# On the support of Betti tables of multiparameter persistent homology modules

Andrea Guidolin111KTH Royal Institute of Technology, Stockholm, Sweden
<EMAIL_ADDRESS>Claudia Landi222University of Modena and Reggio Emilia, Italy
<EMAIL_ADDRESS>
###### Abstract

Persistent homology encodes the evolution of homological features of a
multifiltered cell complex in the form of a multigraded module over a
polynomial ring, called a multiparameter persistence module, and quantifies it
through invariants suitable for topological data analysis.

In this paper, we establish a relation between Betti tables, a standard
invariant for multigraded modules commonly used in multiparameter persistence,
and the critical cells determined by discrete Morse theory on the filtered
cell complex originating the module. In particular, we show that for a
discrete gradient vector field consistent with a given multiparameter sublevel
set filtration, the grades at which its critical cells appear in the
filtration reveal all positions in which the Betti tables are possibly
nonzero. This result is refined in the case of bifiltrations by considering
homological critical grades of a filtered chain complex instead of entrance
grades of critical cells.

MSC: 55N31, 57Q70 (primary); 55U15, 13D02 (secondary)

Keywords: Discrete Morse theory, Multigraded Betti numbers, Koszul complex

##  1 Introduction

One of the main concepts in Topological Data Analysis is persistent homology,
a tool to capture topological information at multiple scales and provide
meaningful topological summaries of the data, as surveyed, for example, in
[Ghr07, Car09, EM12]. In practice, assuming that a data set comes equipped
with measurements like functions or metrics to filter it, persistent homology
transforms the filtered data into a nested family of chain complexes that
depend on as many parameters as the number of different measurements used.
Applying homology with coefficients in a field to such filtered chain complex
produces a parametrized family of vector spaces, connected by linear
transition maps, called a persistent homology module. Algebraic invariants of
persistent homology modules provide the required summaries of the data
topology.

Classically, the development of the theory of persistent homology originated
from two separate roots: Morse theory (as in, e.g., [Bar94, Fro96, Rob00,
ELZ02]), and commutative algebra (as in, e.g., [ZC05, CZ09, Oud15]). These two
perspectives reconcile very elegantly in the case of 1-parameter persistence,
i.e. when the filtration depends on only one parameter. In this case,
persistent homology modules admit a complete invariant, the so-called barcode,
encoding the lifespan of homology classes through the considered filtration.
From the standpoint of Morse theory, the endpoints of bars in a persistence
barcode correspond to cancellation pairs of critical points of the filtering
(Morse) function [ELZ02]. From the algebraic perspective, a persistent
homology module is a representation of a finite linear quiver in the category
of vector spaces. Thus, a 1-parameter persistence module admits a unique
decomposition into interval modules, i.e. indecomposable modules, each
supported on an interval. These intervals are exactly the bars of the
persistence barcode [ZC05].

It is of both theoretical and practical interest to understand persistent
homology in the case of multiple parameters, yielding to the so-called
multiparameter persistence. Indeed, in applications, one often needs to filter
the data using more than only one measurement, obtaining a multiparameter
persistence module. This is the case, for example, when there are different
drivers for a phenomenon [BCGS13], or when one needs to downsize the role of
outliers by adding a co-density measurement to the principal, explanatory,
measurement as in [FL13, BL22].

Unfortunately, the theory of multiparameter persistence modules proved to be
much more elusive than the single-parameter one: in particular, since
multigraded modules are of wild representation type [Gab72], more complicated
indecomposables than just intervals can generally occur, and it is impossible
to list them all or characterize them via discrete invariants. Despite this
difficulty, all the relevant homological events in a multiparameter filtration
are conveniently captured by the Betti tables of the multiparameter persistent
homology module [CZ09]. However, these events cannot be paired to obtain
summaries similar to barcodes, and their mutual dependencies cannot be easily
unveiled.

The main goal of this paper is to relate the events captured by the Betti
tables of a multiparameter persistent homology module to the events captured
by Morse theory, considered in its combinatorial formulation [For98, Koz05].
This attempt to reconnect the algebraic perspective to Morse theory in the
multiparameter situation is not only of theoretical interest in commutative
algebra but also of practical advantage, as it provides a unified perspective
to study persistent homology modules together with the underlying filtered
complexes.

Starting from the observation that for a 1-parameter persistent homology
module the support of the Betti tables coincides with the set of entrance
grades of critical cells in the filtration under consideration, our goal is to
understand whether and to what extent this fact can be generalized to
multiparameter persistence. An indication that this may be the case comes from
the results of [GL23], which establish Morse inequalities involving, on the
one hand, the values of the Betti tables of a multiparameter persistent
homology module, and, on the other hand, the so-called _homological critical
numbers_ of the same filtration. The latter numbers can be viewed as
theoretical lower bounds of the numbers of critical cells entering the
filtration at each filtration grade for any choice of a discrete gradient
vector field consistent with the filtration.

The main result of this paper delimits, in the space of parameters, the
support of the Betti tables of a persistent homology module obtained from a
multiparameter sublevel set filtration of measurement functions. More
precisely, Theorem 4.8 in Section 4 states that the Betti tables can be
nonzero only for parameter values in the closure of the set of entrance grades
of critical cells of a discrete gradient vector field consistent with the
filtration, where the closure is taken with respect to the least upper bound.

The key idea in the proof of this result is the introduction of the
construction of the Koszul complex via mapping cones in the context of
persistence (see Section 3). Using this inductive construction, we are able to
compute Betti tables by looking at the space of parameters only locally and,
more importantly, to disentangle the different parameters of the
multifiltration: the Koszul complex at a fixed grade in an $n$-parameter space
is determined by the Koszul complexes at nearby grades in a $(n-1)$-parameter
space.

Our main result of Section 4 can be strengthened in the case of 2-parameter
persistent homology modules. As a further contribution of this paper, we show
that, in this case, the support of the Betti tables of a persistence module is
contained in the closure of the set of _homological critical grades_ of a
filtration (see Section 5). Although limited to the case of 2-parameter
filtrations, this result improves our main result in two ways: it does not
depend on the choice of a specific discrete gradient vector field and
establishes that all events witnessed by the Betti tables are determined by
homological criticality (Corollary 5.6).

##  2 Preliminaries

Before presenting relevant background material for this article, let us
establish some general notations: $\mathbb{N}$ denotes the set
$\\{0,1,\ldots\\}$ of natural numbers; $[n]$ denotes the set
$\\{1,2,\ldots,n\\}$; $\\{e_{i}\\}_{i=1,\ldots,n}$ is the standard basis of
$\mathbb{N}^{n}$; for any subset $\alpha\subseteq[n]$, we denote
$e_{\alpha}\coloneqq\sum_{j\in\alpha}e_{j}$; $|J|$ denotes the cardinality of
a set $J$; the symbols $\wedge$ and $\vee$ denote the greatest lower bound and
least upper bound, respectively.

###  2.1 Based chain complexes, cell complexes, and homology

Let $\mathbb{F}$ denote a field, arbitrary but fixed. A _based chain complex_
is a chain complex $C_{*}=(C_{q},\partial_{q})_{q\in\mathbb{Z}}$ of vector
spaces over $\mathbb{F}$, which we assume to be of finite dimension, such that
each $C_{q}$ is endowed with a distinguished basis $X_{q}$. Throughout this
article, we assume all chain complexes to be bounded, meaning that $C_{q}=0$
whenever $q<0$ or $q\geq m$ for some integer $m$. Based chain complexes can be
viewed from a combinatorial perspective, as their distinguished bases inherit
the structure of an (abstract) _cell complex_ , in the sense of Lefschetz
[Lef42]. In this work, we call cell complex a finite graded set
$X=\bigsqcup_{q\in\mathbb{N}}X_{q}$, whose elements are called _cells_ ,
endowed with an _incidence function_ $\kappa:X\times X\to\mathbb{F}$. A cell
$\sigma\in X_{q}$ is said to have _dimension_ $q$, denoted $\dim\sigma=q$, or
to be a $q$-_cell_. The incidence function must satisfy two axioms: (i)
$\kappa(\tau,\sigma)\neq 0$ implies $\dim\tau=\dim\sigma+1$, and (ii)
$\sum_{\rho\in X}\kappa(\tau,\rho)\cdot\kappa(\rho,\sigma)=0$, for any pair of
cells $\tau$ and $\sigma$ in $X$. We endow $X$ with the order relation $\leq$,
called the _face partial order_ , generated by the _covering face relation_ :
$\sigma<\tau$ if and only if $\kappa(\tau,\sigma)\neq 0$. Given a cell complex
$X$, we denote $C_{*}(X)=(C_{q}(X),\partial_{q})_{q\in\mathbb{Z}}$ the based
chain complex such that $X_{q}$ is the fixed basis of $C_{q}$, for all $q$,
with differentials $\partial_{q}:C_{q}\to C_{q-1}$ defined on each $\tau\in
X_{q}$ by

$\partial_{q}(\tau)=\sum_{\sigma\in X_{q-1}}\kappa(\tau,\sigma)\sigma.$

We observe that $C_{*}(X)$ is the zero chain complex if $X=\emptyset$.

A graded set $A=\bigsqcup_{q\in\mathbb{N}}A_{q}$ is called a _subcomplex_ of
$X$ if, for all $\tau\in A$, every cell $\sigma\in X$ such that
$\sigma\leq\tau$ is also in $A$. This property makes $A$, endowed with the
restriction of the incidence function of $X$, a cell complex, and is
equivalent to requiring $C_{*}(A)$ to be a chain subcomplex of $C_{*}(X)$. We
denote by $H_{q}(X):=\ker\partial_{q}/\operatorname{im}\partial_{q+1}$ the
homology $\mathbb{F}$-modules of $C_{*}(X)$, and by $H_{q}(X,A)$ the homology
$\mathbb{F}$-modules of the relative chain complex $C_{*}(X,A)$.

We observe that the notion of a cell complex as reviewed above, equivalent to
that of a based chain complex, is general enough to include simplicial
complexes, cubical complexes, and finite regular CW complexes, among other
widely used combinatorial objects admitting a canonically associated chain
complex.

###  2.2 Multifiltrations and multiparameter persistence

One of the main mathematical objects of interest in topological data analysis
are functors from a poset to the category of finite dimensional vector spaces
over a field $\mathbb{F}$. Here, we consider the indexing poset
$\mathbb{N}^{n}$, for some integer $n\geq 1$, equipped with the coordinate-
wise partial order: for $u=(u_{i}),v=(v_{i})\in\mathbb{N}^{n}$, we write
$u\preceq v$ if and only if $u_{i}\leq v_{i}$, for all $1\leq i\leq n$. In
this article, an $n$-_parameter persistence module_ is a functor from the
poset $(\mathbb{N}^{n},\preceq)$ with values in finite-dimensional
$\mathbb{F}$-vector spaces. Morphisms between such functors are the natural
transformations. Explicitly, an $n$-parameter persistence module $V$ consists
of a family $\\{V^{u}\\}_{u\in\mathbb{N}^{n}}$ of $\mathbb{F}$-vector spaces
together with a family $\\{\varphi^{u,v}:V^{u}\to V^{v}\\}_{u\preceq
v\in\mathbb{N}^{n}}$ of linear maps such that
$\varphi^{u,w}=\varphi^{v,w}\circ\varphi^{u,v}$ whenever $u\preceq v\preceq
w$, and $\varphi^{u,u}=\operatorname{id}_{V^{u}}$, for all $u$. A morphism
between two $n$-parameter persistence modules $\\{V^{u},\varphi^{u,v}\\}$ and
$\\{W^{u},\psi^{u,v}\\}$ is a family of linear maps $\\{\nu^{u}:V^{u}\to
W^{u}\\}_{u\in\mathbb{N}^{n}}$ such that
$\nu^{v}\circ\varphi^{u,v}=\psi^{u,v}\circ\nu^{u}$, for all $u\preceq v$ in
$\mathbb{N}^{n}$. A morphism $\nu$ is an isomorphism (monomorphism,
epimorphism, respectively) if, and only if, its components $\nu^{u}$ are
bijective (injective, surjective), for all $u\in\mathbb{N}^{n}$.

In topological data analysis, the typical source of persistence modules are
filtrations of cell complexes associated with the data. An $n$-_filtration_ of
a cell complex $X$ is a family $\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ of
subcomplexes of $X$ such that $u\preceq v$ implies $X^{u}\subseteq X^{v}$. If
a cell $\sigma$ of $X$ is an element of $X^{u}-\bigcup_{j=1}^{n}X^{u-e_{j}}$,
we say that $u$ is an entrance grade of $\sigma$ in the filtration. In this
article we assume, once and for all, that filtrations
$\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ are families of sublevel sets
$X^{u}=\\{\sigma\in X\mid h(\sigma)\preceq u\\}$ of some order-preserving
function $h:(X,\leq)\to(\mathbb{N}^{n},\preceq)$, with $\leq$ denoting the
face partial order on $X$. This assumption is equivalent to requiring every
cell of $X$ to have exactly one entrance grade.

The filtrations we are considering are usually called _one-critical_ [CSZ09]
in topological data analysis. We want to highlight that assuming the
uniqueness of entrance grades is fundamental in order to obtain the results of
this article, which are false for general filtrations of cell complexes. For
instance, in this article we repeatedly use the following fact.

######  Remark 2.1.

Given a one-critical $n$-filtration $\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ and a
finite set of filtration grades
$\\{u_{j}\\}_{j=1,\ldots,k}\subseteq\mathbb{N}^{n}$, with
$u_{j}=(u_{j,1},\ldots,u_{j,n})$ for all $j$, we have
$\bigcap_{j=1}^{k}X^{u_{j}}=X^{w}$, where $w=\bigwedge\\{u_{j}\\}_{j}$ is the
greatest lower bound $(\min\\{u_{j,1}\\}_{j},\ldots,\min\\{u_{j,n}\\}_{j})$ of
the subset $\\{u_{j}\\}_{j=1,\ldots,k}$ in $\mathbb{N}^{n}$. In particular,
for each subset $\alpha\subseteq[n]$, we have the equality
$\bigcap_{j\in\alpha}X^{u-e_{j}}=X^{u-e_{\alpha}}$.

We are interested in persistence modules obtained as the homology of an
$n$-filtration. Given an $n$-filtration $\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ and
applying the $q$th homology functor, one obtains the $n$-_parameter persistent
$q$th-homology module_ ${V}_{q}=\\{{V}_{q}^{u},\iota^{u,v}_{q}\\}_{u\preceq
v\in\mathbb{N}^{n}}$, with ${V}_{q}^{u}:=H_{q}(X^{u})$ and
$\iota^{u,v}_{q}\colon H_{q}(X^{u})\to H_{q}(X^{v})$ induced by the inclusion
maps $X^{u}\hookrightarrow X^{v}$ for $u\preceq v$. We note that it is common
to use the terms _multifiltration_ and _multiparameter_ in place of,
respectively, $n$-filtration and $n$-parameter, to indicate the generic case
when $n>1$. Moreover, $2$-filtrations are also called _bifiltrations_.

The overall purpose of this work is to study the relation between the
homological invariants of multiparameter persistent homology modules called
Betti tables and the multifiltrations from which they are obtained. To this
aim, we adopt some tools and terminology from commutative algebra. An
$n$-_graded module_ over the polynomial ring
$S:=\mathbb{F}[x_{1},\ldots,x_{n}]$ is an $S$-module with a vector space
decomposition $V=\bigoplus_{u\in\mathbb{N}^{n}}V^{u}$ such that $x_{i}\cdot
V^{u}\subseteq V^{u+e_{i}}$, for all $u\in\mathbb{N}^{n}$ and $i\in[n]$. There
is a standard equivalence [CZ09] between the category of $n$-parameter
persistence modules and the category of $n$-graded $S$-modules, allowing us to
view a persistence module $\\{V^{u},\varphi^{u,v}\\}$ as the $n$-graded
$S$-module $\bigoplus_{u\in\mathbb{N}^{n}}V^{u}$, where the action of $S$ is
defined by $x_{i}\cdot z=\varphi^{u,u+e_{i}}(z)$, for all $z\in V^{u}$ and
$i\in[n]$. Standard homological invariants from commutative algebra, like the
_Betti tables_ (also called _multigraded Betti numbers_ , see e.g. [MS05] for
background), were among the first ones studied in multiparameter persistence
[CZ09, Knu08]. Given an $n$-parameter persistent homology module
$\\{V_{q}^{u},\iota_{q}^{u,v}\\}$, obtained as the $q$th homology of an
$n$-filtration, we view it as the $n$-graded $S$-module
$V_{q}=\bigoplus_{u\in\mathbb{N}^{n}}V_{q}^{u}$ and denote its $i$th Betti
table by $\xi_{i}^{q}$, for $i\in\\{0,1,\ldots,n\\}$. We recall that its Betti
tables are functions $\xi_{i}^{q}:\mathbb{N}^{n}\to\mathbb{N}$ defined by

$\xi_{i}^{q}(u)=\dim(\operatorname{Tor}_{i}^{S}(V_{q},\mathbb{F}))^{u},$

for all $u\in\mathbb{N}^{n}$. Explicitly, $\xi^{q}_{i}(u)$ is the dimension
(as an $\mathbb{F}$-vector space) of the piece of grade $u$ of the $n$-graded
$S$-module $\operatorname{Tor}_{i}^{S}(V_{q},\mathbb{F})$. In Section 3 we
give an equivalent definition of the Betti tables based on the Koszul complex.

###  2.3 Discrete Morse theory and multifiltrations

Discrete Morse theory, developed by Forman [For98], is an adaptation of smooth
Morse theory [Mil63] to a combinatorial framework. In its original
formulation, it allows, given a regular CW complex, to construct a homotopy
equivalent CW complex with a smaller number of cells. Building on Forman’s
work, discrete Morse theory has been formulated in purely algebraic terms for
based chain complexes [Koz05] and in more general frameworks [Skö06, JW09]. In
this algebraic setting, the aim is to decompose a chain complex into a smaller
complex and an acyclic complex. As explained in Section 2.1, one can always
take an equivalent combinatorial perspective by considering the cell complexes
associated with based cell complexes. We briefly present here the main ideas
of algebraic discrete Morse theory in the setting of this work.

Let $C_{*}(X)$ be the chain complex associated with a cell complex
$X=\bigsqcup_{q}X_{q}$, and let $<$ be the covering face relation on $X$
introduced in Section 2.1. A pair of cells $(\sigma,\tau)\in X\times X$ with
$\sigma<\tau$ is called a _discrete vector_. A _discrete vector field_
$\mathcal{V}$ on $X$ is a collection of discrete vectors
$\mathcal{V}=\\{(\sigma_{j},\tau_{j})\\}_{j\in J}$ such that all cells
appearing in $\mathcal{V}$ (indifferently as the first or the second component
of a vector) are different. A discrete vector field $\mathcal{V}$ determines a
partition of $X$ into three graded subsets $M,S,T$, where $M$ is the set of
unpaired cells, called _critical cells_ , and $S$ (respectively, $T$) is the
set of cells appearing in $\mathcal{V}$ as first (respectively, second)
components of a discrete vector. The subsets $M,S,T$ inherit the grading by
dimension of the cells of $X$, so that for example $M=\bigsqcup_{q}M_{q}$. A
$\mathcal{V}$-_path_ between two cells $\sigma$ and $\sigma^{\prime}$ is a
sequence
$(\sigma_{0},\tau_{0},\sigma_{1},\tau_{1},\ldots,\sigma_{r-1},\tau_{r-1},\sigma_{r})$
with $r\geq 1$ such that $\sigma_{0}=\sigma$, $\sigma_{r}=\sigma^{\prime}$,
each $(\sigma_{i},\tau_{i})$ is a discrete vector of $\mathcal{V}$, and
$\sigma_{i+1}<\tau_{i}$. The $\mathcal{V}$-path is called _closed_ if
$\sigma_{r}=\sigma_{1}$ and _trivial_ if $r=1$. A discrete vector field
$\mathcal{V}$ is a _discrete gradient vector field_ (also called an _acyclic
matching_ or a _Morse matching_) when all closed $\mathcal{V}$-paths are
trivial.

The core result of discrete Morse theory [For98] can be algebraically stated
as follows [KMS98, Skö06, JW09].

######  Theorem 2.2.

Let $C_{*}(X)=(C_{q}(X),\partial_{q})_{q\in\mathbb{Z}}$ be the chain complex
associated with a cell complex $X=\bigsqcup_{q}X_{q}$ and let
$\mathcal{V}=\\{(\sigma_{j},\tau_{j})\\}_{j\in J}$ be a discrete gradient
vector field on $X$. Then $C_{*}(X)$ is chain homotopy equivalent to
$C_{*}(M)=(C_{q}(M),\partial^{M}_{q})_{q\in\mathbb{Z}}$, where
$M=\bigsqcup_{q}M_{q}$ is the set of critical cells and $\partial^{M}$ is a
differential determined by $\partial$ and $\mathcal{V}$.

We call $C_{*}(M)$ the _(discrete) Morse chain complex_ of $C_{*}(X)$
associated with $\mathcal{V}$. Let us stress that in general $C_{*}(M)$ is not
a chain subcomplex of $C_{*}(X)$, since its differential $\partial^{M}$ is not
simply induced by restriction by the differential $\partial$ of $C_{*}(X)$.
The details on how $\partial^{M}$ is (uniquely) determined by $\partial$ and
$\mathcal{V}$ can be found in [Skö06, JW09]. Equivalently, a cell complex
structure on the set $M=\bigsqcup_{q}M_{q}$, called the _(discrete) Morse
complex_ of $X$ associated with $\mathcal{V}$, is determined by the incidence
function of $X$ and $\mathcal{V}$ [KMS98]. In general, $M$ is not a subcomplex
of $X$.

Discrete Morse theory of filtered chain complexes has been studied in a series
of works related to one-parameter [MN13] or multiparameter persistent homology
[AKL17]. In the remainder of this subsection, we present the main ideas of
discrete Morse theory for multifiltrations.

Consider an $n$-filtration $\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ of a cell
complex $X$, which determines a filtration
$\\{C_{*}(X^{u})\\}_{u\in\mathbb{N}^{n}}$ of the chain complex $C_{*}(X)$.
Given a discrete gradient vector field $\mathcal{V}$ on $X$, there are clearly
induced filtrations $\\{M^{u}\\}_{u\in\mathbb{N}^{n}}$ on the Morse complex
$M$ and $\\{C_{*}(M^{u})\\}_{u\in\mathbb{N}^{n}}$ on the Morse chain complex
$C_{*}(M)=(C_{q}(M),\partial^{M}_{q})$. In general, the former is only a
filtration of sets and the latter is only a filtration of graded
$\mathbb{F}$-vector spaces, as the differential $\partial^{M}$ may fail to be
compatible with the filtration. To avoid this, one can require the discrete
gradient vector field to interact nicely with the multifiltration on $X$.

######  Definition 2.3.

A discrete gradient vector field $\mathcal{V}$ on $X$ is _consistent_ with a
multifiltration $\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ if, for all $(\sigma,\tau)$
in $\mathcal{V}$ and all $u\in\mathbb{N}^{n}$, $\sigma\in X^{u}$ if and only
if $\tau\in X^{u}$.

If $\mathcal{V}$ is consistent with the multifiltration
$\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$, then
$\\{C_{*}(M^{u})\\}_{u\in\mathbb{N}^{n}}$ is a filtration of chain
subcomplexes of $C_{*}(M)$ [MN13, AKL17]. Equivalently,
$\\{M^{u}\\}_{u\in\mathbb{N}^{n}}$ is a filtration of subcomplexes of $M$.
Moreover, the persistent homology modules associated with the multifiltrations
of $X$ and its Morse complex are isomorphic (in the sense of Section 2.2).

######  Proposition 2.4 (Lemma 3.10 in [AKL17]).

Let $\mathcal{V}$ be a discrete gradient vector field on a cell complex $X$
consistent with an $n$-filtration $\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$, and let
$\\{M^{u}\\}_{u\in\mathbb{N}^{n}}$ be the $n$-filtration induced on the Morse
complex $M$. Then, for any $q\in\mathbb{N}$, the persistence modules obtained
as $q$-th homology of the $n$-filtrations $\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$
and $\\{M^{u}\\}_{u\in\mathbb{N}^{n}}$ are isomorphic.

##  3 The Koszul complex of a persistence module

In this section, we describe the Koszul complex associated with an
$n$-parameter persistence module and illustrate some of its properties. In
particular, given an $n$-parameter persistent homology module
$\\{H_{q}(X^{u}),\iota_{q}^{u,v}\\}$, we introduce its Koszul complex at
$u\in\mathbb{N}^{n}$, a chain complex whose $i$th homology module has
dimension equal to the Betti table value $\xi^{q}_{i}(u)$. This chain complex
can be constructed via a repeated procedure which allows us to add one
parameter of the multifiltration at a time.

In Section 3.1, upon briefly recalling general definitions and results, we
provide a more detailed description of Koszul complexes of multiparameter
persistent homology modules. We claim no original results in this subsection,
as the Koszul complex is a standard tool, and the explicit description of its
chain modules and differentials in the case of persistent homology modules is
included, for example, in [GL23, Sect. 3]. Here, besides fixing notations, we
provide further details, especially with regard to bifiltrations, that are
relevant to this work.

In Section 3.2, we explain how the Koszul complex associated with an
$n$-parameter persistence module can be constructed as an iterated mapping
cone, and we highlight the role of this construction for persistent homology
modules, which intuitively allows one to disentangle the different parameters
of the multifiltration and study their impact on the Betti tables. In Section
4, we will apply this technique to study the support of the Betti tables.

###  3.1 The Koszul complex of a multigraded module

Let $S$ denote the polynomial ring $\mathbb{F}[x_{1},\ldots,x_{n}]$. The
_Koszul complex_ $\mathbb{K}_{*}$ is a chain complex of free $n$-graded
$S$-modules whose construction is standard in commutative algebra [MS05, Def.
1.26]. Given an $n$-graded $S$-module $V=\bigoplus_{u\in\mathbb{N}^{n}}V^{u}$,
the _Koszul complex_ $\mathbb{K}_{*}(x_{1},\ldots,x_{n};V)(u)$ _of $V$ at
grade $u\in\mathbb{N}^{n}$_ is the piece of grade $u$ of the ($n$-graded)
chain complex $V\otimes_{S}\mathbb{K}_{*}$. This chain complex of
$\mathbb{F}$-vector spaces can be used to determine the Betti tables
$\xi_{i}(u):=\dim(\operatorname{Tor}_{i}^{S}(V,\mathbb{F}))^{u}$ of $V$ at
grade $u$, for $i\in\\{0,1,\ldots,n\\}$ (see Section 2.2). Indeed, by
definition, $\operatorname{Tor}_{i}^{S}(V,\mathbb{F})$ can be determined by
applying the functor $-\otimes_{S}\mathbb{F}$ to a free resolution of $V$ and
taking $i$th homology of the resulting chain complex. The roles of $V$ and
$\mathbb{F}$ can however be interchanged, by virtue of the isomorphism
$\operatorname{Tor}_{i}^{S}(V,\mathbb{F})\cong\operatorname{Tor}_{i}^{S}(\mathbb{F},V)$
(see, e.g., [Rot09, Thm. 7.1]); in this case, choosing $\mathbb{K}_{*}$ as a
(minimal) free resolution of $\mathbb{F}$ (see [MS05, Prop. 1.28]) yields, for
all $i\in\\{0,1,\ldots,n\\}$, the equality

$\xi_{i}(u)=\dim H_{i}(\mathbb{K}_{*}(x_{1},\ldots,x_{n};V)(u)).$

Let us now provide a more explicit description of the Koszul complex
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ of a persistent homology module
$\\{H_{q}(X^{u}),\iota_{q}^{u,v}\\}$ associated with an $n$-parameter
filtration $\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$, regarded as an $n$-graded
$S$-module $V_{q}=\bigoplus_{u\in\mathbb{N}^{n}}H_{q}(X^{u})$ (as reviewed in
Section 2.2). Even if this description of the Koszul complex can be easily
adapted to any $n$-parameter persistence module, not necessarily built from a
filtered cell complex, we prefer to focus on the case of interest for this
work in order to clearly introduce the notations we will use in what follows.
We recall that, for any subset $\alpha\subseteq[n]$, we set
$e_{\alpha}\coloneqq\sum_{j\in\alpha}e_{j}$.

For each $i\in\\{0,1,\ldots,n\\}$, the chain module in degree $i$ of
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ is

$\mathbb{K}_{i}(x_{1},\ldots,x_{n};V_{q})(u)=\bigoplus_{\alpha\subseteq[n],\;|\alpha|=i}H_{q}(X^{u-e_{\alpha}}).$

The definition can be easily extended if, for some fixed $u\in\mathbb{N}^{n}$
and some $\alpha\subseteq[n]$, it happens that
$u-e_{\alpha}\notin\mathbb{N}^{n}$: throughout this article, by definition, we
set $X^{w}=\emptyset$ whenever the grade $w$ is not in $\mathbb{N}^{n}$. Note
that the modules $\mathbb{K}_{i}(x_{1},\ldots,x_{n};V_{q})(u)$ are zero for
all $i\notin\\{0,1,\ldots n\\}$. The differentials of
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ are defined in terms of the maps
$\iota^{v,w}_{q}:H_{q}(X^{v})\to H_{q}(X^{w})$ as follows: the restriction of

$d_{i}:\mathbb{K}_{i}(x_{1},\ldots,x_{n};V_{q})(u)\to\mathbb{K}_{i-1}(x_{1},\ldots,x_{n};V_{q})(u)$

to each direct summand $H_{q}(X^{u-e_{\alpha}})$ of its domain, with
$\alpha=\\{j_{1}<j_{2}<\ldots<j_{i}\\}$, is

$d_{i|}=\sum_{r=0}^{i-1}(-1)^{r}\iota_{q}^{u-e_{\alpha},\,u-e_{\alpha}+e_{j_{i-r}}}.$

For the sake of a simpler notation, we avoid denoting the grade $u$ in the
differentials $d_{i}$. As we explained, $\xi^{q}_{i}(u)$ coincides with the
dimension (as an $\mathbb{F}$-vector space) of the $i$th homology module of
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$.

Let us detail the cases of $n=1$ and $n=2$ parameters for later convenience.
For a $1$-parameter filtration $\\{X^{u}\\}_{u\in\mathbb{N}}$, the Koszul
complex $\mathbb{K}_{*}(x_{1};V_{q})(u)$ of
$V_{q}=\bigoplus_{u\in\mathbb{N}}H_{q}(X^{u})$ at $u\in\mathbb{N}$ is

$0\xrightarrow{}H_{q}(X^{u-1})\xrightarrow{d_{1}=\iota_{q}^{u-1,u}}H_{q}(X^{u})\xrightarrow{}0.$

The Betti tables at grade $u$ are
$\xi_{0}^{q}(u)=\dim\operatorname{coker}\iota_{q}^{u-1,u}$ and
$\xi_{1}^{q}(u)=\dim\ker\iota_{q}^{u-1,u}$, which correspond respectively to
the number of _births_ and _deaths_ of $q$-homology classes at
$u\in\mathbb{N}$ in the sense of persistence [ELZ02].

For a $2$-parameter filtration $\\{X^{u}\\}_{u\in\mathbb{N}^{2}}$, the Koszul
complex $\mathbb{K}_{*}(x_{1},x_{2};V_{q})(u)$ of the module
$V_{q}=\bigoplus_{u\in\mathbb{N}^{2}}H_{q}(X^{u})$ at $u\in\mathbb{N}^{2}$ is

$0\xrightarrow{}H_{q}(X^{u-e_{1}-e_{2}})\xrightarrow{d_{2}}H_{q}(X^{u-e_{1}})\oplus
H_{q}(X^{u-e_{2}})\xrightarrow{d_{1}}H_{q}(X^{u})\xrightarrow{}0,$

with differentials

$\displaystyle d_{2}$
$\displaystyle=\begin{bmatrix}-\iota_{q}^{u-e_{1}-e_{2},u-e_{1}}\\\
\iota_{q}^{u-e_{1}-e_{2},u-e_{2}}\end{bmatrix}\qquad\text{and}\qquad
d_{1}=[\iota_{q}^{u-e_{1},u}\quad\iota_{q}^{u-e_{2},u}].$

The Betti tables at the grade $u$ are

$\xi_{2}^{q}(u)=\dim\ker d_{2},\qquad\xi_{1}^{q}(u)=\dim(\ker
d_{1}/\operatorname{im}d_{2}),\qquad\xi_{0}^{q}(u)=\dim\operatorname{coker}d_{1}.$

A morphism $\nu=\\{\nu^{u}:V^{u}\to W^{u}\\}_{u\in\mathbb{N}^{n}}$ between
$n$-parameter persistence modules $\\{V^{u},\varphi^{u,v}\\}$ and
$\\{W^{u},\psi^{u,v}\\}$ induces a chain map between the Koszul complexes of
$V=\bigoplus_{u\in\mathbb{N}^{n}}V^{u}$ and
$W=\bigoplus_{u\in\mathbb{N}^{n}}W^{u}$ at $u\in\mathbb{N}^{n}$, the morphism
between the chain modules in degree $i$ being
$\bigoplus_{|\alpha|=i}\nu^{u-e_{\alpha}}$, with $\alpha\subseteq[n]$.
Clearly, an isomorphism between persistence modules induces an isomorphism
between their Koszul complexes.

In what follows, we will apply the previous observation in the particular case
of a multifiltration $\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ of $X$ and the induced
multifiltration $\\{M^{u}\\}_{u\in\mathbb{N}^{n}}$ of its Morse complex. By
virtue of Proposition 2.4, since the modules
$V_{q}:=\bigoplus_{u\in\mathbb{N}^{n}}H_{q}(X^{u})$ and
$V^{\prime}_{q}:=\bigoplus_{u\in\mathbb{N}^{n}}H_{q}(M^{u})$ are isomorphic,
their Koszul complexes $\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ and
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V^{\prime}_{q})(u)$ are also isomorphic, at
all $u\in\mathbb{N}^{n}$. As a consequence, the Betti tables $\xi_{i}^{q}(u)$
can be determined considering the Morse complex instead of the original
complex.

###  3.2 Explicit construction via mapping cones

We now illustrate the explicit construction of the Koszul complex
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ of $V_{q}$ at grade
$u\in\mathbb{N}^{n}$ as an iterated mapping cone. The classical construction
of the Koszul complex via mapping cones can be found in [Eis05, § A2F] and
[BH98, Ch. 1.6]; here we rephrase, adapt, and enrich it with examples, to
provide a complete and explicit treatment for Koszul complexes of persistent
homology modules that conveys the intuition of persistent homology.

Given a chain map $f:B_{*}\to C_{*}$, the _mapping cone_
$\operatorname{Cone}(f)_{*}$ of $f$ is the chain complex with
$\operatorname{Cone}(f)_{i}\coloneqq B_{i-1}\oplus C_{i}$ and differential
$\delta_{i}:B_{i-1}\oplus C_{i}\to B_{i-2}\oplus C_{i-1}$ defined by

$\delta_{i}(b,c)\coloneqq(-\partial^{B}_{i-1}(b),\;\partial^{C}_{i}(c)+f_{i-1}(b)),$
(3.1)

for all $i$, with $b\in B_{i-1},c\in C_{i}$ and $\partial^{B},\partial^{C}$
respectively denoting the differentials of $B_{*}$ and $C_{*}$, see [Eis95, §
A3.12].

Let $\mathcal{F}:=\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ be an $n$-filtration of a
cell complex $X$. As is evident from the definitions in Section 3.1, the
Koszul complex $\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ of the associated
persistent homology module $V_{q}=\bigoplus_{u\in\mathbb{N}^{n}}H_{q}(X^{u})$
at the fixed grade $u\in\mathbb{N}^{n}$ only depends on the subcomplexes
$X^{u-e_{\alpha}}$ of the filtration, with $\alpha\subseteq[n]$. In other
words, to determine $\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ it is enough
to consider the smaller $n$-filtration
$\mathcal{F}^{u}:=\\{X^{u-e_{\alpha}}\\}_{\alpha\subseteq[n]}$, containing
$2^{n}$ subcomplexes of the original $n$-filtration $\mathcal{F}$. We observe
that, fixed any $j\in[n]$, the $n$-filtration $\mathcal{F}^{u}$ can be
partitioned into $2^{n-1}$ $1$-filtrations $X^{u-e_{\alpha}-e_{j}}\subseteq
X^{u-e_{\alpha}}$, one for each $\alpha\subseteq[n]\smallsetminus\\{j\\}$.
More generally, fixed any non-empty subset
$J\coloneqq\\{j_{1},\ldots,j_{t}\\}\subseteq[n]$, there is a partition of
$\mathcal{F}^{u}$ consisting of $2^{n-t}$ $t$-filtrations of the form
$\\{X^{u-e_{\alpha}-e_{\gamma}}\\}_{\gamma\subseteq J}$, one for each
$\alpha\subseteq[n]\smallsetminus J$. Every such $t$-filtration has an
associated Koszul complex
$\mathbb{K}_{*}(x_{j_{1}},\ldots,x_{j_{t}};V_{q})(u-e_{\alpha})$ that
intuitively only encodes information on the parameters $j_{1},\ldots,j_{t}$ of
the $n$-filtration $\mathcal{F}^{u}$. Given $k\in[n]\smallsetminus J$,
regarded here as an additional parameter to be taken into account, one can
consider the $(t+1)$-filtration given by the union of two $t$-filtrations
$\\{X^{u-e_{\alpha}-e_{\gamma}}\\}_{\gamma\subseteq J}$ and
$\\{X^{u-e_{\alpha}-e_{k}-e_{\gamma}}\\}_{\gamma\subseteq J}$, for any
$\alpha\subseteq[n]\smallsetminus(J\cup\\{k\\})$. Below, we will explain how
the Koszul complex associated with such $(t+1)$-filtration can be constructed
as the mapping cone of a chain map between the two Koszul complexes associated
with the $t$-filtrations.

We begin by illustrating in detail the first few steps of the procedure based
on iterated mapping cones to construct the Koszul complex
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ starting from “$1$-parameter”
Koszul complexes “in direction $e_{j}$”

$\mathbb{K}_{*}(x_{j};V_{q})(w)=\left(0\xrightarrow{}H_{q}(X^{w-e_{j}})\xrightarrow{d_{1}=\iota_{q}^{w-e_{j},w}}H_{q}(X^{w})\xrightarrow{}0\right),$

for any fixed $j\in[n]$ and for $w=u-e_{\alpha}$ with
$\alpha\subseteq[n]\smallsetminus\\{j\\}$, and from specific chain maps
between them. The chain maps are those induced by inclusions “in direction
$e_{k}$”, for any fixed $k\in[n]\setminus\\{j\\}$, that is

$f^{k}(x_{j};V_{q})(w-e_{k}):\mathbb{K}_{*}(x_{j};V_{q})(w-e_{k})\to\mathbb{K}_{*}(x_{j};V_{q})(w),$

with
$f_{i}^{k}(x_{j};V_{q})(w-e_{k}):\mathbb{K}_{i}(x_{j};V_{q})(w-e_{k})\to\mathbb{K}_{i}(x_{j};V_{q})(w)$
defined, for degrees $i=0,1$, as

$\displaystyle f_{0}^{k}(x_{j};V_{q})(w-e_{k})$
$\displaystyle=\iota_{q}^{w-e_{k},w}:H_{q}(X^{w-e_{k}})\xrightarrow{}H_{q}(X^{w}),$
$\displaystyle f_{1}^{k}(x_{j};V_{q})(w-e_{k})$
$\displaystyle=\iota_{q}^{w-e_{j}-e_{k},w-e_{j}}:H_{q}(X^{w-e_{j}-e_{k}})\xrightarrow{}H_{q}(X^{w-e_{j}}).$

The mapping cone $\operatorname{Cone}(f^{k}(x_{j};V_{q})(w-e_{k}))_{*}$ is the
Koszul complex $\mathbb{K}_{*}(x_{j},x_{k};V_{q})(w)$, associated with the
$2$-filtration $\\{X^{w-e_{\gamma}}\\}_{\gamma\subseteq\\{j,k\\}}$.
Intuitively, it is obtained from the previous step, where only the $j$th
parameter was considered, by adding one parameter more, namely the $k$th
parameter of the original $n$-filtration. Explicitly,
$\mathbb{K}_{*}(x_{j},x_{k};V_{q})(w)$ is the chain complex

$0\xrightarrow{}H_{q}(X^{w-e_{j}-e_{k}})\xrightarrow{\;d_{2}\;}H_{q}(X^{w-e_{j}})\oplus
H_{q}(X^{w-e_{k}})\xrightarrow{\;d_{1}\;}H_{q}(X^{w})\xrightarrow{}0$

where the differentials, applying the definition (3.1), are

$\displaystyle d_{2}$
$\displaystyle=\begin{bmatrix}-\iota_{q}^{w-e_{j}-e_{k},w-e_{j}}\\\
\iota_{q}^{w-e_{j}-e_{k},w-e_{k}}\end{bmatrix}\qquad\text{and}\qquad
d_{1}=[\iota_{q}^{w-e_{j},w}\quad\iota_{q}^{w-e_{k},w}].$

The process we just described can be repeated, by choosing a new “direction”
$e_{\ell}$ corresponding to a new parameter
$\ell\in[n]\smallsetminus\\{j,k\\}$ and constructing
$\mathbb{K}_{*}(x_{j},x_{k},x_{\ell};V_{q})(w)$ as the mapping cone of the
chain map $f^{\ell}(x_{j},x_{k};V_{q})(w-e_{\ell})$ induced by inclusions in
direction $e_{\ell}$, for each $w=u-e_{\alpha}$ with
$\alpha\subseteq[n]\smallsetminus\\{j,k,\ell\\}$. Explicitly,
$f^{\ell}(x_{j},x_{k};V_{q})(w-e_{\ell})$ is defined by the following maps, in
degrees $i=0,1,2$:

$\displaystyle f_{0}^{\ell}(x_{j},x_{k};V_{q})(w-e_{\ell})$
$\displaystyle=\iota_{q}^{w-e_{\ell},w},$ $\displaystyle
f_{1}^{\ell}(x_{j},x_{k};V_{q})(w-e_{\ell})$
$\displaystyle=\iota_{q}^{w-e_{j}-e_{\ell},w-e_{j}}\oplus\iota_{q}^{w-e_{k}-e_{\ell},w-e_{k}},$
$\displaystyle f_{2}^{\ell}(x_{j},x_{k};V_{q})(w-e_{\ell})$
$\displaystyle=\iota_{q}^{w-e_{j}-e_{k}-e_{\ell},w-e_{j}-e_{k}}.$

If the order in which the indeterminates are added is changed, one obtains
isomorphic chain complexes: for example,
$\mathbb{K}_{*}(x_{j},x_{k},x_{\ell};V_{q})(w)$ is isomorphic to
$\mathbb{K}_{*}(x_{j},x_{\ell},x_{k};V_{q})(w)$. At the last step, one obtains
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ as the mapping cone of the chain
map $f^{m}(x_{1},\ldots,\hat{x}_{m},\ldots,x_{n};V_{q})(u-e_{m})$ between
$\mathbb{K}_{*}(x_{1},\ldots,\hat{x}_{m},\ldots,x_{n};V_{q})(u-e_{m})$ and
$\mathbb{K}_{*}(x_{1},\ldots,\hat{x}_{m},\ldots,x_{n};V_{q})(u)$.

Thanks to the iterative nature of the process, we can provide an explicit
description of the chain complex
$\mathbb{K}_{*}(x_{j_{1}},\ldots,x_{j_{t}};V_{q})(u)$ for any
$u\in\mathbb{N}^{n}$ and any non-empty subset
$J\coloneqq\\{j_{1},\ldots,j_{t}\\}\subseteq[n]$. For each
$i\in\\{0,1,\ldots,|J|\\}$, the chain module in degree $i$ is

$\mathbb{K}_{i}(x_{j_{1}},\ldots,x_{j_{t}};V_{q})(u)=\bigoplus_{\gamma\subseteq
J,\;|\gamma|=i}H_{q}(X^{u-e_{\gamma}}).$

The modules $\mathbb{K}_{i}(x_{j_{1}},\ldots,x_{j_{t}};V_{q})(u)$ are zero for
all $i\notin\\{0,1,\ldots,|J|\\}$. The differentials of the chain complex
$\mathbb{K}_{*}(x_{j_{1}},\ldots,x_{j_{t}};V_{q})(u)$ can be described as
follows: the restriction of

$d_{i}:\mathbb{K}_{i}(x_{j_{1}},\ldots,x_{j_{t}};V_{q})(u)\to\mathbb{K}_{i-1}(x_{j_{1}},\ldots,x_{j_{t}};V_{q})(u)$

to each direct summand $H_{q}(X^{u-e_{\gamma}})$ of its domain, with
$\gamma=\\{j_{s(1)},\ldots,j_{s(i)}\\}$ and $s(1)<\cdots<s(i)$, is

$d_{i|}=\sum_{r=0}^{i-1}(-1)^{r}\iota_{q}^{u-e_{\gamma},\,u-e_{\gamma}+e_{j_{s(i-r)}}}.$

For any $k\in[n]\smallsetminus J$, the Koszul complex
$\mathbb{K}_{*}(x_{j_{1}},\ldots,x_{j_{t}},x_{k};V_{q})(u)$ is the mapping
cone of the chain map induced by inclusions in direction $e_{k}$,

$f^{k}(x_{j_{1}},\ldots,x_{j_{t}};V_{q})(u-e_{k}):\mathbb{K}_{*}(x_{j_{1}},\ldots,x_{j_{t}};V_{q})(u-e_{k})\to\mathbb{K}_{*}(x_{j_{1}},\ldots,x_{j_{t}};V_{q})(u),$

which for each degree $i\in\\{0,1,\ldots,|J|\\}$ is defined by

$f_{i}^{k}(x_{j_{1}},\ldots,x_{j_{t}};V_{q})(u-e_{k})=\bigoplus_{\gamma\subseteq
J,\;|\gamma|=i}\iota_{q}^{u-e_{\gamma}-e_{k},\,u-e_{\gamma}}.$

In Section 4, several results will be obtained by showing certain mapping
cones to be acyclic, i.e. having vanishing homology in all degrees. We recall
the following immediate consequence of [Eis95, Prop. A3.19] (see also [Wei94,
Corollary 1.5.4]), which gives an equivalent condition to the acyclicity of a
mapping cone.

######  Proposition 3.1.

A chain map $f:B_{*}\to C_{*}$ is a quasi-isomorphism (i.e., it induces
isomorphisms $H_{q}(B_{*})\cong H_{q}(C_{*})$ in homology, for all
$q\in\mathbb{Z}$) if and only if $\operatorname{Cone}(f)_{*}$ is acyclic.

######  Corollary 3.2.

Let $f:B_{*}\to C_{*}$ be a chain map, and let $B_{*}$ and $C_{*}$ be acyclic.
Then $\operatorname{Cone}(f)_{*}$ is acyclic.

###### Proof.

If $B_{*}$ and $C_{*}$ are acyclic, the chain map $f$ must be a quasi-
isomorphism. ∎

##  4 Entrance grades of critical cells and support of Betti tables

In this section, we are interested in relating the set of parameter grades at
which new critical cells enter the filtration of the Morse complex, on the one
hand, and the set of parameter grades where the Betti tables are nonzero, on
the other hand. Additionally, we look for a relation between the dimension of
the critical cells and the degree of the persistent homology module whose
Koszul complex they impact on.

Our fixed setting for the whole section will be as follows. Let
$\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ be an $n$-parameter filtration of a cell
complex $X$, let $\mathcal{V}$ be a fixed discrete gradient vector field
consistent with the filtration (see Section 2.3) and let
$\\{M^{u}\\}_{u\in\mathbb{N}^{n}}$ be the associated $n$-parameter filtration
of the Morse complex $M$. We assume the multifiltration
$\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ to be _exhaustive_ , that is,
$X=\bigcup_{u\in\mathbb{N}^{n}}X^{u}$. Clearly, since $X$ is also graded by
the dimension $q$ of cells, this means that
$X_{q}=\bigcup_{u\in\mathbb{N}^{u}}X^{u}_{q}$ for all $q$, and it is immediate
to show that, as a consequence, $M_{q}=\bigcup_{u\in\mathbb{N}^{u}}M^{u}_{q}$,
for all $q$.

Notationally, for any non-empty subset $A$ of cells of $X$, we set

$\mathcal{G}(A)\coloneqq\\{\text{entrance grades of the cells of
$A$}\\}\subseteq\mathbb{N}^{n}.$

We recall that each cell has a unique entrance grade because we are assuming
multifiltrations to be one-critical (cf. Section 2.2). We denote by
$\overline{G}$ the closure of a non-empty subset $G\subseteq\mathbb{N}^{n}$
with respect to the least upper bound in $\mathbb{N}^{n}$, which is the set
$\overline{G}:=\\{\bigvee B\mid B\subseteq
G,B\neq\emptyset\\}\subseteq\mathbb{N}^{n}$, with $\bigvee B$ denoting the
least upper bound of $B$ in $(\mathbb{N}^{n},\preceq)$. Moreover, we denote by
$\operatorname{supp}\xi^{q}_{i}:=\\{u\in\mathbb{N}^{n}\mid\xi^{q}_{i}(u)\neq
0\\}$ the support of the Betti table
$\xi^{q}_{i}:\mathbb{N}^{n}\to\mathbb{N}$. We finally establish the following
notation to be used throughout this section.

######  Notation 4.1.

Having fixed a grade $u\in\mathbb{N}^{n}$, for any $\alpha\subseteq[n]$ we set
$w(\alpha):=u-e_{\alpha}$, where $e_{\alpha}:=\sum_{j\in\alpha}e_{j}$.

Our goal for this section is to prove that
$\bigcup_{i=0}^{n}\operatorname{supp}\xi^{q}_{i}\subseteq\overline{\mathcal{G}(M_{q})}\cup\overline{\mathcal{G}(M_{q+1})}$
and, moreover,
$\operatorname{supp}\xi^{q}_{0}\subseteq\overline{\mathcal{G}(M_{q})}$ and
$\operatorname{supp}\xi^{q}_{n}\subseteq\overline{\mathcal{G}(M_{q+1})}$, for
all $q\in\mathbb{N}$ (Theorem 4.8). We observe that the first inclusion is
clearly equivalent to the following statement: if
$u\notin\overline{\mathcal{G}(M_{q})}\cup\overline{\mathcal{G}(M_{q+1})}$,
then $\xi^{q}_{i}(u)=0$, for all $i\in\\{0,1,\ldots,n\\}$. To start with, we
prove a result that allows us to rephrase the hypothesis of this statement.

######  Proposition 4.2.

Let $A$ be any subset of cells of $X$ and let $u\in\mathbb{N}^{n}$. Then
$u\notin\overline{\mathcal{G}(A)}$ if and only if there exists $j\in[n]$ such
that for any subset $\alpha_{j}\subseteq[n]\smallsetminus\\{j\\}$ it holds
$(X^{w(\alpha_{j})}\smallsetminus X^{w(\alpha_{j})-e_{j}})\cap A=\emptyset$,
where $w(\alpha_{j})$ is defined as in Notation 4.1.

###### Proof.

We prove Proposition 4.2 by contraposition, showing the equivalence of the
following statements:

  1. 1.

$u\in\overline{\mathcal{G}(A)}$.

  2. 2.

For all $j\in[n]$, there exists a subset
$\alpha_{j}\subseteq[n]\smallsetminus\\{j\\}$ such that
$\left(X^{w(\alpha_{j})}\smallsetminus X^{w(\alpha_{j})-e_{j}}\right)\cap
A\neq\emptyset$.

Assume that $u\in\overline{\mathcal{G}(A)}$. If $u\in\mathcal{G}(A)$, we are
done by taking $\alpha_{j}=\emptyset$, for all $j$. If
$u\notin\mathcal{G}(A)$, then $u=\bigvee\\{v_{1},\ldots,v_{r}\\}$ with $r\geq
2$ and $v_{1},\ldots v_{r}\in\mathcal{G}(A)$. In this case, by definition of
the least upper bound, for all $j\in[n]$ there exists $\ell(j)\in[r]$ such
that $u-e_{j}\not\succeq v_{\ell(j)}$. Therefore, taking a cell
$\sigma_{\ell(j)}\in A$ with entrance grade $v_{\ell(j)}$, we have
$\sigma_{\ell(j)}\in\left(X^{u}\smallsetminus X^{u-e_{j}}\right)\cap A$, since
$u-e_{j}\not\succeq v_{\ell(j)}$ implies $\sigma_{\ell(j)}\notin X^{u-e_{j}}$
by one-criticality of the multifiltration (Section 2.2). The second statement
follows again by taking $\alpha_{j}=\emptyset$, for all $j$.

Conversely, assume that the second statement holds. For each $j\in[n]$, let
$v(j)$ denote the entrance grade of a cell
$\sigma_{j}\in\left(X^{w(\alpha_{j})}\smallsetminus
X^{w(\alpha_{j})-e_{j}}\right)\cap A$, for some
$w(\alpha_{j})=u-\sum_{i\in\alpha_{j}}e_{i}$. Let
$v=\bigvee\\{v(1),\ldots,v(n)\\}$. From $v(j)\preceq w(\alpha_{j})\preceq u$,
for all $j$, we see that $v\preceq u$. Let us show now that $v=u$, which
concludes the proof. If $v\neq u$, then there exists $j\in[n]$ such that
$v\preceq u-e_{j}$. Since $\sigma_{j}$ has entrance grade $v(j)$ and
$v(j)\preceq v\preceq u-e_{j}$, we have $\sigma_{j}\in X^{u-e_{j}}$. On the
other hand, we are assuming that $\sigma_{j}\in X^{w(\alpha_{j})}$ with
$w(\alpha_{j})=u-\sum_{i\in\alpha_{j}}e_{i}$ and $j\notin\alpha_{j}$. The
latter condition implies that $w(\alpha_{j})$ and $u-e_{j}$ are not
comparable. More precisely, the greatest lower bound of $w(\alpha_{j})$ and
$u-e_{j}$ is $w(\alpha_{j})-e_{j}$. Hence, the one-criticality assumption on
the multifiltration yields a contradiction (see Remark 2.1), since we are
assuming that $\sigma_{j}\notin X^{w(\alpha_{j})-e_{j}}$. ∎

We underline that the one-criticality assumption on the $n$-filtration
$\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ plays a key role in the proof of
Proposition 4.2.

The particular case of Proposition 4.2 where $A$ is the set $M_{q}$ of
critical $q$-cells, for some $q$, is the most relevant for our purposes.

######  Corollary 4.3.

For any $u\in\mathbb{N}^{n}$, we have $u\notin\overline{\mathcal{G}(M_{q})}$
if and only if there exists $j\in[n]$ such that
$M_{q}^{w(\alpha_{j})-e_{j}}=M_{q}^{w(\alpha_{j})}$, for all subsets
$\alpha_{j}\subseteq[n]\smallsetminus\\{j\\}$.

Proposition 4.2 also yields information on the maps of the persistent homology
modules $\\{H_{q}(X^{u}),\iota_{q}^{u,v}\\}$ and
$\\{H_{q-1}(X^{u}),\iota_{q-1}^{u,v}\\}$ in the “vicinity” of a fixed grade
$u\notin\overline{\mathcal{G}(M_{q})}$.

######  Corollary 4.4.

If $u\notin\overline{\mathcal{G}(M_{q})}$, then there exists $j\in[n]$ such
that, for all $\alpha_{j}\subseteq[n]\smallsetminus\\{j\\}$, the inclusion
$X^{w(\alpha_{j})-e_{j}}\hookrightarrow X^{w(\alpha_{j})}$ induces a
surjection

$\iota_{q}^{w(\alpha_{j})-e_{j},w(\alpha_{j})}:H_{q}(X^{w(\alpha_{j})-e_{j}})\to
H_{q}(X^{w(\alpha_{j})})$

and an injection

$\iota_{q-1}^{w(\alpha_{j})-e_{j},w(\alpha_{j})}:H_{q-1}(X^{w(\alpha_{j})-e_{j}})\to
H_{q-1}(X^{w(\alpha_{j})}).$

###### Proof.

By Proposition 4.2, if $u\notin\overline{\mathcal{G}(M_{q})}$, then there
exists $j\in[n]$ such that, for all
$\alpha_{j}\subseteq[n]\smallsetminus\\{j\\}$, we have
$M_{q}^{w(\alpha_{j})-e_{j}}=M_{q}^{w(\alpha_{j})}$, which implies
$H_{q}(M^{w(\alpha_{j})},M^{w(\alpha_{j})-e_{j}})=0$. By the natural
isomorphism of the long exact sequences of relative homology of
$(X^{w},X^{v})$ and $(M^{w},M^{v})$ for each $v\preceq w$ (see Proposition
2.4), the sequence

$H_{q}(X^{w(\alpha_{j})-e_{j}})\xrightarrow{\quad}H_{q}(X^{w(\alpha_{j})})\xrightarrow{\quad}0\xrightarrow{\quad}H_{q-1}(X^{w(\alpha_{j})-e_{j}})\xrightarrow{\quad}H_{q-1}(X^{w(\alpha_{j})}),$

where the first and last maps are
$\iota_{q}^{w(\alpha_{j})-e_{j},w(\alpha_{j})}$ and
$\iota_{q-1}^{w(\alpha_{j})-e_{j},w(\alpha_{j})}$, respectively, is exact.
Hence, our claim follows. ∎

######  Remark 4.5.

Moving towards the proof of our main result, let us note that the hypothesis
$u\notin\overline{\mathcal{G}(M_{q})}\cup\overline{\mathcal{G}(M_{q+1})}$
implies, applying Corollary 4.3 twice, that the following properties hold
simultaneously:

  1. (i)

there exists $j\in[n]$ such that
$M_{q}^{w(\alpha_{j})-e_{j}}=M_{q}^{w(\alpha_{j})}$, for all subsets
$\alpha_{j}\subseteq[n]\smallsetminus\\{j\\}$.

  2. (ii)

there exists $\ell\in[n]$ such that
$M_{q+1}^{w(\alpha_{\ell})-e_{\ell}}=M_{q+1}^{w(\alpha_{\ell})}$, for all
subsets $\alpha_{\ell}\subseteq[n]\smallsetminus\\{\ell\\}$.

Clearly, the indices $j$ and $\ell$ of properties (i) and (ii) in Remark 4.5
can either coincide or not. We next prove that both cases imply the acyclicity
of certain Koszul complexes, addressing the case $j=\ell$ in Lemma 4.6 and the
case $j\neq\ell$ in Lemma 4.7.

######  Lemma 4.6.

If properties (i) and (ii) in Remark 4.5 are verified with $j=\ell$, then the
Koszul complex $\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ is acyclic.

###### Proof.

Corollary 4.4 implies that the maps

$i_{q}^{w(\alpha_{j})-e_{j},w(\alpha_{j})}:H_{q}(X^{w(\alpha_{j})-e_{j}})\to
H_{q}(X^{w(\alpha_{j})})$

are isomorphisms, for all subsets
$\alpha_{j}\subseteq[n]\smallsetminus\\{j\\}$. Therefore, the induced chain
map

$f^{j}(x_{1},\ldots,\hat{x}_{j},\ldots,x_{n};V_{q})(u-e_{j}):\mathbb{K}_{*}(x_{1},\ldots,\hat{x}_{j},\ldots,x_{n};V_{q})(u-e_{j})\to\mathbb{K}_{*}(x_{1},\ldots,\hat{x}_{j},\ldots,x_{n};V_{q})(u)$

is an isomorphism of chain complexes. Hence, the claim follows from
Proposition 3.1 because the chain complex
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ is the mapping cone of
$f^{j}(x_{1},\ldots,\hat{x}_{j},\ldots,x_{n};V_{q})(u-e_{j})$. ∎

######  Lemma 4.7.

Let $u\in\mathbb{N}^{u}$ and suppose that properties (i) and (ii) of Remark
4.5 hold with $j\neq\ell$. Then, for any $w\coloneqq
w(\alpha)=u-\sum_{i\in\alpha}e_{i}$ with
$\alpha\subseteq[n]\smallsetminus\\{j,\ell\\}$, the Koszul complex
$\mathbb{K}_{*}(x_{j},x_{\ell};V_{q})(w)$ is acyclic.

###### Proof.

By the isomorphism of Koszul complexes (Section 3), it is enough to show that
$\mathbb{K}_{*}(x_{j},x_{\ell};V^{\prime}_{q})(w)$ is acyclic, with
$V^{\prime}_{q}\coloneqq\bigoplus_{u\in\mathbb{N}^{n}}H_{q}(M^{u})$. In order
to apply Proposition 3.1, we regard
$\mathbb{K}_{*}(x_{j},x_{\ell};V^{\prime}_{q})(w)$ as the mapping cone of the
chain map

$f^{\ell}(x_{j};V^{\prime}_{q})(w-e_{\ell}):\mathbb{K}_{*}(x_{j};V^{\prime}_{q})(w-e_{\ell})\to\mathbb{K}_{*}(x_{j};V^{\prime}_{q})(w).$

We want to prove that $f^{\ell}(x_{j};V^{\prime}_{q})(w-e_{\ell})$ induces
isomorphisms between the homology modules of

$\mathbb{K}_{*}(x_{j};V^{\prime}_{q})(w-e_{\ell})=\left(0\to
H_{q}(M^{w-e_{j}-e_{\ell}})\xrightarrow{d_{1}=\iota_{q}^{w-e_{j}-e_{\ell},w-e_{\ell}}}H_{q}(M^{w-e_{\ell}})\to
0\right)$

and

$\mathbb{K}_{*}(x_{j};V^{\prime}_{q})(w)=\left(0\to
H_{q}(M^{w-e_{j}})\xrightarrow{d_{1}=\iota_{q}^{w-e_{j},w}}H_{q}(M^{w})\to
0\right).$

Since $\iota_{q}^{w-e_{j}-e_{\ell},w-e_{\ell}}$ and $\iota_{q}^{w-e_{j},w}$
are surjective (see proof of Corollary 4.4), homology in degree $0$ is zero
for both Koszul complexes. Hence, we only have to show that

$f^{\prime}:\ker\iota_{q}^{w-e_{j}-e_{\ell},w-e_{\ell}}\to\ker\iota_{q}^{w-e_{j},w}$

is an isomorphism, where $f^{\prime}$ denotes the restriction of
$\iota_{q}^{w-e_{j}-e_{\ell},w-e_{j}}$ to
$\ker\iota_{q}^{w-e_{j}-e_{\ell},w-e_{\ell}}$. The map $f^{\prime}$ is
injective because $\iota_{q}^{w-e_{j}-e_{\ell},w-e_{j}}$ is injective (see
proof of Corollary 4.4). A quick way to show that $f^{\prime}$ is an
isomorphism is therefore showing that
$\ker\iota_{q}^{w-e_{j}-e_{\ell},w-e_{\ell}}$ and $\ker\iota_{q}^{w-e_{j},w}$
are vector spaces of the same (finite) dimension. We use here the notations
$Z_{q}(M^{v})$ and $B_{q}(M^{v})$ respectively for the submodules of cycles
and boundaries of $C_{q}(M^{v})$, for all $v\in\mathbb{N}^{n}$. By Remark
4.5(i), $M_{q}^{w-e_{j}-e_{\ell}}=M_{q}^{w-e_{\ell}}$ and
$M_{q}^{w-e_{j}}=M_{q}^{w}$, which implies
$Z_{q}(M^{w-e_{j}-e_{\ell}})=Z_{q}(M^{w-e_{\ell}})$ and
$Z_{q}(M^{w-e_{j}})=Z_{q}(M^{w})$. Similarly, by Remark 4.5(ii),
$M_{q+1}^{w-e_{j}-e_{\ell}}=M_{q+1}^{w-e_{j}}$ and
$M_{q+1}^{w-e_{\ell}}=M_{q+1}^{w}$, which implies
$B_{q}(M^{w-e_{j}-e_{\ell}})=B_{q}(M^{w-e_{j}})$ and
$B_{q}(M^{w-e_{\ell}})=B_{q}(M^{w})$. Expressing the dimension of the kernels
of the surjective maps $\iota_{q}^{w-e_{j}-e_{\ell},w-e_{\ell}}$ and
$\iota_{q}^{w-e_{j},w}$ with respect to the dimension of these submodules, one
can easily check that they coincide. ∎

We underline that to conclude the proof we use an argument based on the
equality of some subsets of critical cells. This part of the proof cannot be
replaced by using only the properties of the induced maps in homology (as in
Corollary 4.4). As a counterexample, consider the diagram

${0}$${0}$${V}$${0}$

of vector spaces, with $\dim V\neq 0$. We can regard the rows as two chain
complexes with surjective differentials, and the vertical arrows as an
injective chain map between them, as in our proof. However, the mapping cone
of this chain map is clearly not acyclic.

We can now complete the proof of our main result for this section.

######  Theorem 4.8.

Let $\\{X^{u}\\}_{u\in\mathbb{N}^{n}}$ be an $n$-parameter exhaustive
filtration of a cell complex $X$, let $\mathcal{V}$ be a fixed discrete
gradient vector field consistent with the filtration, and let
$\\{M^{u}\\}_{u\in\mathbb{N}^{n}}$ be the associated $n$-parameter filtration
of the Morse complex $M$. Then

$\bigcup_{i=0}^{n}\operatorname{supp}\xi^{q}_{i}\subseteq\overline{\mathcal{G}(M_{q})}\cup\overline{\mathcal{G}(M_{q+1})},$

for all $q\in\mathbb{N}$. Furthermore,
$\operatorname{supp}\xi^{q}_{0}\subseteq\overline{\mathcal{G}(M_{q})}$ and
$\operatorname{supp}\xi^{q}_{n}\subseteq\overline{\mathcal{G}(M_{q+1})}$, for
all $q\in\mathbb{N}$.

###### Proof.

To prove that
$\bigcup_{i=0}^{n}\operatorname{supp}\xi^{q}_{i}\subseteq\overline{\mathcal{G}(M_{q})}\cup\overline{\mathcal{G}(M_{q+1})}$,
let $u\notin\overline{\mathcal{G}(M_{q})}\cup\overline{\mathcal{G}(M_{q+1})}$.
As we have seen, properties (i) and (ii) of Remark 4.5 hold, which involve
indices $j,\ell\in[n]$. If $j=\ell$, the Koszul complex
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ is acyclic by Lemma 4.6. If
$j\neq\ell$, consider the Koszul complexes
$\mathbb{K}_{*}(x_{j},x_{\ell};V_{q})(w)$, for any $w\coloneqq
w(\alpha)=u-\sum_{i\in\alpha}e_{i}$ with
$\alpha\subseteq[n]\smallsetminus\\{j,\ell\\}$, which are acyclic by Lemma
4.7. The Koszul complex $\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ can be
obtained from the chain complexes $\mathbb{K}_{*}(x_{j},x_{\ell};V_{q})(w)$ by
iterating the mapping cone construction (see Section 3). At each iteration of
this process, by Corollary 3.2, one obtains acyclic Koszul complexes, hence we
can conclude that $\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ is acyclic,
that is, $\xi^{q}_{i}(u)=0$ for all $i\in\\{0,\ldots,n\\}$.

To prove that
$\operatorname{supp}\xi^{q}_{0}\subseteq\overline{\mathcal{G}(M_{q})}$, we
observe that if $u\notin\overline{\mathcal{G}(M_{q})}$ then by Corollary 4.4
there exists $j\in[n]$ such that $H_{q}(X^{u-e_{j}})\to H_{q}(X^{u})$ is
surjective. This implies that the differential $d_{1}$ of the Koszul complex
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ is surjective, hence
$\xi_{0}^{q}(u)=\dim(H_{q}(X^{u})/\operatorname{im}d_{1})=0$.

Similarly, to prove that
$\operatorname{supp}\xi^{q}_{n}\subseteq\overline{\mathcal{G}(M_{q+1})}$, we
observe that if $u\notin\overline{\mathcal{G}(M_{q+1})}$ then by Corollary 4.4
there exists $j\in[n]$ such that $H_{q}(X^{w-e_{j}})\to H_{q}(X^{w})$ is
injective, where $w:=u-\sum_{i\in[n]\setminus\\{j\\}}e_{i}$. This implies that
the differential $d_{n}$ of the Koszul complex
$\mathbb{K}_{*}(x_{1},\ldots,x_{n};V_{q})(u)$ is injective, hence
$\xi_{n}^{q}(u)=\dim\ker d_{n}=0$. ∎

##  5 Homological critical grades and support of Betti tables for
bifiltrations

In this section, we fix $n=2$ and study the support of the Betti tables of
persistent homology modules associated with a bifiltration
$\\{X^{u}\\}_{u\in\mathbb{N}^{2}}$ of a cell complex $X$. In what follows, we
make use of the notations introduced at the beginning of Section 4.
Additionally, for $q\in\mathbb{N}$, let us set

$\mathcal{C}_{q}(X):=\\{u\in\mathbb{N}^{2}\mid\dim H_{q}(X^{u},X^{u-e_{1}}\cup
X^{u-e_{2}})\neq 0\\}$

and call it the set of _$q$ -homological critical grades_ (see [GL23]). For
any fixed $u\in\mathbb{N}^{2}$ and any $q\in\mathbb{N}$, let us recall the
following known inequalities (see [LS22, Corollary 1], and [GL23] for a
generalization to the case $n\geq 2$):

$\xi_{0}^{q}(u)+\xi_{1}^{q-1}(u)-\xi_{2}^{q-1}(u)\leq\dim
H_{q}(X^{u},X^{u-e_{1}}\cup
X^{u-e_{2}})\leq\xi_{0}^{q}(u)+\xi_{1}^{q-1}(u)+\xi_{2}^{q-2}(u).$ (5.1)

To interpret the results of this section, we remark that, if $M$ is the Morse
complex associated with any discrete gradient vector field consistent with the
filtration $\\{X^{u}\\}_{u\in\mathbb{N}^{2}}$, by [LS22, Prop. 1] we have
$\mathcal{C}_{q}(X)\subseteq\mathcal{G}(M_{q})$. As we will show (Proposition
5.5 and Corollary 5.6), for bifiltrations we are able to bound the support of
the Betti tables using the sets $\mathcal{C}_{q}(X)$ instead of the sets
$\mathcal{G}(M_{q})$, thus strengthening our general results of Section 4.

First, we prove a technical result that crucially depends on the one-
criticality assumption (Section 2.2) on the bifiltration.

######  Lemma 5.1.

Let $v\in\mathbb{N}^{2}$ and let $j\neq\ell$ in $\\{1,2\\}$. Then, there is a
short exact sequence of chain complexes

$0\xrightarrow{}C_{*}(X^{v-e_{\ell}},X^{v-e_{1}-e_{2}})\xrightarrow{}C_{*}(X^{v},X^{v-e_{j}})\xrightarrow{}C_{*}(X^{v},X^{v-e_{1}}\cup
X^{v-e_{2}})\xrightarrow{}0.$

######  Remark 5.2.

The statement has to be interpreted by setting $X^{v-e_{1}}=\emptyset$ if
$v-e_{1}$ is not in $\mathbb{N}^{2}$, and similarly for $X^{v-e_{2}}$ and
$X^{v-e_{1}-e_{2}}$. We use this convention throughout this section.

###### Proof.

Without loss of generality, we prove the statement for $j=1$ and $\ell=2$. The
sequence

$0\xrightarrow{}C_{*}(X^{v-e_{1}}\cup
X^{v-e_{2}},X^{v-e_{1}})\xrightarrow{}C_{*}(X^{v},X^{v-e_{1}})\xrightarrow{}C_{*}(X^{v},X^{v-e_{1}}\cup
X^{v-e_{2}})\xrightarrow{}0$

associated with the triple $X^{v-e_{1}}\subseteq X^{v-e_{1}}\cup
X^{v-e_{2}}\subseteq X^{v}$ is exact. Now we observe that, for any
$q\in\mathbb{N}$, the relative chain modules of the pair $(X^{v-e_{1}}\cup
X^{v-e_{2}},X^{v-e_{1}})$ are

$\displaystyle C_{q}(X^{v-e_{1}}\cup
X^{v-e_{2}},X^{v-e_{1}}):=\frac{C_{q}(X^{v-e_{1}}\cup
X^{v-e_{2}})}{C_{q}(X^{v-e_{1}})}=\frac{C_{q}(X^{v-e_{1}})+C_{q}(X^{v-e_{2}})}{C_{q}(X^{v-e_{1}})}$
$\displaystyle\cong\frac{C_{q}(X^{v-e_{2}})}{C_{q}(X^{v-e_{1}})\cap
C_{q}(X^{v-e_{2}})}=\frac{C_{q}(X^{v-e_{2}})}{C_{q}(X^{v-e_{1}-e_{2}})}=:C_{q}(X^{v-e_{2}},X^{v-e_{1}-e_{2}}),$

where we used the classical isomorphism theorem for modules and, in the
penultimate equality, the fact that $C_{q}(X^{v-e_{1}})\cap
C_{q}(X^{v-e_{2}})=C_{q}(X^{v-e_{1}-e_{2}})$ as a consequence of the equality
$X^{v-e_{1}}\cap X^{v-e_{2}}=X^{v-e_{1}-e_{2}}$ given by the one-criticality
assumption on the filtration (see Remark 2.1). These isomorphisms between
chain modules commute with the differentials of the chain complexes
$C_{*}(X^{v-e_{1}}\cup X^{v-e_{2}},X^{v-e_{1}})$ and
$C_{*}(X^{v-e_{2}},X^{v-e_{1}-e_{2}})$, since they are induced by the
differential of $C_{*}(X)$. ∎

######  Corollary 5.3.

Let $v\in\mathbb{N}^{2}$, $q\in\mathbb{N}$ and $j\neq\ell$ in $\\{1,2\\}$, and
suppose that $H_{q}(X^{v},X^{v-e_{1}}\cup X^{v-e_{2}})=0$. Then
$H_{q}(X^{v},X^{v-e_{j}})\neq 0$ implies
$H_{q}(X^{v-e_{\ell}},X^{v-e_{1}-e_{2}})\neq 0$.

###### Proof.

By Lemma 5.1, the following is a portion of a long exact sequence in homology:

$H_{q}(X^{v-e_{\ell}},X^{v-e_{1}-e_{2}})\xrightarrow{}H_{q}(X^{v},X^{v-e_{j}})\xrightarrow{}H_{q}(X^{v},X^{v-e_{1}}\cup
X^{v-e_{2}}).$

Since $H_{q}(X^{v},X^{v-e_{1}}\cup X^{v-e_{2}})=0$, the first map is
surjective, and the claim follows immediately. ∎

To prove the final result of this section (Proposition 5.5), we first show
directly that the support of the Betti table $\xi_{2}^{q-1}$ is contained in
$\overline{\mathcal{C}_{q}(X)}$.

######  Lemma 5.4.

For all $q\in\mathbb{N}$, we have
$\operatorname{supp}\xi_{2}^{q-1}\subseteq\overline{\mathcal{C}_{q}(X)}$.

###### Proof.

Let $u\in\operatorname{supp}\xi_{2}^{q-1}$. We prove that there exists
$\lambda\in\mathbb{N}$ such that

$H_{q}(X^{u-\lambda e_{1}},X^{u-(\lambda+1)e_{1}}\cup X^{u-\lambda
e_{1}-e_{2}})\neq 0.$ (5.2)

If condition (5.2) holds for $\lambda=0$, then $u\in\mathcal{C}_{q}(X)$.
Otherwise, since the same property can be proven with the roles of $e_{1}$ and
$e_{2}$ interchanged, our claim follows by observing that $(u-\lambda
e_{1})\vee(u-\mu e_{2})=u$, for every $\lambda,\mu\in\mathbb{N}$.

Assume that (5.2) is false (i.e. it is an equality) for all
$\lambda\in\mathbb{N}$; then $H_{q}(X^{u-\lambda e_{1}},X^{u-\lambda
e_{1}-e_{2}})\neq 0$ implies
$H_{q}(X^{u-(\lambda+1)e_{1}},X^{u-(\lambda+1)e_{1}-e_{2}})\neq 0$ by
Corollary 5.3 (applied with $v:=u-\lambda e_{1}$), and we can therefore use an
inductive argument. The base case of the induction is
$H_{q}(X^{u-e_{1}},X^{u-e_{1}-e_{2}})\neq 0$ for $\lambda=1$, which holds
because the hypothesis $u\in\operatorname{supp}\xi_{2}^{q-1}$ implies that
$i_{q-1}^{u-e_{1}-e_{2},u-e_{1}}:H_{q-1}(X^{u-e_{1}-e_{2}})\to
H_{q-1}(X^{u-e_{1}})$ has nonzero kernel (see Section 3.1). Since
$X^{u-\lambda e_{1}}=\emptyset=X^{u-\lambda e_{1}-e_{2}}$ for a sufficiently
large $\lambda$, we see that the induction leads to a contradiction. ∎

######  Proposition 5.5.

For all $q\in\mathbb{N}$, we have
$\operatorname{supp}\xi_{0}^{q}\cup\operatorname{supp}\xi_{1}^{q-1}\cup\operatorname{supp}\xi_{2}^{q-1}\subseteq\overline{\mathcal{C}_{q}(X)}$.

###### Proof.

Let us assume that $u\notin\overline{\mathcal{C}_{q}(X)}$. In the first
inequality of (5.1), the term $\dim H_{q}(X^{u},X^{u-e_{1}}\cup X^{u-e_{2}})$
is zero by definition of $\mathcal{C}_{q}(X)$. By Lemma 5.4,
$\xi_{2}^{q-1}(u)=0$, hence we have $\xi_{0}^{q}(u)+\xi_{1}^{q-1}(u)=0$, which
is equivalent to $\xi_{0}^{q}(u)=\xi_{1}^{q-1}(u)=0$. ∎

We observe that the inclusion
$\operatorname{supp}\xi_{0}^{q}\subseteq\overline{\mathcal{C}_{q}(X)}$ can be
proven directly, in a similar way to the proof of Lemma 5.4. Contrarily, a
direct proof of the inclusion
$\operatorname{supp}\xi_{1}^{q-1}\subseteq\overline{\mathcal{C}_{q}(X)}$
eludes us.

In conclusion, for bifiltrations, we can bound the support of Betti tables as
follows.

######  Corollary 5.6.

For all $q\in\mathbb{N}$, the Betti tables of degree $q$ satisfy

$\operatorname{supp}\xi_{0}^{q}\cup\operatorname{supp}\xi_{1}^{q}\cup\operatorname{supp}\xi_{2}^{q}\,\subseteq\,\overline{\mathcal{C}_{q}(X)}\cup\overline{\mathcal{C}_{q+1}(X)}.$

Furthermore, the union of the supports of all Betti tables satisfies

$\bigcup_{q}\mathcal{C}_{q}(X)\,\subseteq\,\bigcup_{q,i}\operatorname{supp}\xi_{i}^{q}\,\subseteq\,\bigcup_{q}\overline{\mathcal{C}_{q}(X)}.$

###### Proof.

The first statement holds by Proposition 5.5 and implies the second inclusion
of the second statement. The first inclusion of the second statement follows
from the second inequality of (5.1), which implies that
$\mathcal{C}_{q}(X)\,\subseteq\,\operatorname{supp}\xi_{0}^{q}\cup\operatorname{supp}\xi_{1}^{q-1}\cup\operatorname{supp}\xi_{2}^{q-2}$,
for all $q\in\mathbb{N}$. ∎

We remark that the first statement of Corollary 5.6 is not a consequence of
Theorem 4.8, as for 2-parameter persistent homology modules it is known that
$\mathcal{C}_{q}(X)$ can be strictly contained in $\mathcal{G}(M_{q})$, for
any choice of a discrete gradient vector field to determine the latter set of
grades (see [LS22, p. 2369] for an example).

For $n>2$ parameters, we believe that exact sequences like those of Lemma 5.1,
along with those induced in homology, can still be useful to study the
relation between Betti tables and homological critical grades. In this case,
however, these sequences assemble in much more complicated systems, and
appropriately disentangling them would require a different approach.

## Acknowledgments

This work was partially supported by the Wallenberg AI, Autonomous Systems and
Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
This work was partially carried out within the activities of ARCES (University
of Bologna) and under the auspices of INdAM-GNSAGA.

## References

  * [AKL17] Madjid Allili, Tomasz Kaczynski, and Claudia Landi.  Reducing complexes in multidimensional persistent homology theory.  Journal of Symbolic Computation, 78:61–75, 2017.
  * [Bar94] Serguei Barannikov.  Framed Morse complexes and its invariants.  Advances in Soviet Mathematics, 21:93–116, 1994.
  * [BCGS13] Silvia Biasotti, Andrea Cerri, Daniela Giorgi, and Michela Spagnuolo.  PHOG: Photometric and geometric functions for textured shape retrieval.  Computer Graphics Forum, 32(5):13–22, 2013.
  * [BH98] Winfried Bruns and H Jürgen Herzog.  Cohen-Macaulay rings.  Number 39 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1998.
  * [BL22] Andrew J. Blumberg and Michael Lesnick.  Stability of 2-parameter persistent homology.  Foundations of Computational Mathematics, 2022.
  * [Car09] Gunnar E. Carlsson.  Topology and data.  Bulletin of the American Mathematical Society, 46:255–308, 2009\.
  * [CSZ09] Gunnar Carlsson, Gurjeet Singh, and Afra Zomorodian.  Computing multidimensional persistence.  In International Symposium on Algorithms and Computation, pages 730–739. Springer, 2009.
  * [CZ09] Gunnar Carlsson and Afra Zomorodian.  The theory of multidimensional persistence.  Discrete & Computational Geometry, 42(1):71–93, 2009.
  * [Eis95] David Eisenbud.  Commutative Algebra: with a view toward algebraic geometry, volume 150 of Graduate Texts in Mathematics.  Springer-Verlag New York, 1995.
  * [Eis05] David Eisenbud.  The geometry of syzygies: a second course in algebraic geometry and commutative algebra, volume 229 of Graduate Texts in Mathematics.  Springer-Verlag New York, 2005.
  * [ELZ02] Herbert Edelsbrunner, David Letscher, and Afra Zomorodian.  Topological persistence and simplification.  Discrete & Computational Geometry, 28:511–533, 2002.
  * [EM12] Herbert Edelsbrunner and Dmitriy Morozov.  Persistent homology: theory and practice.  In Proceedings of the European Congress of Mathematics, pages 31–50, 2012.
  * [FL13] Patrizio Frosini and Claudia Landi.  Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval.  Pattern Recognition Letters, 34(8):863 – 872, 2013.
  * [For98] Robin Forman.  Morse theory for cell complexes.  Advances in Mathematics, 134:90–145, 1998.
  * [Fro96] Patrizio Frosini.  Connections between size functions and critical points.  Mathematical Methods in the Applied Sciences, 19(7):555–569, 1996\.
  * [Gab72] Peter Gabriel.  Unzerlegbare darstellungen. i.  Manuscripta Mathematica, 6:71–103, 1972.
  * [Ghr07] Robert Ghrist.  Barcodes: the persistent topology of data.  Bulletin of the American Mathematical Society, 45(1):61–75, 2007\.
  * [GL23] Andrea Guidolin and Claudia Landi.  Morse inequalities for the Koszul complex of multi-persistence.  Journal of Pure and Applied Algebra, 227(7):107319, 2023.
  * [JW09] Michael Jöllenbeck and Volkmar Welker.  Resolution of the residue class field via algebraic discrete Morse theory, volume 197 of Memoirs of the AMS.  American Mathematical Society, Providence, 2009.
  * [KMS98] Tomasz Kaczyński, Marian Mrozek, and Maciej Ślusarek.  Homology computation by reduction of chain complexes.  Computers & Mathematics with Applications, 35(4):59–70, 1998.
  * [Knu08] Kevin P Knudson.  A refinement of multi-dimensional persistence.  Homology, Homotopy and Applications, 10(1):259–281, 2008.
  * [Koz05] Dmitry N Kozlov.  Discrete Morse theory for free chain complexes.  Comptes Rendus Mathematique, 340(12):867–872, 2005.
  * [Lef42] Solomon Lefschetz.  Algebraic topology, volume 27.  American Mathematical Soc., 1942.
  * [LS22] Claudia Landi and Sara Scaramuccia.  Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology.  Journal of Combinatorial Optimization, 44(4):2347–2374, 2022.
  * [Mil63] John Milnor.  Morse theory.  Princeton University Press, 1963.
  * [MN13] Konstantin Mischaikow and Vidit Nanda.  Morse theory for filtrations and efficient computation of persistent homology.  Discrete & Computational Geometry, 50(2):330–353, 2013.
  * [MS05] Ezra Miller and Bernd Sturmfels.  Combinatorial commutative algebra, volume 227 of Graduate Texts in Mathematics.  Springer-Verlag New York, 2005.
  * [Oud15] Steve Y Oudot.  Persistence theory: from quiver representations to data analysis, volume 209.  American Mathematical Society, 2015.
  * [Rob00] Vanessa Robins.  Computational topology at multiple resolutions: Foundations and applications to fractals and dynamics.  PhD thesis, University of Colorado at Boulder, 2000.
  * [Rot09] Joseph Rotman.  An Introduction to Homological Algebra.  Springer-Verlag New York, 2nd edition, 2009.
  * [Skö06] Emil Sköldberg.  Combinatorial discrete Morse theory from an algebraic viewpoint.  Transactions of the American Mathematical Society, 358(1):115–129, 2006.
  * [Wei94] Charles A Weibel.  An introduction to homological algebra.  Cambridge University Press, 1994.
  * [ZC05] Afra Zomorodian and Gunnar Carlsson.  Computing persistent homology.  Discrete & Computational Geometry, 33(2):249–274, 2005.