operatorname{lcm}(\sigma\cup m_{i})=x_{1}\cdots
x_{n}$, establishing the assertion. ∎

######  Proposition 4.7.

Let $\sigma\in V(G_{u})$, and $m_{i}\in{\mathcal{G}}(I_{p}(C_{n}))$ be a
monomial. Then the monomial $m_{i}$ is a bridge of $\sigma$ if and only if the
following conditions are met:

  1. (1)

$m_{i}\in\sigma$.

  2. (2)

There exist monomials $m_{j}$ and $m_{k}$ in $\sigma$ for $1\leq j<k\leq n$
with $i\neq j,k$ such that the distance between these two monomials along
$m_{i}$ is at most $p$, i.e.,

    1. (a)

$k-j\leq p$ when $j<i<k$.

    2. (b)

$j+n-k\leq p$ otherwise.

######  Remark 4.8.

As in Remark 3.5, the monomials $m_{j}$ and $m_{k}$ from Proposition 4.7 are
often chosen to be those closest to $m_{i}$ among the monomials in $\sigma$ in
terms of their distance to $m_{i}$.

We now turn our attention to characterizing the true gaps for vertices in
$G_{u}$ that are $A_{u}$-critical.

######  Proposition 4.9.

Let $\sigma\in V(G_{u})$ be an $A_{u}$-critical cell and let
$m_{i}\in{\mathcal{G}}(I_{p}(C_{n}))$. Assume $m_{i}$ does not dominate any
bridge of $\sigma$. Then $m_{i}$ is a true gap of $\sigma$ if and only if the
following statements hold:

  1. (a)

$m_{i}\notin\sigma$.

  2. (b)

If there exists a monomial $m_{k}\in\sigma\cap M_{i+p-1}$, then none of the
other monomials in $M_{i+p-1}$ belongs to $\sigma$. Furthermore,
$m_{i+p}\notin\sigma$.

  3. (c)

If $1\leq i\leq p-1$ and $\sigma\cap\\{m_{1},\ldots,m_{i-1}\\}$ is empty, then
the only monomial in $M_{i}$ that belongs to $\sigma$ is $m_{n}$. Furthermore,
$m_{i-p}\notin\sigma$.

###### Proof.

Suppose $m_{i}$ does not dominate any bridges of $\sigma$, i.e.,
$\operatorname{sb}(\sigma)\succ m_{i}$. We start our proof by observing
$m_{n}\in\sigma$ by Proposition 4.4 since $\sigma$ is an $A_{u}$-critical
cell.

We first deal with the forward direction. Assume that $m_{i}$ is a true gap of
$\sigma$. Then, $m_{i}$ is also a gap of $\sigma$ and $m_{i}\notin\sigma$ by
Proposition 4.6, which verifies condition (a). For condition (b), the
arguments align closely with those presented in Proposition 3.6 for parts (b)
and (c). Thus, the details are omitted to avoid redundancy.

Now, to verify condition (c), assume that $1\leq i\leq p-1$ and that
$\sigma\cap\\{m_{1},\ldots,m_{i-1}\\}$ is empty. Given that $m_{n}\in M_{i}$
for the stated range of $i$, there exists a monomial $m_{t}\in\sigma$ that
also belongs to $M_{i}$, but with $t\notin\\{1,\ldots,i\\}$. In this scenario,
unless $m_{t}=m_{n}$, the monomial $m_{n}$ would emerge as a bridge for
$\sigma\cup m_{i}$. This implies that $\operatorname{sb}(\sigma\cup
m_{i})=m_{n}$, a contradiction to [8, Proposition 2.21]. Hence, the only
monomial in $\sigma\cap M_{i}$ is $m_{n}$. For the last segment of (c), if we
assume $m_{i-p}\in\sigma$, it leads to the conclusion that
$\operatorname{sb}(\sigma\cup m_{i})=m_{n}$. This is, yet again, contradictory
to [8, Proposition 2.21], thereby confirming that $m_{i-p}\notin\sigma$.

For the reverse direction, assume conditions (a), (b), and (c) hold. If we
argue by contradiction and assume that $m_{i}$ is not a true gap of $\sigma$,
then there exists a monomial $m_{t}$ that is a bridge of $\sigma\cup m_{i}$
but is not a bridge of $\sigma$, with $m_{i}\succ m_{t}$. Using Proposition
4.7, we deduce $m_{t}\in\sigma$. Furthermore, there exist monomials
$m_{j_{t}}$ and $m_{k_{t}}$ in $\sigma\cup m_{i}$ with $1\leq j_{t}<k_{t}\leq
n$ and $t\notin\\{j_{t},k_{t}\\}$ such that

  1. (1)

$k_{t}-j_{t}\leq p$ when $j_{t}<t<k_{t}$ or

  2. (2)

$j_{t}+n-k_{t}\leq p$ otherwise.

As discussed in 4.5, we may assume that $j_{t}$ and $k_{t}$ are the indices of
monomials in $\sigma$ that are closest to $m_{t}$. Because $m_{t}$ is not a
bridge of $\sigma$, either $j_{t}=i$ or $k_{t}=i$. If $j_{t}<t<k_{t}$, we must
have $j_{t}=i$ since $m_{i}\succ m_{t}$. The reasoning for this case mirrors
the corresponding portion of Proposition 3.6, eventually leading to a
contradiction.

Note that we must have $j_{t}<k_{t}<t$ given $i<t$ when we consider other
orderings of $j_{t},t$, and $k_{t}$ on the cycle. Based on (2), we have:

($\star$) $j_{t}<k_{t}<t<j_{t}+n\leq k_{t}+p\leq n+p.$

In the scenario where $k_{t}=i$, we have $m_{j_{t}}\in\sigma$. and considering
($\star$ ‣ 4), $m_{t}$ is in $M_{i+p-1}$. However, due to condition (b) and
the fact that $m_{j_{t}+n}\in\sigma$, it must be that $m_{j_{t}+n}=m_{i+p}$, a
contradiction to (b). For the final case, we can assume $j_{t}=i$ and with
$j_{t}$ and $k_{t}$ being the closest monomials in $\sigma$ to $m_{t}$, the
intersection $\sigma\cap\\{m_{1},\ldots,m_{i-1}\\}$ is empty. Given ($\star$ ‣
4), we deduce that $i\leq p$. Since the only monomial in $\sigma\cap M_{i}$ is
$m_{n}$ by (c) and $(i+n)-p\leq k_{t}\leq n$, we have $m_{t}=m_{n}$ and
$m_{k_{t}}=m_{i-p}$. Consequently, $m_{k_{t}}=m_{i-p}\in\sigma$, a
contradiction to condition (c). Thus, $m_{i}$ is indeed a true gap of
$\sigma$.∎

Having addressed the true gaps of $A_{u}$-critical cells in the preceding
proposition, our focus now shifts to the distinct classes of critical cells
introduced in Definition 2.7: the absolutely critical and the fortunately
critical cells. As noted in Corollary 2.10, the absolutely critical cells are
uniquely characterized by the lack of both bridges and true gaps. On the other
hand, the fortunately critical cells stand out. Their first element serves as
their smallest bridge while still having no true gaps.

######  Lemma 4.10.

Let $\sigma\in V(G_{u})$ be an $A_{u}$-critical cell. If $\sigma$ is
fortunately critical, then:

  1. (1)

The smallest bridge of $\sigma$ satisfies $\operatorname{sb}(\sigma)\succ
m_{p}$.

  2. (2)

If $\sigma=\\{m_{i_{1}},\ldots,m_{i_{t}}\\}$ where $i_{1}<\cdots<i_{t}$, then
$\operatorname{sb}(\sigma)=m_{i_{1}}.$

Furthermore, no monomial from the set $\\{m_{p},m_{p+1},\ldots,m_{n}\\}$
serves as a bridge or a true gap of $\sigma$ (irrespective of whether $\sigma$
is fortunately or absolutely critical).

###### Proof.

Let $\sigma$ be a vertex in $G_{u}$ that is fortunately critical. By
definition, there exists another vertex $\sigma^{\prime}$ such that
$\sigma\backslash\operatorname{sb}(\sigma)=\sigma^{\prime}\backslash\operatorname{sb}(\sigma^{\prime})$
and $\operatorname{sb}(\sigma)\succ\operatorname{sb}(\sigma^{\prime})$.

We start by noting that the monomial $\operatorname{sb}(\sigma^{\prime})$ is a
true gap of $\sigma\backslash\operatorname{sb}(\sigma)$ by [8, Proposition
2.22]. Given that every true gap of $\sigma$ dominates
$\operatorname{sb}(\sigma)$ by Remark 2.9, it follows that
$\operatorname{sb}(\sigma^{\prime})$ cannot be a true gap of $\sigma$. This
implies that there exists a monomial $m_{i}\in\sigma^{\prime}$ for which
$\operatorname{sb}(\sigma\cup\operatorname{sb}(\sigma^{\prime}))=m_{i}$, with
$\operatorname{sb}(\sigma^{\prime})\succ m_{i}$ and $m_{i}$ not being a bridge
of $\sigma$. By Proposition 4.7, there exist monomials $m_{j}$ and $m_{k}$ in
$\sigma\cup\operatorname{sb}(\sigma^{\prime})$ with $1\leq j<k\leq n$ and
$i\notin\\{j,k\\}$ such that either:

  1. (a)

$k-j\leq p$ and $j<i<k$ holds, or

  2. (b)

$j+n\leq k+p$.

Inevitably, we have
$\\{\operatorname{sb}(\sigma),\operatorname{sb}(\sigma^{\prime})\\}=\\{m_{j},m_{k}\\}$
since $m_{i}$ is neither a bridge of $\sigma$ nor $\sigma^{\prime}$. We
dissect this further: If $j<i<k$, then
$\operatorname{sb}(\sigma^{\prime})=m_{j}$ since
$\operatorname{sb}(\sigma^{\prime})\succ m_{i}\succ m_{k}$. In this case, it
is immediate that $m_{k}\in\sigma^{\prime}$. Since
$m_{j},m_{i},m_{k}\in\sigma^{\prime}$ and $k-j\leq p$ by (a), the monomial
$m_{i}$ is a bridge of $\sigma^{\prime}$ by Proposition 4.7, which is a
contradiction. Thus, (b) holds. Moreover, since
$\\{m_{j},m_{k}\\}=\\{\operatorname{sb}(\sigma),\operatorname{sb}(\sigma^{\prime})\\}$
and both $\operatorname{sb}(\sigma),\operatorname{sb}(\sigma^{\prime})$
dominate $m_{i}$, we must have $j<k<i$. Consequently,
$m_{j}=\operatorname{sb}(\sigma)$ and
$m_{k}=\operatorname{sb}(\sigma^{\prime})$. Since
$m_{n}\in\sigma\cup\operatorname{sb}(\sigma^{\prime})$, we can conclude that
$m_{n}$ is a bridge of $\sigma\cup\operatorname{sb}(\sigma^{\prime})$ by
Proposition 4.7 and (b). This means $m_{i}=m_{n}$.

  1. (1)

To prove $\operatorname{sb}(\sigma)\succ m_{p}$, we first note that
$\operatorname{sb}(\sigma)=m_{j}\succeq m_{p}$ due to the inequality $j+n\leq
k+p\leq n+p$ by (b). If $j=p$, (b) from above implies that we have $n\leq k$,
which is not possible. So, $\operatorname{sb}(\sigma)\succ m_{p}$.

  2. (2)

Let $\sigma=\\{m_{i_{1}},\ldots,m_{i_{t}}\\}$ where $i_{1}<\cdots<i_{t}$. Our
goal is to demonstrate $\operatorname{sb}(\sigma)=m_{i_{1}}$. On the contrary,
suppose there is a monomial $m_{s}$ in $\sigma$ such that
$m_{s}\succ\operatorname{sb}(\sigma)=m_{j}$. Then, by Proposition 4.7,
monomial $m_{n}$ is a bridge of $\sigma^{\prime}$ since $m_{k},m_{n},$ and
$m_{s+n}$ are all members of $\sigma^{\prime}$ and the inequality $s+n-k<p$
holds true by (b) above. This posits a contradiction, as it means
$\operatorname{sb}(\sigma^{\prime})=m_{k}\succ m_{n}$ while $m_{n}$ is a
bridge of $\sigma^{\prime}$. Thus, $\operatorname{sb}(\sigma)=m_{i_{1}}$.

For the final part of the statement, consider a vertex $\sigma$ in $G_{u}$. If
$\sigma$ is absolutely critical, it lacks both bridges and true gaps, by
Corollary 2.10, thereby satisfying the given statement. When $\sigma$ is
fortunately critical, the statement remains valid due to
$\operatorname{sb}(\sigma)\succ m_{p}$, and the fact that each true gap of
$\sigma$ dominates $\operatorname{sb}(\sigma)$ by Remark 2.9. ∎

Our primary objective is to comprehensively identify every element within an
$A_{u}$-critical cell. We begin our identification with a series of
observations and initiate the process by pinpointing specific values of $j$
for which $m_{j}\in\sigma$.

######  Lemma 4.11.

Let $\sigma=\\{m_{i_{1}},\ldots,m_{i_{t}}\\}$ be an $A_{u}$-critical cell.
Then

  1. (a)

$i_{t}=n$, and $m_{n}$ is not a bridge for $\sigma$.

  2. (b)

$i_{t-1}=n-k$ for some $1\leq k\leq p$.

  3. (c)

$i_{t-2}<n-p$.

  4. (d)

$i_{1}=p-k+1$ where $k$ is number from the expression of $i_{t-1}$ in (a).

###### Proof.

Assume that $\sigma=\\{m_{i_{1}},\ldots,m_{i_{t}}\\}$ is an $A_{u}$-critical
cell.

  1. (a)

We begin by noting that $m_{i_{t}}=m_{n}$ as per Lemma 4.4. If $\sigma$ is
absolutely critical, then, according to Corollary 2.10, it has no bridges. On
the other hand, for a fortunately critical $\sigma$, Lemma 4.10 dictates that
$\operatorname{sb}(\sigma)=m_{i_{1}}$. Given that $m_{i_{1}}\succ m_{n}$, it
is evident that $m_{n}$ is not a bridge for $\sigma$.

  2. (b)

Recall from Proposition 4.3 that $i_{t}-i_{t-1}\leq p$. Then, $i_{t-1}=n-k$
for some $1\leq k\leq p$.

  3. (c)

For the sake of contradiction, suppose $m_{n-j}\in\sigma$ for $1\leq k<j\leq
p$. Then, $m_{n-k}$ emerges as a bridge for $\sigma$. However, this
contradicts the nature of $\sigma$, as it is either absolutely critical (hence
having no bridges) or fortunately critical (where
$\operatorname{sb}(\sigma)=m_{i_{1}}$).

  4. (d)

Our initial step is to derive $p-k+1\leq i_{1}\leq p$. The upper bound
$i_{1}\leq p$ is a direct consequence of Proposition 4.3. To establish the
lower bound, if there exists an $m_{i}\in\sigma$ for $i\leq p-k$, then $m_{n}$
is a bridge of $\sigma$ by Proposition 4.7. This assertion, however, yields a
contradiction by (a). Thus, $p-k+1\leq i_{1}\leq p$.

If $k=1$, then it is immediate that $i_{1}=p$, satisfying the statement of
(d). For $2\leq k\leq p$, it remains to show $i_{1}\leq p-k+1$. An significant
insight here is that by Proposition 4.9 (b), $m_{n-k+1}$ cannot be a true gap
of $\sigma$ since $M_{p-k}$ contains both $m_{n-k}$ and $m_{n}$ from $\sigma$.
This means $\sigma\cup m_{n-k+1}$ has a bridge $m_{b}\in\sigma$ such that
$m_{n-k+1}\succ m_{b}$. Since the only monomial in $\sigma$ that is dominated
by $m_{n-k+1}$ is $m_{n}$, we have $m_{b}=m_{n}$. Using Proposition 4.7 for
the bridge $m_{b}$, we conclude $i_{1}\leq p-k+1$. ∎

To identify the other elements of $\sigma$, we examine them in relation to the
possible values of $k$. Here, $n-k$ is the penultimate element of $\sigma$
when $1\leq k\leq p$. The next two lemmas address the $k=1$ case and the
$2\leq k\leq p$ case separately due to nuanced variations in their proofs.
Together, these lemmas give a complete overview of all $A_{u}$-critical cells.

######  Lemma 4.12.

Let $\sigma=\\{m_{i_{1}},\ldots,m_{i_{t}}\\}$ be an $A_{u}$-critical cell with
$i_{t-1}=n-1$. Then, the distance between consecutive elements of $\sigma$
alternates between $1$ and $p$. Specifically,

$\sigma=\\{m_{p},\ldots,m_{n-(p+1)},m_{n-(p+1)},m_{n-1},m_{n}\\}.$

Moreover, we have $p\equiv n$ or $n-1\pmod{p+1}$.

###### Proof.

The main idea of the proof revolves around retracing our steps from $i_{t-1}$,
pinpointing the preceding indices of elements in $\sigma$ until we arrive at
$i_{1}=p$. A recurring and instrumental point from Proposition 4.3 to note
during the proof is: $i_{j}-i_{j-1}\leq p$ for
$m_{i_{j-1}},m_{i_{j}}\in\sigma$.

Before we identify elements of $\sigma$, note that $\sigma$ is an absolutely
critical cell by Lemma 4.10 because $i_{1}=p$. Recall from Lemma 4.11 that
$i_{t-2}\leq n-1-p$. The first step of our analysis is immediate as
$i_{t-1}-i_{t-2}=p$ by Lemma 4.11 (b) and (c). If $i_{t-2}=i_{1}$, we are
done; otherwise, we move on to $i_{t-3}$.

Next, we show that $i_{t-2}-i_{t-3}=1$ which is equivalent to obtaining
$m_{n-p-2}\in\sigma$. For the sake of contradiction, suppose
$m_{n-p-2}\notin\sigma$. Since $\sigma$ is absolutely critical, $m_{n-p-2}$
cannot be a true gap of $\sigma$. This means $\sigma\cup m_{n-p-2}$ has a new
bridge $m_{b}\in\sigma$ such that $m_{n-p-2}\succ m_{b}$. Then, by Proposition
4.7, there exists $m_{s}\in\sigma$ such that the distance between $m_{s}$ and
$m_{n-p-2}$ along $m_{b}$ is at most $p$. Given that $m_{n-p-2}\succ m_{b}$,
by Proposition 4.7, we have either

  1. (1)

$s-(n-p-2)\leq p$ when $1\leq n-p-2<b<s\leq n$, or

  2. (2)

$s+n-(n-p-2)\leq p$ when $1\leq s<n-p-2<b\leq n.$

The latter case is not possible because it implies that $s\leq-2$. Thus,
$m_{b}=m_{n-1}$ and $m_{s}=m_{n}$. On the other hand, by (1), the distance
between $m_{n-p-2}$ and $m_{n}$ must be at most $p$ but it is $p+2$, a
contradiction. Thus, we conclude that $i_{t-3}=n-p-2$. If $i_{t-3}=i_{1}$, our
task is complete; otherwise, our focus shifts to $i_{t-4}$.

Towards showing $i_{t-3}-i_{t-4}=p$, recall that $i_{t-3}-i_{t-4}\leq p$, or
equivalently, $i_{t-4}\geq n-2(p+1)$. If $m_{n-2(p+1)}\notin\sigma$, then
$i_{t-2}-i_{t-4}\leq p$, implying that $m_{i_{t-3}}$ is a bridge of $\sigma$
Proposition 4.7. This leads to a contradiction because $\sigma$ is absolutely
critical. Hence, $i_{t-4}=n-2(p+1)$. If $i_{t-4}=i_{1}$, our investigation is
complete; otherwise, we continue in the same fashion for $i_{t-5}$.

Subsequent distances between consecutive elements of $\sigma$ can be concluded
by employing similar arguments in an alternating way until we reach $i_{1}=p$.
This results in the congruence $p\equiv n\text{ or }n-1\pmod{p+1}$, concluding
the proof. ∎

######  Lemma 4.13.

Let $\sigma=\\{m_{i_{1}},\ldots,m_{i_{t}}\\}$ be an $A_{u}$-critical cell with
$i_{t-1}=n-k$ for some $2\leq k\leq p$. Then, the distance between consecutive
elements of $\sigma$ alternates between $k$ and $(p+1)-k$. Specifically,

$\sigma=\\{m_{p-k+1},\ldots,m_{n-k-(p+1)},m_{n-(p+1)},m_{n-k},m_{n}\\}$

Moreover, we have $p-k+1\equiv n$ or $n-k\pmod{p+1}$.

###### Proof.

The idea behind this proof is similar to that of Lemma 4.12. Recalling that
$i_{t-1}=n-k$ and $i_{t-2}\leq n-p-1$ by Lemma 4.11 (c), our immediate goal is
to confirm that $i_{t-2}=n-p-1$.

Assume, for the sake of contradiction, that $m_{n-p-1}\notin\sigma$. We claim
that $m_{n-p-1}$ cannot be a true gap of $\sigma$. For contradiction, suppose
that it is a true gap of $\sigma$. Contradiction is evident if $\sigma$ is
absolutely critical because it lacks true gaps. If $\sigma$ is fortunately
critical, it is potentially-type-2 and we have
$m_{n-p-1}\succ\operatorname{sb}(\sigma)$ by Remark 2.9. Given that
$\operatorname{sb}(\sigma)=m_{i_{1}}$ as stated in Lemma 4.10 and the only
elements in $\sigma$ dominated by $m_{n-p-1}$ are $m_{n-k}$ and $m_{n}$, we
have $\operatorname{sb}(\sigma)=m_{i_{1}}=m_{n-k}$, and therefore
$|\sigma|=2$. This yields a contradiction because $\sigma$ cannot have any
bridge in this case. Thus, we conclude that $m_{n-p-1}$ cannot be a true gap
of $\sigma$. This means adding $m_{n-p-1}$ to $\sigma$ creates a new bridge
$m_{b}\in\sigma$ with $m_{n-p-1}\succ m_{b}$. Using Proposition 4.7, there
exists a monomial $m_{s}\in\sigma$ such that its distance to $m_{n-p-1}$ along
$m_{b}$ is at most $p$. In other words, given that $m_{n-p-1}\succ m_{b}$, we
have

  1. (1)

$s-(n-p-1)\leq p$ when $1\leq n-1-p<b<s\leq n$ or

  2. (2)

$s+n-(n-p-1)\leq p$ when $1\leq s<n-1-p<b\leq n$.

The latter is untenable since it implies $s<-1$. For the former,
$m_{b}=m_{n-k}$ and $m_{s}=m_{n}$, contradicting (1) as the distance between
$m_{n-1-p}$ and $m_{n}$ is $p+1$. Hence, $i_{t-2}=n-1-p$. If $i_{t-2}$ equals
$i_{1}$, our investigation concludes. Suppose $i_{t-2}\neq i_{1}$.

The next step is to show $i_{t-3}=n-k-(p+1)$. We first claim that
$m_{i_{t-2}}$ cannot be a bridge of $\sigma$. If it is a bridge of $\sigma$,
then $\operatorname{sb}(\sigma)=m_{i_{1}}$ by Lemma 4.10 because $\sigma$ must
be fortunately critical. This means $i_{1}=i_{t-2}$, a contradiction to our
assumption. Since $m_{i_{t-2}}$ cannot be a bridge of $\sigma$, we have
$i_{t-3}\leq n-k-(p+1)$ by Proposition 4.7. So, it suffices to show
$m_{n-k-(p+1)}\in\sigma$.

For the sake of contradiction, suppose $m_{n-k-(p+1)}\notin\sigma$. If
$m_{n-k-(p+1)}$ is a true gap of $\sigma$, then $\sigma$ is potentially-type-2
and $m_{n-k-(p+1)}\succ\operatorname{sb}(\sigma)=m_{i_{1}}$ by Remark 2.6 and
Lemma 4.10. This means $i_{1}=i_{t-2}$, a contradiction. Since $m_{n-k-(p+1)}$
is not a true gap of $\sigma$, the cell $\sigma\cup m_{n-k-(p+1)}$ has a new
bridge $m_{b}\in\sigma$ where $m_{n-k-(p+1)}\succ m_{b}$. Then, by Proposition
4.7, there exists $m_{s}\in\sigma$ such that the distance between
$m_{n-k-(p+1)}$ and $m_{s}$ along $m_{b}$ is at most $p$. In other words,
given that $m_{n-k-(p+1)}\succ m_{b}$, we have

  1. (1)

$s-(n-k-(p+1))\leq p$ when $1\leq n-k-(p+1)<b<s\leq n$ or

  2. (2)

$s+n-(n-k-(p+1))\leq p$ when $1\leq s<n-k-(p+1)<b\leq n$

One can verify that (2) cannot happen. For the first case, the monomial
$m_{s}$ is either $m_{n-k}$ or $m_{n}$. For either value of $s$, one can see
that the distance is at least $p+1$, a contradiction to (1). Therefore,
$i_{t-3}=n-k-(p+1)$. If $i_{t-3}=i_{1}$, the process terminates here. If
$i_{t-3}\neq i_{1}$, one can repeat the above arguments for the next steps
until reaching $i_{1}=p-k+1$. This results in the congruence $p-k+1\equiv
n\text{ or }n-k\pmod{p+1}$, concluding the proof. ∎

Below, we describe all $A_{u}$-critical cells. This proposition serves as the
centerpiece of this chapter’s main result.

######  Proposition 4.14.

Let $n=(p+1)q+r$ where $0\leq r\leq p$. Then, we have:

  1. (1)

For $r=0$, the only $A_{u}$-critical cells are the cells $\sigma_{i}$, where

$\sigma_{i}:=\\{m_{j}\mid j\equiv i\pmod{p+1}\\}$

for each $1\leq i\leq p$. Moreover, each $\sigma_{i}$ is absolutely critical.

  2. (2)

For $r\neq 0$, the only $A_{u}$-critical cell is

$\tau_{r}:=\\{m_{j}\mid j\geq r,j\equiv r,2r\pmod{p+1}\\}.$

Moreover, the cell $\tau_{r}$ is absolutely critical if and only if $2r>p$.

###### Proof.

Let $\sigma=\\{m_{i_{1}},\ldots,m_{i_{t}}\\}$ be an $A_{u}$-critical cell with
$t\geq 2$. To identify all $A_{u}$-critical cells, we first address the
immediate case where $n=p+1$. It is evident that $t\geq 3$ is not possible, as
this would lead to $\operatorname{sb}(\sigma)=m_{n}$, a contradiction by Lemma
4.11 (a). Consequently, $\sigma$ must be of the form
$\sigma_{i}=\\{m_{i},m_{n}\\}$ for all $1\leq i\leq p$. A straightforward
check confirms that every such $\sigma_{i}$ is absolutely critical. Having
settled this case, we proceed under the assumption $n>p+1$.

Recall from Lemma 4.12 and Lemma 4.13 that the distance between consecutive
elements of $\sigma$ alternates between $k$ and $(p+1)-k$. Specifically, for
$1\leq k\leq p$, we have

$\sigma=\\{m_{p-k+1},\ldots,m_{n-k-(p+1)},m_{n-(p+1)},m_{n-k},m_{n}\\}$

where $p-k+1\equiv n$ or $n-k\pmod{p+1}$. It is important to clarify that the
distance between the first and last elements of $\sigma$ is not considered.

Observe that the conditions $p-k+1\equiv n\pmod{p+1}$ and $n-k\equiv
0\pmod{p+1}$ are equivalently expressed as $n\equiv p+1-k\pmod{p+1}$ and
$n\equiv 0\pmod{p+1}$, respectively. Consequently, the cell $\sigma$ can be
expressed as follows:

$\sigma=\begin{cases}\\{m_{(p+1-k)+i(p+1)},m_{(i+1)(p+1)}\leavevmode\nobreak\
:\leavevmode\nobreak\ 0\leq i\leq q-1\\}&\text{if }r=0,\\\
\\{m_{r+i(p+1)},m_{2r+i(p+1)}\leavevmode\nobreak\ :\leavevmode\nobreak\ 0\leq
i\leq q-1\\}\cup\\{m_{n}\\}&\text{if }r\neq 0,\end{cases}$

where the former corresponds to $\sigma_{p+1-k}$ for $1\leq k\leq p$ and the
latter corresponds to $\tau_{r}$ in the statement of the proposition. It is
easily verified that each $\sigma_{p+1-k}$ is absolutely critical, as it lacks
both bridges and true gaps for $1\leq k\leq p$. Furthermore, one can verify
that $\tau_{r}$ is absolutely critical if and only if $2r>p$. ∎

The conclusions drawn below directly arise from the descriptions of
$A_{u}$-critical cells given in Proposition 4.14.

######  Corollary 4.15.

Let $\sigma\in V(G_{u})$ be a critical cells of ${\mathcal{G}}(I_{p}(C_{n}))$.
Then

$|\sigma|=\begin{cases}2q&\text{ if }n\equiv 0\pmod{(p+1)},\\\ 2q+1&\text{
otherwise}.\\\ \end{cases}$

In particular, all $A_{u}$-critical cells have the same cardinality.

We now present the main theorem of this section, which follows immediately
from the preceding corollary, and the description of the differentials of the
corresponding generalized Barile-Macchia resolution in Theorem 2.2.

######  Theorem 4.16.

A generalized Barile-Macchia resolution of $I_{p}(C_{n})$ is minimal.

We conclude this section with a brief discussion. In [8], the first two
authors introduced and examined Barile-Macchia resolutions, which are cellular
and independent of $\operatorname{char}(\Bbbk)$. Though effective for many
classes of ideals, this method does not always produce a minimal free
resolution. Specifically, the edge ideal of a 9-cycle, $I_{2}(C_{9})$, cannot
be minimally resolved by a Barile-Macchia resolution, as highlighted in [8].
To the best of our knowledge, no previous construction has been developed to
minimally resolve edge ideals of cycles; we introduce the first such
construction, ensuring a cellular minimal resolution.

######  Corollary 4.17.

Edge ideals of cycles have cellular minimal free resolutions.

Echoing the approach of the preceding section, we conclude with an example
presenting a CW complex that supports minimal free resolutions of the path
ideals of cycles, where the projective dimensions equal $2$. Specifically,
these correspond to the ideals $I_{p}(C_{n})$ for which $p+2\leq n\leq 2p+1$.

We consider the minimal generalized Barile-Macchia resolutions of these
ideals. One can obtain the following list of all critical cells of
$I_{p}(C_{n})$:

$\emptyset,\\{m_{1}\\},\dots,\\{m_{p+2}\\},\\{m_{1},m_{2}\\},\\{m_{2},m_{3}\\},\dots,\\{m_{n-1},m_{n}\\},\\{m_{n},m_{1}\\},\\{m_{r},m_{2r},m_{n}\\},$

where $n=(p+1)+r$ for $1\leq r\leq p$. The following $n$-gon $\Delta$ is in
correspondence with the critical cell of $I_{p}(C_{n})$:

$\\{m_{1},m_{2}\\}$$\\{m_{2},m_{3}\\}$$\\{m_{3},m_{4}\\}$$\\{m_{4},m_{5}\\}$$\\{m_{1},m_{n}\\}$$\\{m_{1}\\}$$\\{m_{2}\\}$$\\{m_{3}\\}$$\\{m_{4}\\}$$\\{m_{5}\\}$$\\{m_{n}\\}$$\\{m_{r},m_{2r},m_{n}\\}$

Drawing parallels with Example 3.17, and using [5, Lemma 2.2], one can
establish that all restricted subcomplexes of $\Delta$ are either empty, a
line segment, or the $n$-gon itself, which are all acyclic. Thus, $\Delta$
support the minimal free resolution of $R/I_{p}(C_{n})$.

Acknowledgements. We thank Sara Faridi for helpful discussions about the
manuscript. The first author was supported by NSF grants DMS 1801285, 2101671,
and 2001368 .

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#  Algebraic Morse theory via Homological Perturbation Lemma with two
applications

Jun Chen  (Jun Chen and Guodong Zhou) School of Mathematical Sciences, Key
Laboratory of Mathematics and Engineering Applications (Ministry of
Education), Shanghai Key Laboratory of PMMP, East China Normal University,
Shanghai 200241, P.R.China<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Yuming
Liu  (Yuming Liu) School of Mathematical Sciences, Laboratory of Mathematics
and Complex Systems, Beijing Normal University, Beijing 100875, P.R.China
<EMAIL_ADDRESS>and  Guodong Zhou

###### Abstract.

As a generalization of the classical killing-contractible-complexes lemma, we
present algebraic Morse theory via homological perturbation lemma, in a form
more general than existing presentations in the literature. Two-sided Anick
resolutions due to E. Sköldberg are generalised to algebras given by quivers
with relations and a minimality criterion is provided as well. Two
applications of algebraic Morse theory are presented. It is shown that the
Chinese algebra of rank $n\geq 1$ is homologically smooth and of global
dimension $\frac{n(n+1)}{2}$, and the minimal two-sided projective resolution
of a Koszul algebra is constructed.

###### Key words and phrases:

Algebraic Morse theory; Chinese algebra; Homological perturbation lemma;
Killing-contractible-complexes lemma; Two-sided Anick resolution

###### 2010 Mathematics Subject Classification:

18G35, 16E05, 16E10

###### Contents

  1. 1 Homological Perturbation Lemma

## Introduction

Discrete Morse theory has its origin in the work of K. S. Brown, R. Geoghegan
[BG]. In that paper, one encountered a cell complex with one vertex and
infinitely many cells in each positive dimension. The authors, using ad hoc
method, collapsed this cell complex to a quotient complex with only two cells
in each positive dimension. K. S. Brown formalized and applied the collapsing
method scheme to groups with a rewriting system [Brown]. Motivated by
differential topology, R. Forman [Forman] rediscovered and developed this
theory as a discrete version of the usual smooth Morse theory, hence the name
“discrete Morse theory”. Since then, this subject has received much attention
in combinatorial and computational topology; see [Forman02, Kozlov08, Knudson,
Scoville] etc.

An algebraic version of discrete Morse theory has been developed by E.
Sköldberg [S06], D. Kozlov [Kozlov], M. Jöllenbeck and V. Welker [JW]. It has
many applications in algebra, such as combinatorial commutative algebra [JW],
cohomology of Lie algebras [S05, LV16, LV17b, LV19], Hochschild cohomology
[Lopatkin, LV17a], and operad theory [DK13].

E. Sköldberg [S18] studied algebraic Morse theory from the viewpoint of the
Homological Perturbation Lemma [EM, Shih, Brown64, HK, Crainic]. This paper
pushes this idea further, generalising and complementing [S06, S18] as well as
[Kozlov, JW] and presents an application to Chinese algebras.

The layout of this paper is as follows.

The first section contains an introduction to the Homological Perturbation
Lemma (HPL) by providing some new proofs and observations which seem to be of
independent interest. In the second section we use HPL to generalise the
classical killing-contractible-complexes lemma in homological algebra.

Algebraic Morse theory is presented in the third section and our definition of
the key concept “Morse matching” is more general than those in [S06, Kozlov,
JW].

One of the most important advantages of algebraic Morse theory is that it
provides a method to construct two-sided Anick resolutions, compared with the
original one-sided version in [Anick, AG]. This is particularly useful for
Hochschild cohomology. In the fourth section, using the corresponding Gröbner-
Shirshov basis theory [Green], we furnish a construction of two-sided Anick
resolutions for algebras given by quivers with relations, generalising [S06,
Section 3.2]. A similar resolution with a different construction of the
differentials is provided by S. Chouhy and A. Solotar [CS], which might be
closely related to two-sided Anick resolutions.

The fifth section contains a criterion which gives a sufficient condition when
the two-sided Anick resolution is minimal; see Theorem
LABEL:Thm:_minimal_criterion. We also provide a counter-example to a result
that appeared in [JW].

The sixth section contains an application of algebraic Morse theory to Chinese
algebras [DK92, CEKNH]. Using the Gröbner-Shirshov basis for Chinese algebras
discovered by Y. Chen and J. Qiu [CQ], we prove that a Chinese algebra of rank
$n\geq 1$ is homologically smooth, that is, it admits a finite length
resolution by finitely generated projective bimodules, and its global
dimension is $\frac{n(n+1)}{2}$; see Theorem LABEL:Thm:_Chinese_algebras.

In last section, we consider the quadratic algebra $A=k\langle x,y,z\ |\
x^{2}+yx,xz,zy\rangle$ first studied by N. Iyudu and S. Shkarin [IS] in their
classification result for Hilbert series of Koszul algebras with three
generators and three relations with the goal to answer two questions from the
book [PP]. In [Dots], V. Dotsenko and S. R. Chowdhury constructed the one-
sided Anick resolution of $A$. We will use the algebraic Morse theory to
construct the two-sided minimal resolution of $A$ through its two-sided Anick
resolution.

##  1\. Homological Perturbation Lemma

In this section, we give an introduction to the Homological Perturbation Lemma
(HPL).

We first introduce some relevant notions which appear to be well known,
although to our knowledge, some of them do not appear in the literature.

######  Definition 1.1.

  * (a)

A homotopy retract datum (aka. HR datum)

(1)
$\textstyle{{(L_{*},b)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{{(M_{*},b)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{h}$

consists of the following

    * (i)

two chain maps $i$ and $p$ between complexes $(L_{*},b)$ and $(M_{*},b)$, and

    * (ii)

a homotopy $h$ between $ip$ and $1$ (so $ip=1+bh+hb$).

  * (b)

An HR datum is a strong quasi-isomorphism datum (aka. SQI datum) if moreover,

    * (iii)

$p$ and $i$ are quasi-isomorphisms.

  * (c)

An SQI datum is a homotopy equivalence datum (aka. HE datum) if moreover,

    * (iv)

$pi$ is homotopic to $1$.

  * (d)

An HE datum is a deformation retract datum (aka. DR datum), if moreover,

    * (v)

$pi=1$.

  * (e)

A DR datum is a strong deformation retract datum (aka. SDR datum), if
moreover,

    * (vi)

$h^{2}=0,hi=0,ph=0$.

######  Remark 1.2.

  * (a)

DR data and SDR data exist even from the beginning of HPL [EM, Shih, Brown64].
The name HE data appeared in the work of M. Crainic [Crainic] with a different
meaning, namely that which we call SQI data.

  * (b)

A DR datum can be always modified to an SDR datum, as explained in [LS].

We now introduce a condition which lies at the heart of the HPL.

A perturbation $\delta$ of Datum (1) is a graded map on $M_{*}$ of the same
degree as $b$ such that $b+\delta$ is a new differential. We call it small, if
$1-\delta h$ is invertible. In this case, put

$A=(1-\delta h)^{-1}\delta$

and consider a new perturbed datum

(2)
$\textstyle{(L_{*},b_{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\infty}}$$\textstyle{(M_{*},b+\delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\infty}}$$\scriptstyle{h_{\infty}}$

with

$i_{\infty}=i+hAi,p_{\infty}=p+pAh,h_{\infty}=h+hAh,b_{\infty}=b+pAi.$

######  Remark 1.3.

In a heuristic manner,

$\begin{array}[]{rclcl}A&=&(1-\delta
h)^{-1}\delta&=&\sum_{n=0}^{\infty}(\delta
h)^{n}\delta=\sum_{n=0}^{\infty}\delta(h\delta)^{n},\\\
i_{\infty}&=&i+hAi&=&\sum_{n=0}^{\infty}(h\delta)^{n}i,\\\
p_{\infty}&=&p+pAh&=&\sum_{n=0}^{\infty}p(\delta h)^{n},\\\
h_{\infty}&=&h+hAh&=&\sum_{n=0}^{\infty}h(\delta h)^{n},\\\
b_{\infty}&=&b+pAi&=&b+\sum_{n=0}^{\infty}p(\delta h)^{n}\delta
i=b+\sum_{n=0}^{\infty}p\delta(h\delta)^{n}i.\end{array}$

The following result summarizes known versions of the HPL.

######  Theorem 1.4 (Homological Perturbation Lemma).

Assume that $\delta$ is a small perturbation of Datum (1). If Datum (1) is an
HR datum (resp. SQI datum, HE datum, SDR datum), then so is Datum (2).

######  Remark 1.5.

  * (a)

It is well known that Theorem 1.4 does NOT hold for DR data; see for example,
[Crainic, 2.3 Remarks (i)].

  * (b)

The SDR version is the first version of the HPL that appeared in the
literature [EM, Shih, Brown64]; the HR version is essentially contained in the
proof of the SDR version, in particular, we have

$i_{\infty}p_{\infty}=1+(b+\delta)h_{\infty}+h_{\infty}(b+\delta);$

the HE version was first shown in [HK]; the SQI version is contained in
[Crainic]. For a detailed historic account, see [Hue].

We will present a direct proof for the HE version compared with the original
proof of [HK], which used a mapping cylinder construction to deduce the HE
version from the SDR version, and we will also provide a streamlined proof for
the SQI version. In the course of the proofs, we add some which appear to be
new observations.

In the proofs of this section, in order to facilitate the understanding of the
reader, we often underline some terms indicating that the terms will be
changed in the next equality.

For the rest of this section, assume that we are given an HR datum (1) which
is endowed with a small perturbation $\delta$.

The following three equalities are crucial in the proof of the HR version of
HPL.

######  Lemma 1.6 (see [Crainic, Main Perturbation Lemma]).

We have the following equalities:

(3) $\delta hA=Ah\delta=A-\delta,$ (4) $(1-\delta
h)^{-1}=1+Ah,(1-h\delta)^{-1}=1+hA,$ (5) $AipA+Ab+bA=0.$

From now on we assume both the HR version and the SDR version of HPL. We are
going to deduce the SQI and the HE version of HPL from these assumptions. The
following interesting observations are useful in the proof of the SQI and the
HE version of HPL.

######  Lemma 1.7.

The maps $p_{\infty}i-1:(L_{*},b)\longrightarrow(L_{*},b_{\infty})$ and
$pi_{\infty}-1:(L_{*},b_{\infty})\longrightarrow(L_{*},b)$ are chain maps,
that is,

(6) $b_{\infty}(p_{\infty}i-1)=(p_{\infty}i-1)b$ (7)
$(pi_{\infty}-1)b_{\infty}=b(pi_{\infty}-1).$

  * Proof

We will show Equality (6) and Equality (7) can be proved similarly. In fact,
we have

$\begin{array}[]{rcl}b_{\infty}(p_{\infty}i-1)&=&\underline{b_{\infty}p_{\infty}}i-b_{\infty}\\\
&=&p_{\infty}bi+\underline{p_{\infty}}\delta i-b_{\infty}\\\
&=&p_{\infty}bi+p\underline{(\delta+Ah\delta)}i-b_{\infty}\\\
&\stackrel{{\scriptstyle(\ref{Eq:
Ah8})}}{{=}}&p_{\infty}\underline{bi}+\underline{pAi-b_{\infty}}\\\
&=&p_{\infty}ib-b\\\ &=&(p_{\infty}i-1)b.\end{array}$

$\Box$

######  Lemma 1.8.

The chain maps

$i\circ(pi_{\infty}-1):(L_{*},b_{\infty})\longrightarrow(M_{*},b),$
$(p_{\infty}i-1)\circ p:(M_{*},b)\longrightarrow(L_{*},b_{\infty}),$
$(pi_{\infty}-1)\circ p_{\infty}:(M_{*},b+\delta)\longrightarrow(L_{*},b),$

and

$i_{\infty}\circ(p_{\infty}i-1):(L_{*},b)\longrightarrow(M_{*},b+\delta)$

are all null-homotopic.

    * Proof

In fact, we have the following equalities:

(8) $i(pi_{\infty}-1)=hi_{\infty}b_{\infty}+bhi_{\infty},$ (9)
$(p_{\infty}i-1)p=b_{\infty}p_{\infty}h+p_{\infty}hb,$ (10)
$(pi_{\infty}-1)p_{\infty}=ph_{\infty}(b+\delta)+bph_{\infty},$ (11)
$i_{\infty}(p_{\infty}i-1)=(b+\delta)h_{\infty}i+h_{\infty}ib.$

whose proofs can be verified directly.