# Foundations of differential calculus for modules over posets

Jacek Brodzki  School of Mathematical Sciences, Southampton University,
Southampton, UK<EMAIL_ADDRESS>,  Ran Levi  Institute of Mathematics,
University of Aberdeen, Aberdeen, UK<EMAIL_ADDRESS>and  Henri Riihimäki
Department of Mathematics, KTH Royal Institute of Technology, Stockholm,
Sweden<EMAIL_ADDRESS>
###### Abstract.

Persistence modules were introduced in the context of topological data
analysis. Generalised persistence module theory is the study of functors from
an arbitrary poset, or more generally an arbitrary small category, to some
abelian target category. In other words, a persistence module is simply a
representation of the source category in the target abelian category.
Unsurprisingly, it turns out that when the source category is more general
than a linear order, then its representation type is generally wild. In this
paper we introduce a new set of ideas for local analysis of persistence module
by methods borrowed from spectral graph theory and multivariable calculus.

The term _persistence module_ emerged as a concept in early work of Carlsson
and Zomorodian [8], and motivated much of the theoretical developments in
persistent homology and topological data analysis. Generalised persistence
modules were introduced by Bubenik, de Silva and Scott in [7], where a
persistence module is defined to be any functor $F$ from a preordered set
$\mathcal{P}$ to an arbitrary category $\mathcal{D}$. In most setups the
preordered set $\mathcal{P}$ is assumed to be a poset (i.e. an antisymmetric
preordered set) and the target category $\mathcal{D}$ is taken to be abelian -
typically the category ${\mathrm{Vect}_{k}}$ of finite dimensional vector
spaces over a field $k$. Working with arbitrary posets, particularly such that
have a natural topology like $(\mathbb{R},\leq)$, is in practice too general,
and from a computational point of view, impossible. Chazal et. al. [11,
Section 3.8] introduced the concept of _tame persistence modules_ for functors
$M\colon(\mathbb{R},\leq)\to{\mathrm{Vect}_{k}}$. The concept was generalised
by Scolamiero et. al. [29] to functors on $(\mathbb{Q}^{n},\leq)$, and
generalised further to arbitrary posets by Miller [26], who defines a
persistence module $M\colon\mathcal{Q}\to{\mathrm{Vect}_{k}}$ to be tame, if
there exists a finite poset $\mathcal{P}$, a poset map
$\alpha\colon\mathcal{Q}\to\mathcal{P}$, and a persistence module
$N\colon\mathcal{P}\to{\mathrm{Vect}_{k}}$, such that $M=N\circ\alpha$.
Specialising Miller’s definition to modules on $(\mathbb{R},\leq)$, such a
module $M$ has the property that there is a finite set of real numbers
$t_{1}<t_{2}<\cdots<t_{n}$, depending on $M$, such that for any $t_{i}\leq
a\leq b\leq t_{i+1}$, the homomorphism $M(a\leq b)$ is an isomorphism.

Functors from finite categories to the category of vector spaces are referred
to in the literature as _representations_ of the categories in question. The
functor category from a finite category $\mathcal{C}$ to an abelian category
is itself an abelian category. It is in fact isomorphic to the module
category, in the ordinary sense, over the category algebra $k\mathcal{C}$
[31]. Thus persistence modules over finite posets can be thought of as
ordinary modules over the poset algebra $k\mathcal{P}$ (see Section 1.1). For
category algebras over finite posets one has Drozd trichotomy theorem [15],
which states that any finite dimensional algebra is either of _finite
representation type_ , i.e. has finitely many isomorphism classes of
indecomposable modules, or otherwise of _infinite representation type_ , in
which case they are either of _tame representation type_ or of _wild
representation type_. Notice that the term “tame representation type” refers
to the algebra and not any module over it. This is not to be confused with the
notion of a “tame persistence module”, as defined above.

Tame persistence modules over $(\mathbb{R},\leq)$ can be factored, by
definition, through a finite linearly ordered poset, and poset algebras over
such posets are of finite representation type. Hence modules over
$(\mathbb{R},\leq)$ are classifiable by their _persistence diagrams_ (or
equivalently persistence barcodes, or persistence landscapes) up to
isomorphism. Traditional algebraic tools employed in the study of generalised
persistence modules, such as decomposition into indecomposable modules, where
possible, free presentations, and resolutions (see for example [1, 5, 6, 14,
21]) all share the same difficulty. Arbitrary finite posets are typically of
infinite (tame or wild) representation type, which makes classification
practically impossible. Thus, from the point of view of topological data
analysis, finding alternative methods of extracting computable information out
of persistence modules is desirable; some notable examples of this line of
work are [16, 26, 29].

In this article we introduce a new approach to the study of persistence
modules. Our starting point is essentially representation theoretic. Namely we
consider isomorphism classes of modules over the category algebra of a finite
poset. However, instead of attempting to understand persistence modules
globally, we propose a _calculus of persistence modules_ , that is a
methodology that enables one to extract local information. Indeed, in
exploring properties of a nice real valued function of several variables one
typically employs standard techniques of multivariable calculus, which allow
studying the function locally. The notions of gradient, divergence and
Laplacian come to mind in this context. Inspired by these ideas, there is a
discrete version of multivariable calculus for weighted directed graphs. Our
treatment of persistence module theory is a powerful generalisation of the
ideas of discrete calculus on graphs. In particular we shall define the notion
of a gradient for persistence modules, as well as concepts corresponding to
the divergence and the Laplacian.

By contrast to standard representation theoretic analysis of persistence
modules, where the ground poset remains fixed, as does the representation type
of its poset algebra, the construction of the gradient and other operations,
analogous to classical multivariable calculus, for persistence modules allows
one to consider modules locally, namely on sub-posets of the original poset,
informed by the behaviour of the gradient. For instance, in [16] the authors
study modules over _commutative ladders_. They show that a commutative ladder
of any type is representation-finite if and only if its length is at most 4.
In Section 7.1 we consider two types of commutative ladders of any length and
show that the gradient of any module over such ladders can be written as a sum
of modules over posets of finite representation type. Another example is
motivated by [2], where the authors study certain module categories over
finite 2-dimensional grids which they refer to as _filtered hierarchical
clustering_. An important corollary of their main results [2, Corollary 1.6]
states that $m\times n$ grid posets are of finite or tame representation type
only for a very small number of cases, and are of wild representation type in
all other cases. In Section 7.2 we demonstrate how our approach easily gives
computable information about modules over grid posets of any size.

To state our main theorems some preparation is required. We provide a brief
description here, and more details in Section 1. _Calculus on weighted
directed graphs_ is a discrete calculus for functions whose domain is the
vertex set of a finite graph with weighted directed edges [24]. In this
context the gradient is an operation which takes functions on the vertex set
of a graph to functions on its edge set. The _line digraph_ of a directed
graph $\mathcal{G}$ is a directed graph $\widehat{\mathcal{G}}$, whose
vertices are the edges of $\mathcal{G}$. Thus the gradient can be thought of
as an operator that takes functions on the vertices of $\mathcal{G}$ to
functions on the vertices of $\widehat{\mathcal{G}}$. We take a similar point
of view. Our gradient will take a persistence module on a finite poset
$\mathcal{P}$ to a difference of persistence module on a poset
$\widehat{\mathcal{P}}$ associated to $\mathcal{P}$, where
$\widehat{\mathcal{P}}$ is obtained from $\mathcal{P}$ using the line digraph
of the Hasse diagram of $\mathcal{P}$.

The graph theoretic gradient is defined on an edge as the difference between
the value at its target vertex and the value at its source vertex [24]. To
bring this idea to the universe of persistence modules we therefore need an
additive structure with additive inverses. Since we are only interested in
persistence modules up to isomorphism, it makes sense to consider the
Grothendieck ring $\EuR{Gr}(k\mathcal{P})$ of isomorphism classes of modules
over the poset algebra $k\mathcal{P}$, where the sum and product operations
are given by direct sum and tensor product (over the field $k$), respectively.
This allows us to define the gradient as the difference
$\nabla\overset{\text{{def}}}{=}\phi^{*}-\beta^{*}$ of two natural
homomorphisms
$\phi^{*},\beta^{*}\colon\EuR{Gr}(k\mathcal{P})\to\EuR{Gr}(k\widehat{\mathcal{P}})$,
(the _front_ and _back_ morphisms) that captures the variation of the module
along each indecomposable edge in $\mathcal{P}$, i.e. a relation in the Hasse
diagram of $\mathcal{P}$. The indecomposable relations in $\mathcal{P}$ can be
thought of as discrete analogs of infinitesimally small moves in a metric
space. Our categorical setup fits directly with the classical definition (See
Example 2.7). With this setup we can now state our first result.

######  Theorem A (Theorem 4.2).

Let $\mathcal{P}$ be a finite poset. Then the gradient operator
$\nabla\colon\EuR{Gr}(k\mathcal{P})\to\EuR{Gr}(k\widehat{\mathcal{P}})$
satisfies the following properties:

  1. a)

$\nabla$ is a well defined group homomorphism.

  2. b)

If $[M]\in\EuR{Gr}(k\mathcal{P})$ is locally constant then $\nabla[M]=0$.

  3. c)

$\nabla$ satisfies a Leibniz type rule, i.e. for all
$[M],[N]\in\EuR{Gr}(k\mathcal{P})$, the identity

$\nabla([M]\cdot[N])=\nabla[M]\cdot\phi^{*}[N]+\beta^{*}[M]\cdot\nabla[N]$

holds in $\EuR{Gr}(k\widehat{\mathcal{P}})$.

Furthermore, $\nabla$ is natural with respect to restrictions to sub-posets,
namely, if $\iota\colon\mathcal{Q}\xrightarrow{\subseteq}\mathcal{P}$ is a
sub-poset, then
$\iota^{*}\circ\nabla_{\mathcal{P}}=\nabla_{\mathcal{Q}}\circ\iota^{*}$.

Here by a locally constant module we mean a functor
$M\colon\mathcal{P}\to{\mathrm{Vect}_{k}}$ that takes every morphism in
$\mathcal{P}$ to an isomorphism of vector spaces.

By analogy to ordinary calculus, an obvious question is whether a vanishing
gradient of a module $M$ implies that it is locally constant. The answer turns
out to be not quite so straightforward. We say that a digraph $\mathcal{G}$ is
a _directed tree_ if between any two distinct vertices $a$ and $b$ in
$\mathcal{G}$ there is at most one directed path. Recall that for any digraph
$\mathcal{G}=(V,E)$, a _maximal tree in $\mathcal{G}$_ is a subgraph
$\mathcal{T}\subseteq\mathcal{G}$ on the same vertex set $V$ and with edges
$E^{\prime}\subseteq E$ such that $\mathcal{T}=(V,E^{\prime})$ is a directed
tree and such that $E^{\prime}$ is maximal with respect to this property, i.e.
adding extra edges from $E$ to $E^{\prime}$ will produce a subgraph that is
not a directed tree. We say that a poset $\mathcal{P}$ is _line connected_ if
the line digraph of its Hasse diagram is connected.

######  Theorem B (Theorem 4.5).

Let $\mathcal{P}$ be a finite line connected poset, and let $\mathcal{T}$ be a
line connected maximal tree in its Hasse diagram $\mathcal{H}_{\mathcal{P}}$.
Let $\mathcal{P}_{\mathcal{T}}\subseteq\mathcal{P}$ denote the sub-poset
generated by $\mathcal{T}$. Let $M\in k\mathcal{P}\mbox{-}\EuR{mod}$ be a
module, and let $M_{\mathcal{T}}$ denote the restriction of $M$ to
$\mathcal{P}_{\mathcal{T}}$. Assume that $\nabla[M_{\mathcal{T}}]=0$ in
$\EuR{Gr}(k\mathcal{P}_{\mathcal{T}})$. Then the following statements hold.

  1. (1)

For any objects
$u,v\in\mathrm{Obj}(\mathcal{P})=\mathrm{Obj}(\mathcal{P}_{\mathcal{T}})$,
there is an isomorphism

$\alpha_{u,v}\colon M(u)\to M(v),$

such that $\alpha_{u,u}=1_{M(u)}$ and $\alpha_{v,w}\alpha_{u,v}=\alpha_{u,w}$.

  2. (2)

For every pair of indecomposable morphisms $u\leq w$ and $s\leq t$ in
$\mathcal{P}_{\mathcal{T}}$,

$\alpha_{w,t}\circ M(u\leq w)=M(s\leq t)\circ\alpha_{u,s}.$

  3. (3)

$M_{\mathcal{T}}$ is locally constant if and only if $M(u\leq v)$ is an
isomorphism for some indecomposable relation $u\leq v$ in
$\mathcal{P}_{\mathcal{T}}$.

Furthermore, if $M\in k\mathcal{P}\mbox{-}\EuR{mod}$ is a module such that for
any indecomposable relation $u\leq v$ in $\mathcal{P}_{\mathcal{T}}$ there
exists an isomorphism $\alpha_{u,v}\colon M(u)\to M(v)$ that satisfy
Conditions (1) and (2), then $\nabla[M_{\mathcal{T}}]=0$.

If $\mathcal{P}$ is not generated by a tree, it is rather easy to find
examples of modules on $\widehat{\mathcal{P}}$ that are not the gradient of
any $k\mathcal{P}$-module. Thus such modules are, in the appropriate sense,
not integrable. In particular, in this situation it is possible to construct
$k\mathcal{P}$-modules with a vanishing gradient that do not satisfy
Conditions (1) and (2) of Theorem B. This is the reason why we restrict
modules to a maximal sub-tree. Of course similar statements can be made for
any sub-tree of $\mathcal{P}$ with the obvious modification to the
conclusions. This stands in sharp contrast to ordinary calculus, where any
differentiable function whose gradient vanishes on a domain is constant in the
interior of that domain, and any real valued continuous function on a
reasonable domain is integrable there.

Theorem B thus provides a sharp consequence of the vanishing of the gradient
of a module $M\in k\mathcal{P}\mbox{-}\EuR{mod}$ on a maximal tree
$\mathcal{T}$. It implies that for such a module all point modules (the values
on objects) are abstractly isomorphic, and all morphisms induced by applying
$M$ to indecomposable morphisms in $\mathcal{P}_{\mathcal{T}}$ can be
identified through a collection of non-canonical isomorphisms between point
modules.

We next examine the implication for a pair of $k\mathcal{P}$-modules of having
isomorphic gradients. A typical element in the Grothendieck ring
$\EuR{Gr}(k\mathcal{P})$, which is ordinarily referred to as a _virtual
module_ , is a difference of two equivalence classes of genuine modules. For
$M\in k\mathcal{P}\mbox{-}\EuR{mod}$, define the _rank function_
$\operatorname{rk}\nolimits(M)\colon\operatorname{Mor}(\mathcal{P})\to\mathbb{N}$
to be the function that takes a relation $x\leq y$ to the rank of the
homomorphism $M(x\leq y)$. The rank function clearly extends by additivity to
$\EuR{Gr}(k\mathcal{P})$, since it depends only on the isomorphism type of a
module. Notice also that our definition of the rank function includes also the
dimensions of point modules, which appear as the rank of $M$ applied to
identity morphisms.

It is well known that the rank invariant is a complete invariant for modules
over $(\mathbb{R},\leq)$ or any other linear order, but this fails for more
general posets [9]. However the rank invariant is still an important invariant
of modules. The next two theorems give new conditions on some of the local
behaviour of the rank invariant.

######  Theorem C (Theorem 4.13).

Let $\mathcal{P}$ be a finite poset. Let
$[X]=[M]-[N]\in\EuR{Gr}(k\mathcal{P})$ be an element with $M,N\in
k\mathcal{P}\mbox{-}\EuR{mod}$.

  1. (1)

Assume that $\nabla[X]=0$. Then
$\operatorname{rk}\nolimits[X](u_{0}<v_{0})=\operatorname{rk}\nolimits[X](u_{1}<v_{1})$
for any pair of comparable objects $(u_{0},u_{1})<(v_{0},v_{1})$ in
$\widehat{\mathcal{P}}$.

Assume in addition that $\mathcal{P}$ is line connected, let $\mathcal{T}$ be
a line connected maximal tree for $\mathcal{H}_{\mathcal{P}}$, and let
$\mathcal{P}_{\mathcal{T}}\subseteq\mathcal{P}$ be the sub-poset generated by
$\mathcal{T}$. If $[X]$ has a vanishing gradient on $\mathcal{T}$, then
$\operatorname{rk}\nolimits[X]$ has the following properties:

  1. (2)

It is constant on all identity morphisms in $\mathcal{P}$.

  2. (3)

It is constant on all indecomposable relations in $\mathcal{P}_{\mathcal{T}}$,
namely for any pair of indecomposable relations $u_{0}<u_{1}$, and
$v_{0}<v_{1}$ in $\mathcal{P}_{\mathcal{T}}$, one has
$\operatorname{rk}\nolimits[X](u_{0}<u_{1})=\operatorname{rk}\nolimits[X](v_{0}<v_{1})$.

Theorem C falls short of stating that isomorphic gradients imply equality of
rank invariants. Indeed Example 4.14 shows that this is not the case. A
natural question is whether a converse implication is true, namely whether
equal rank invariants imply that the modules in question have isomorphic
gradients. This question is answered in the negative in Example 4.19.

In discrete calculus for finite weighted digraphs one considers real valued
functions on vertices or edges as finite dimensional real vectors spaces, and
as such one has the ordinary inner product defined on the spaces of vertex and
edge functions, $\langle-,-\rangle_{V}$ and $\langle-,-\rangle_{E}$
respectively. This pairing allows one to define divergence and Laplacian for
digraphs. The adjoint operator $\nabla^{*}$, also called the _divergence
operator_ is then defined by the requirement that the relation

$\langle\nabla^{*}(f),g\rangle_{V}=\langle f,\nabla(g)\rangle_{E}$

holds.

The theory we propose here can be applied to a certain setup in standard
calculus on graphs (See Example 2.9). For modules $M,N\in
k\mathcal{P}\mbox{-}\EuR{mod}$ define a pairing

$\langle[M],[N]\rangle_{\mathcal{P}}\overset{\text{{def}}}{=}\dim_{k}(\mathrm{Hom}_{k\mathcal{P}}(M,N)).$

Extend the definition to a pairing on $\EuR{Gr}(k\mathcal{P})$ and similarly
to a pairing $\langle-,-\rangle_{\widehat{\mathcal{P}}}$ on
$\EuR{Gr}(k\widehat{\mathcal{P}})$. A related pairing, that in a sense appears
more natural, is the _Euler pairing_. For two modules $M,N\in
k\mathcal{P}\mbox{-}\EuR{mod}$, consider the graded vector space
$\mathrm{Ext}^{*}_{k\mathcal{P}}(M,N)$. Since $\mathcal{P}$ is assumed to be a
finite poset and modules in $k\mathcal{P}\mbox{-}\EuR{mod}$ are assumed
finitely generated, $\mathrm{Ext}^{i}_{k\mathcal{P}}(M,N)$ is a finite
dimensional vector space for all $i$ and it vanishes for $i$ sufficiently
large. Hence it makes sense to consider its Euler characteristic. Thus we
define an Euler pairing

$\chi_{\mathcal{P}}([M],[N])=\chi(\mathrm{Ext}^{*}_{k\mathcal{P}}(M,N)),$

and extend to the corresponding Grothendieck group by additivity. If the Hasse
diagram of $\mathcal{P}$ is a directed tree, thus in particular an acyclic
quiver, then $\mathrm{Ext}^{i}_{k\mathcal{P}}(M,N)$ vanish for $i>1$ [13,
Theorem 2.3.2], and the Euler pairing can be computed as an ordinary inner
product of dimension vectors [13, Proposition 2.5.2]. The following
proposition gives an easy relation between the two types of pairings.

######  Proposition D (Proposition 5.3).

Let $\mathcal{P}$ be a finite poset, and let $M,N\in
k\mathcal{P}\mbox{-}\EuR{mod}$ be modules. Let
$\langle-,-\rangle_{\mathcal{P}}$ and $\chi_{\mathcal{P}}(-,-)$ be the Hom
pairing and the Euler pairing respectively. Let

$0\to P_{n}\to\cdots\to P_{0}\to M\to 0,\quad\text{and}\quad 0\to N\to
I_{0}\to\cdots I_{n}\to 0$

be a projective resolution for $M$ and an injective resolution for $N$. Then

$\chi_{\mathcal{P}}([M],[N])=\sum_{i=0}^{n}(-1)^{i}\langle[P_{i}],[N]\rangle_{\mathcal{P}}=\sum_{i=0}^{n}(-1)^{i}\chi_{\mathcal{P}}([P_{i}],[N]),$

and

$\chi_{\mathcal{P}}([M],[N])=\sum_{j=0}^{n}(-1)^{j}\langle[M],[I_{k}]\rangle_{\mathcal{P}}=\sum_{j=0}^{n}(-1)^{j}\chi_{\mathcal{P}}([M],[I_{k}]).$

Furthermore, write $P_{i}\cong\bigoplus\epsilon^{i}_{v}F_{v}$ and
$I_{j}=\bigoplus\delta_{u}^{i}G_{u}$, with
$\epsilon_{v},\delta^{j}_{u}\in\mathbb{N}$ and $v,u\in\mathcal{P}$, and where
$F_{v}$ and $G_{u}$ are the indecomposable projective determined by $v$ and
the indecomposable injective determined by $u$. Then

$\chi_{\mathcal{P}}([M],[N])=\sum_{v\in\mathcal{P}}\sum_{i=0}^{n}(-1)^{i}\epsilon_{v}^{i}\dim_{n}N(v)=\sum_{u\in\mathcal{P}}\sum_{j=0}^{n}(-1)^{j}\delta_{u}^{j}\dim_{n}M(u).$

We next offer our analogs of the divergence and the laplacian in our context.
Considering the (non-symmetric) Hom pairing as an analog of an inner product,
the left and right Kan extensions offer themselves as a natural way of
constructing a left adjoint $\nabla^{*}$ and a right adjoint $\nabla_{*}$ of
the gradient. Thus we define _left and right divergence_. Example 2.9
demonstrates that with the right categorical setup for ordinary weighted
digraphs, the left and right divergence operators coincide with each other and
with the ordinary definition of the divergence. In general we have the
following.

$\langle\nabla[X],[Y]\rangle_{\widehat{\mathcal{P}}}=\langle[X],\nabla_{*}[Y]\rangle_{\mathcal{P}}\quad\text{and}\quad\langle\nabla^{*}[X],[Y]\rangle_{\mathcal{P}}=\langle[X],\nabla[Y]\rangle_{\widehat{\mathcal{P}}}.$

In particular

$\langle\nabla^{*}\nabla[X],[X]\rangle_{\mathcal{P}}=\langle\nabla[X],\nabla[X]\rangle_{\widehat{\mathcal{P}}}=\langle[X],\nabla_{*}\nabla[X]\rangle_{\mathcal{P}}.$

While the Euler pairing offers some better general properties, the adjointness
relations above do not hold for it in general. We shall elaborate on this
point in Section 3, where we show in fact that our basic constructions apply
in a much more general context than modules categories over finite posets.

Composing the gradient with the left and the right divergence, we obtain left
and right Laplacians $\Delta^{0}$ and $\Delta_{0}$ , respectively, for
persistence modules. This allows us to define left and right harmonic modules,
namely such that their corresponding Laplacians vanish. Laplacians, higher
dimensional generalisation and a possible setup for Hodge theory for
persistence modules will be studied in future work.

The paper is organised as follows. Section 1 contains all the technical
background material we use throughout the paper. In Section 2 we prepare the
general setup for the construction of the gradient and the divergence in the
context of module categories. In Section 3 we study some properties of the Hom
pairing and the Euler pairing in the context of modules over category
algebras. Section 4 is dedicated to the study of the gradient of virtual
modules over finite posets and the proof of Theorems A, B and C. In Section 5
we specialise the Hom and Euler pairings in for modules over posets and prove
Proposition D. Section 6 is dedicated to the (left and right) divergence, the
corresponding Laplacians, and adjointness relations with the gradient. Finally
in Section 7 we present applications to modules over commutative ladder
posets, and to filtered hierarchical clustering modules over commutative grid
posets.

The authors are grateful to E. Meir for many helpful conversations on modules
over category algebras and for finding an error in an early version of this
paper.

##  1\. Preliminaries

In this section we record the definitions, notation and all preliminary
material that will be used throughout the paper.

###  1.1. Persistence Modules

Let $\mathcal{C}$ be a small category associated to a poset. Modern theory of
persistence modules studies the structure and invariants of functors from
$\mathcal{C}$ to some target category, often the category
${\mathrm{Vect}_{k}}$ of vector spaces over a field $k$. For the purpose of
this preliminary discussion the target category may be any abelian category
$\mathcal{A}$. A category $\mathcal{C}$ is said to be _finite_ if the set of
all its morphisms forms a finite set; this implies that the object set is
likewise finite.

If $\mathcal{A}$ is an abelian category, then the functor category
$\mathcal{A}^{\mathcal{C}}$ whose objects are functors from $\mathcal{C}$ to
$\mathcal{A}$ and whose morphisms are natural transformation is also an
abelian category. If $\Phi\colon\mathcal{C}\to\mathcal{D}$ is a functor then
pre-composition with $\Phi$ induces a functor

$\Phi^{*}\colon\mathcal{A}^{\mathcal{D}}\to\mathcal{A}^{\mathcal{C}}.$

This functor, which is sometimes referred to as _restriction along $\Phi$_,
will be used in our definition of the gradient in Section 4. The restriction
$\Phi^{*}$ generally has left and right adjoints given by the left and right
Kan extensions, respectively (See Section 1.4). These will be used in Section
6. A good reference for the general theory of functor categories is [18].

######  Definition 1.1.

Let $\mathcal{P}$ be a finite poset, let $k$ be a field, and let
${\mathrm{Vect}_{k}}$ be the category of finite dimensional vector spaces over
$k$. A _persistence module_ on $\mathcal{P}$ is a functor
$M\colon\mathcal{P}\to{\mathrm{Vect}_{k}}$.

Let $\mathcal{C}$ be a small category. A functor
$M\colon\mathcal{C}\to{\mathrm{Vect}_{k}}$ may be thought of as a
representation of the category $\mathcal{C}$ over $k$. This is a particularly
useful approach when the category $\mathcal{C}$ is finite.

######  Definition 1.2.

Let $k$ be a field and let $\mathcal{C}$ be a finite category. The _category
algebra_ $k\mathcal{C}$ is the unital algebra generated as a $k$-vector space
by all morphisms $x\to y$ in $\mathcal{C}$ (including identities). Two
morphisms multiply by composition, and non-composable morphisms multiply to 0
[27]. The unit is the $k$-algebra map $\eta\colon k\to k\mathcal{C}$ that
sends $1\in k$, to the element $\mathbf{1}\in k\mathcal{C}$, that is the sum
over all objects in $\mathcal{C}$ of the identity relations
$1_{x}\overset{\text{{def}}}{=}x\to x$.

The following theorem due to Mitchell allows us to alternate between functor
categories and categories of modules, when the category in question is finite.

######  Theorem 1.3 ([27]).

Let $\mathcal{C}$ be a finite category and let $k$ be a field. The category
$k\mathcal{C}\mbox{-}\EuR{mod}$ of modules over the category algebra
$k\mathcal{C}$ and $k\mathcal{C}$-linear homomorphisms is equivalent to the
category of functors $M\colon\mathcal{C}\to{\mathrm{Vect}_{k}}$ and natural
transformations between them.

The equivalence in Mitchell’s theorem is given as follows. If
$M\colon\mathcal{C}\to{\mathrm{Vect}_{k}}$ is a functor, let

$\mathbf{M}\overset{\text{{def}}}{=}\bigoplus_{x\in\mathrm{Obj}(\mathcal{C})}M(x).$

The (left) action of $k\mathcal{C}$ on $\mathbf{M}$ is determined by

$\varphi\cdot\mathbf{a}=M(\varphi)(a_{y}),\quad\text{where}\quad\mathbf{a}=\sum_{x\in\mathcal{C}}a_{x}\in\mathbf{M},\quad
a_{x}\in M(x).$

Conversely, if $\mathbf{N}$ is a left $k\mathcal{C}$-module, let
$N\colon\mathcal{C}\to{\mathrm{Vect}_{k}}$ be the functor that takes an object
$x\in\mathcal{C}$ to $N(x)\overset{\text{{def}}}{=}1_{x}\cdot\mathbf{N}$ and a
morphism $\varphi\colon y\to z$ to the homomorphism $N(\varphi)$ that maps
$1_{y}\cdot\mathbf{a}\in 1_{y}\cdot\mathbf{N}$ to $\varphi\cdot\mathbf{a}\in
1_{z}\cdot\mathbf{N}$.

Any poset $\mathcal{P}$ may be considered as a small category, and thus admits
a category algebra $k\mathcal{P}$, which we will refer to as the _poset
algebra_. If $\mathcal{P}$ is a finite poset, then Mitchell’s theorem gives an
alternative way of thinking of persistence modules, namely as ordinary modules
over the poset algebra $k\mathcal{P}$. Both approaches are useful.

The standard modern treatment of one parameter persistence studies modules
over the poset $(\mathbb{R},\leq)$, i.e. the poset given by the real numbers
with their natural order. Higher dimensional analogs consider modules where
the parametrising poset is $\mathbb{R}^{n}$. These posets are not discrete and
certainly not finite. However, for all practical purposes persistence modules
are studied in a _tame_ setup. This concept was introduced by Chazal et al.
[11, Sections 3.8-3.9]. We will use tameness as defined by Miller in [26, Def.
2.12] for general posets. A persistence module $M$ is said to be tame if
$M(p)$ is finite dimensional for every object $p\in\mathcal{P}$ and if,
roughly speaking, $\mathcal{P}$ admits a finite partition into sub-posets on
which $M$ is constant.

Figure 1. Tame persistence module $M$ on $(\mathbb{R}^{2},\leq)$ and its
encoding into a finite poset $\mathcal{Q}$. The shaded rectangles correspond
to objects of $\mathcal{Q}$ and represent subsets of $\mathbb{R}^{2}$ on which
all the morphisms of $M$ are isomorphisms, i.e. $M$ is constant on those
subsets.

More precisely, if $M$ is a tame persistence module on an arbitrary poset
$\mathcal{P}$ (possibly infinite and with nontrivial topology) Miller defines
an _encoding_ of $M$ to be a module $N\in k\mathcal{Q}\mbox{-}\EuR{mod}$ for
some poset $\mathcal{Q}$, together with a functor
$\pi\colon\mathcal{P}\to\mathcal{Q}$, such that $M=\pi^{*}(N)$; see Figure 1.
The encoding is said to be _finite_ , if the poset $\mathcal{Q}$ is finite and
the values of $N$ on its objects are finite dimensional [26, Def. 4.1]. Miller
also gives a necessary and sufficient condition for a persistence module to
admit a finite encoding [26, Thm 4.22]. This essentially means that much of
the information encoded in the module can be read off from a finite encoding.
This point of view fits perfectly with our setup, in which posets will always
be assumed to be finite.

The following general terminology is rather standard and will become useful in
our analysis.

######  Definition 1.4.

Let $\mathcal{C}$ be a small category. We say that a module $M\in
k\mathcal{C}\mbox{-}\EuR{mod}$ is

  * •

_locally constant_ if $k\mathcal{C}$ acts on $M$ by isomorphisms.

  * •

_virtually trivial_ if any non-scalar element in $k\mathcal{C}$ acts on $M$
trivially.

Equivalently, if modules are considered as functors
$M\colon\mathcal{C}\to{\mathrm{Vect}_{k}}$, then $M$ is locally constant if it
takes any morphism in $\mathcal{C}$ to an isomorphism, and virtually trivial
if any non-identity morphism in $\mathcal{C}$ induces the zero homomorphism.

######  Lemma 1.5.

Let $\mathcal{C}$ be any small category and let $M,N\in
k\mathcal{C}\mbox{-}\EuR{mod}$ be virtually trivial modules. Let
$\mathrm{Obj}_{\mathcal{C}}$ denote the discrete category (only identity
morphisms) with objects set $\mathrm{Obj}(\mathcal{C})$, and let
$j_{\mathcal{C}}\colon\mathrm{Obj}_{\mathcal{C}}\to\mathcal{C}$ denote the
inclusion. Then there is a natural isomorphism of groups

$J\colon\mathrm{Hom}_{k\mathcal{C}\mbox{-}\EuR{mod}}(M,N)\to\mathrm{Hom}_{k\mathrm{Obj}_{\mathcal{C}}}(j_{\mathcal{C}}^{*}(M),j_{\mathcal{C}}^{*}(N)),$

where on both sides $\mathrm{Hom}$ denotes the set of natural transformations
between the respective modules, and $j_{\mathcal{C}}^{*}$ denotes pre-
composition with $j_{\mathcal{C}}$.

###### Proof.

Since $M$ and $N$ are virtually trivial, a natural transformation from $M$ to
$N$ amounts exactly to a homomorphism of vector spaces $M(x)\to N(x)$ for each
$x\in\mathrm{Obj}(\mathcal{C})$. Thus $J$ is an isomorphism of sets. Since the
functor categories are abelian and $j^{*}_{\mathcal{C}}$ is additive, the
lemma follows. ∎

######  Corollary 1.6.

Let $\mathcal{C}$ be any small category and let $M,N\in
k\mathcal{C}\mbox{-}\EuR{mod}$ be virtually trivial modules. Then $M\cong N$
if and only if $M(x)\cong N(x)$ for each object
$x\in\mathrm{Obj}(\mathcal{C})$.

###  1.2. Line digraphs

A central idea in our work is to define operators on persistence modules whose
output also lies in persistence modules but possibly over a different
parameter poset. This is where the classical graph theoretic construction of
the line digraph becomes useful.

######  Definition 1.7.

A _directed graph_ , or a _digraph_ , $\mathcal{G}$ is given by a vertex set
$V$ and an edge set $E\subset V\times V$. A digraph has two maps $s,t\colon
E\rightarrow V$ called the _source_ and _target_ , respectively, such that if
$e=(u,v)$ is an edge, then $s(e)=u$ and $t(e)=v$.

Definition 1.7 excludes the possibility of two vertices connected by more than
one edge in any direction, but it does allow reciprocal connections. A digraph
is said to be _acyclic_ if there is no directed path that starts and ends at
the same vertex. While acyclicity is not assumed throughout, in the context of
this work we do assume that all digraphs are _loop-free_ , namely that if
$e=(u,v)\in E$, then $u\neq v$.

######  Definition 1.8.

Given a digraph $\mathcal{G}=(V,E)$, the associated _line digraph_
$\widehat{\mathcal{G}}=(\widehat{V},\widehat{E})$ is the digraph with vertices
$\widehat{V}=E$ and with a directed edge $(u,v)\to(w,z)$ in $\widehat{E}$
whenever $v=w$. The directed edges in $\widehat{E}$ are denoted by ordered
triples $(u,v,z)$.

The line digraph construction is clearly functorial with respect to inclusions
of subgraphs. Namely, if $\mathcal{G}^{\prime}\subseteq\mathcal{G}$, then
$\widehat{\mathcal{G}^{\prime}}\subseteq\widehat{\mathcal{G}}$.

A digraph $\mathcal{G}$ is said to be _connected_ if, ignoring edge direction
and collapsing bidirectional edges into a single undirected edge, one obtains
a connected graph. The line digraph associated to a digraph $\mathcal{G}$ is
not generally connected. But if $\widehat{\mathcal{G}}$ is a connected line
digraph of some digraph $\mathcal{G}$, then $\mathcal{G}$ is connected. The
following concept is useful for our purposes.

######  Definition 1.9.

Let $\mathcal{G}$ be a digraph with a line digraph $\widehat{\mathcal{G}}$. A
subgraph $\mathcal{G}_{0}\subseteq\mathcal{G}$ is said to be a _line component
of $\mathcal{G}$_ if there exists a connected component
$\widehat{\mathcal{G}_{0}}\subseteq\widehat{\mathcal{G}}$ such that
$\widehat{\mathcal{G}_{0}}$ is the line digraph associated to
$\mathcal{G}_{0}$. A digraph $\mathcal{G}$ is said to be _line connected_ if
$\widehat{\mathcal{G}}$ is connected, or equivalently if the only line
component of $\mathcal{G}$ is $\mathcal{G}$ itself.

---
$\textstyle{d}$$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---
$\textstyle{bd}$$\textstyle{cd}$$\textstyle{ab\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{ac\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Figure 2. A digraph (left) and its line digraph (right) with the line
components and their associated line digraphs in red and blue.

Clearly any digraph $\mathcal{G}$ is a union of its line components (which are
not generally disjoint in $\mathcal{G}$). The line components of $\mathcal{G}$
can be constructed from the connected components of $\widehat{\mathcal{G}}$ by
considering for each component $\widehat{\mathcal{G}}_{0}$, the subgraph of
$\mathcal{G}$ consisting of all edges corresponding to the vertices of
$\widehat{\mathcal{G}}_{0}$ (See Figure 2).

Any poset $\mathcal{P}$ is uniquely determined by its associated _Hasse
diagram_ $\mathcal{H}_{\mathcal{P}}$, which is the transitive reduction of the
acyclic digraph underlying the poset $\mathcal{P}$. Namely the digraph
$\mathcal{H}_{\mathcal{P}}$ has as vertices the objects of $\mathcal{P}$, and
directed edges are the non-identity indecomposable relations in $\mathcal{P}$,
namely those relations $u<v$ in $\mathcal{P}$ such that there is no
intermediate relation $u<y<v$. The corresponding edge in the digraph
$\mathcal{H}_{\mathcal{P}}$ is then denoted by $(u,v)$. By taking the
transitive closure of $\mathcal{H}_{\mathcal{P}}$ one recovers $\mathcal{P}$.
_The Hasse diagram $\mathcal{H}_{\mathcal{P}}$ can be thought of as encoding
the analog in $\mathcal{P}$ of infinitesimally small moves in a metric space_.

######  Definition 1.10.

A poset $\mathcal{P}$ is said to be _line connected_ if its Hasse diagram
$\mathcal{H}_{\mathcal{P}}$ is line connected.

###  1.3. The Grothendieck ring

As is the case for any category of modules, many questions about persistence
modules can be reduced to questions about isomorphism classes. Hence we work
specifically with isomorphism classes of modules over poset algebras. In this
context the concept of the Grothendieck ring is very useful.

Let $k$ be a field, let $A$ be a unital $k$-algebra and let
$A\mbox{-}\EuR{mod}$ denote the category of finitely generated left
$A$-modules. Then $A\mbox{-}\EuR{mod}$ is a monoid with operation given by
direct sum, and thus the set of isomorphism classes of $A$-modules forms a
commutative monoid where

(1) $[M]+[N]\overset{\text{{def}}}{=}[M\oplus N].$

The Grothendieck group $\EuR{Gr}(A)$ of finitely generated $A$-modules is the
group completion of this monoid. Namely, it is the quotient of the free
abelian group generated by isomorphism classes of $A$-modules, subject to the
relation (1). Thus any element $[X]\in\EuR{Gr}(A)$ can be written uniquely as
a difference $[X]=[M]-[N]$, where $M,N\in A\mbox{-}\EuR{mod}$. In particular,
an equation of the form $[M]-[N]=[U]-[V]$ in $\EuR{Gr}(A)$ where $M,N,U,V\in
A\mbox{-}\EuR{mod}$ should be interpreted as $M\oplus V\cong N\oplus U$. We
refer to elements of $\EuR{Gr}(A)$ as _virtual modules_.

Tensor product over $k$, with the diagonal action of $A$ on the factors, turns
$\EuR{Gr}(A)$ into a ring, known as the Grothendieck ring, where
$[M]\cdot[N]=[M\otimes N]$. If $f^{*}\colon B\mbox{-}\EuR{mod}\to
A\mbox{-}\EuR{mod}$ is a functor that preserves tensor products, then $f^{*}$
induces a ring homomorphism

$f^{*}\colon\EuR{Gr}(B)\to\EuR{Gr}(A).$

This is the case, for instance, if $f\colon A\to B$ is a homomorphism of
$k$-algebras that induces the restriction $f^{*}$ on module categories, but
for our applications this will not generally be the case (See Remark 1.14).

An important example of this construction is given by the Grothendieck ring
$\EuR{Gr}(k)$ of finite dimensional vector spaces over $k$, which is
isomorphic to the ring of integers $\mathbb{Z}$. A much more interesting
family of examples that is a subject of extensive study arises in
representation theory. If $G$ is a group and $kG$ its group algebra, then
$\EuR{Gr}(kG)$ is the Grothendieck ring of all virtual linear representations
of the group $G$ over $k$.

######  Definition 1.11.

Let $A$ be a unital $k$-algebra. Define the _reduced Grothendieck ring_
$\EuR{Gr}_{e}(A)$ to be the quotient of the Grothendieck ring $\EuR{Gr}(A)$,
by an additional family of relations:

$[M]=[M^{\prime}]+[M^{\prime\prime}]$

if there is a short exact sequence of $A$-modules $0\to M^{\prime}\to M\to
M^{\prime\prime}\to 0$.

The reduced Grothendieck ring $\EuR{Gr}_{e}(A)$ inherits a ring structure from
that of $\EuR{Gr}(A)$, since the tensor product over the ground field $k$ is
an exact functor, and the obvious projection $\EuR{Gr}(A)\to\EuR{Gr}_{e}(A)$
is a ring homomorphism. However, $\EuR{Gr}_{e}(A)$ is a much less interesting
object than $\EuR{Gr}(A)$, since the class of any module is equal to the sum
of all simple modules in a composition series. Thus $\EuR{Gr}_{e}(A)$ is
isomorphic to the free commutative algebra generated by the simple
$A$-modules.

######  Definition 1.12.

For a finite poset $\mathcal{P}$ with object set $V$, define
$\mathbf{dim}_{k}\colon\EuR{Gr}(k\mathcal{P})\to\mathbb{Z}^{V}$ by

$\mathbf{dim}_{k}[M](v)\overset{\text{{def}}}{=}\dim_{k}(M(v)),$

for any $M\in k\mathcal{P}\mbox{-}\EuR{mod}$ and for $[X]=[M]-[N]$ with
$M,N\in k\mathcal{P}\mbox{-}\EuR{mod}$, let

$\mathbf{dim}_{k}[X]\overset{\text{{def}}}{=}\mathbf{dim}_{k}[M]-\mathbf{dim}_{k}[N].$

For $[X]\in\EuR{Gr}(k\mathcal{P})$ we refer to $\mathbf{dim}_{k}[X]$ as the
_dimension vector of $[X]$_.

Since the addition operation in $\EuR{Gr}(k\mathcal{P})$ is induced by direct
sum, the function $\mathbf{dim}_{k}$ is clearly a group homomorphism, whose
kernel is the subgroup of all virtual modules $[X]=[M]-[N]$ such that
$\dim_{k}M(v)=\dim_{k}N(v)$ for every $v\in\mathcal{P}$. Clearly for $M\in
k\mathcal{P}\mbox{-}\EuR{mod}$, $\mathbf{dim}_{k}[M]=0$ if and only if $M=0$.

######  Definition 1.13.

Let $\mathcal{C}$ be a small category. An element
$[X]\in\EuR{Gr}(k\mathcal{C})$ is said to be _locally constant_ , if it can be
represented as a difference of the isomorphism classes of two locally constant
modules, and similarly for _virtually trivial_.

######  Remark 1.14.

Recall that if $\mathcal{C}$ is a category with finite set of objects and $k$
is a field, then the functor category $k\mathcal{C}\mbox{-}\EuR{mod}$ is
equivalent to the category of modules over the category algebra $k\mathcal{C}$
(see [31]). If $F\colon\mathcal{C}\to\mathcal{D}$ is a functor, then one
obtains a homomorphism $kF\colon k\mathcal{C}\to k\mathcal{D}$, which is a
ring homomorphism if and only if $F$ is injective on object sets [31,
Proposition 2.2.3], and otherwise is only a homomorphism of vector spaces. In
particular $F^{*}$ preserves tensor products over $k$, and hence induces a
ring homomorphism
$F^{*}\colon\EuR{Gr}(k\mathcal{D})\to\EuR{Gr}(k\mathcal{C})$.

###  1.4. Functor categories and Kan extensions

Kan extensions will be used to define the divergence operators in Section 2,
and we recall here the basic notions. Let $\mathcal{C},\mathcal{D}$ be small
categories, and let $\mathcal{A}$ be a category that is bicomplete (i.e. all
small limits and colimits exist in $\mathcal{A}$). Let
$\mathcal{A}^{\mathcal{C}}$ and $\mathcal{A}^{\mathcal{D}}$ denote the
categories of functors from $\mathcal{C}$ and $\mathcal{D}$ to $\mathcal{A}$,
respectively, with morphisms given by natural transformations. If
$F\colon\mathcal{C}\to\mathcal{D}$ is a functor, then the restriction
$F^{*}\colon\mathcal{A}^{\mathcal{D}}\to\mathcal{A}^{\mathcal{C}}$ admits left
and right adjoints, given by the left and right Kan extensions [25, X.3,
Corollary 2],

$L_{F},R_{F}\colon\mathcal{A}^{\mathcal{C}}\to\mathcal{A}^{\mathcal{D}}.$

For the sake of completeness, we briefly describe these constructions. The
reader is referred to [25, Chapter X] for details. For any object
$d\in\mathcal{D}$, let $F\downarrow d$ denote the _overcategory of $d$ with
respect to $F$_, whose objects are pairs $(c,\varphi)$, where $c$ is an object
in $\mathcal{C}$ and $\varphi\colon F(c)\to d$ is a morphism in $\mathcal{D}$.
A morphism $(c,\varphi)\to(c^{\prime},\varphi^{\prime})$ in $F\downarrow d$ is
a morphism $\beta\colon c\to c^{\prime}$, such that $\varphi^{\prime}\circ
F(\beta)=\varphi$. There is a forgetful functor $\\#\colon F\downarrow
d\to\mathcal{C}$, sending an object $(c,\varphi)$ to $c$. By analogy one
defines $d\downarrow F$, the _undercategory of $d$ with respect to $F$_, and
similarly a forgetful functor to $\mathcal{C}$. For
$M\in\mathcal{A}^{\mathcal{C}}$, left and right Kan extensions of $M$ along
$F$ are defined by

$L_{F}(M)(d)\overset{\text{{def}}}{=}\mathop{\textup{colim}}\limits_{F\downarrow
d}M_{\\#},\quad\text{and}\quad
R_{F}(M)(d)\overset{\text{{def}}}{=}\mathop{\textup{lim}}\limits_{d\downarrow
F}M_{\\#},$

where in both cases $M_{\\#}$ denotes the composition of $M$ with the
respective forgetful functor. Thus if $N\in\mathcal{A}^{\mathcal{C}}$ and
$M\in\mathcal{A}^{\mathcal{D}}$, we have the adjointness relations:

$\mathrm{Hom}_{\mathcal{A}^{\mathcal{D}}}(L_{F}(N),M)\cong\mathrm{Hom}_{\mathcal{A}^{\mathcal{C}}}(N,F^{*}(M)),\quad\text{and}\quad\mathrm{Hom}_{\mathcal{A}^{\mathcal{C}}}(F^{*}(M),N)\cong\mathrm{Hom}_{\mathcal{A}^{\mathcal{D}}}(M,R_{F}(N)).$

##  2\. A categorical setting for the discrete gradient and divergence

In this section we define the gradient and the divergence on digraphs in a
very general setting. With the right setup, the standard notions of these
concepts then occur as special cases. In sections 4 and 6 we will specialise
to the case of persistence modules, but these constructions can be carried out
in a much more general context.

We start by recalling the graph theoretic definitions of gradient and
divergence. Let $\mathcal{G}=(V,E)$ be a digraph. Let $A$ be an abelian group,
and let $f\colon V\to A$ and $F\colon E\to A$ be arbitrary functions. Then one
defines the gradient $\nabla(f)\colon E\to A$ and $\nabla^{*}(F)\colon V\to A$
by

(2) $\nabla(f)(v\to
w)\overset{\text{{def}}}{=}f(w)-f(v),\quad\text{and}\quad\nabla^{*}(F)(v)=\sum_{u\to
v}F(u\to v)-\sum_{v\to w}F(v\to w).$

The reader is referred to [24] for a comprehensive discussion of the gradient
and the divergence on digraphs.

We now describe a categorical setup for these concepts. A multi-digraph is a
more general analog of what is meant by a digraph in this article. The
difference is that in a multi-digraph one allows the set of directed edges
between any pair of vertices to have arbitrary cardinality, while the concept
we use of a digraph allows at most one edge in each direction. Any small
category $\mathcal{C}$ is a quotient category of the path category of a multi-
digraph whose vertex set is in $1-1$ correspondence with the set of objects in
$\mathcal{C}$ and whose directed edges form a subset of its morphisms, where
the equivalence relations are given by the commutativity relations in
$\mathcal{C}$ (See [4, Section 5] for more detail). A canonical example of
this type is given by considering the multi-digraph of all morphisms in a
category $\mathcal{C}$. Then $\mathcal{C}$ is reconstructed from that multi-
digraph by declaring two paths equivalent if the corresponding morphisms in
$\mathcal{C}$ coincide. Clearly, restricting to digraph rather than multi-
digraph puts some restrictions on the type of categories that appear as
quotient categories of the corresponding path categories. For instance, a
poset $\mathcal{P}$ can be thought of as a quotient of the path category of
its Hasse diagram $\mathcal{H}_{\mathcal{P}}$ by relations imposed by the
requirement that between any two objects there is at most one morphism.
However, quotient categories of path categories of digraphs form a much more
general family of categories than just posets. All of those, as we shall see
below, share some important properties.

######  Definition 2.1.

If $\mathcal{C}$ is a category obtained as a quotient category of the path
category $P(\mathcal{G})$ of some digraph $\mathcal{G}$, we say that
$\mathcal{G}$ is a _generating digraph_ for $\mathcal{C}$, or that
$\mathcal{C}$ is _generated_ by $\mathcal{G}$.

Notice that the choice of a generating digraph for a small category
$\mathcal{C}$ is generally not unique.

Next we associate with a small category $\mathcal{C}$ and a generating digraph
$\mathcal{G}$ another small category $\widehat{\mathcal{C}}$ that is generated
by the associated line digraph $\widehat{\mathcal{G}}$. The category
$\widehat{\mathcal{C}}$ depends on the choice of the generating digraph, but
for any such choice one has two functors $\widehat{\mathcal{C}}\to\mathcal{C}$
that will allow us to define a gradient and two divergence operators. While we
will later restrict to the case where $\mathcal{C}$ is a poset with a
canonical generating digraph given by its Hasse diagram, here we study these
constructions in greater generality with a view to other possible
applications.

######  Definition 2.2.

Let $\mathcal{C}$ be a small category generated by a digraph $\mathcal{G}$,
and let $\widehat{\mathcal{G}}$ denote its associated line digraph. Let
$P(\mathcal{G})$ and $P(\widehat{\mathcal{G}})$ denote the associated path
categories. Define a category $\widehat{\mathcal{C}}$ with the same object set
as $P(\widehat{\mathcal{G}})$. Assume that

$\displaystyle\alpha\overset{\text{{def}}}{=}$
$\displaystyle((u,t)\to(t,a_{1})\to\cdots\to(a_{m},s)\to(s,v)),\quad\text{and}\quad$
$\displaystyle\gamma\overset{\text{{def}}}{=}$
$\displaystyle((u,t)\to(t,b_{1})\to\cdots\to(b_{k},s)\to(s,v))$

are two morphisms in $P(\widehat{\mathcal{G}})$. For each pair of objects
$(u,t)$ and $(s,v)$, define an equivalence relation on the morphism set
$P(\widehat{\mathcal{G}})((u,t),(s,v))$, to be the transitive closure of the
relation $\alpha\sim\gamma$ if the compositions

$t\to a_{1}\to\cdots\to a_{m}\to s,\quad\text{and}\quad t\to b_{1}\to\cdots\to
b_{k}\to s$

coincide in $\mathcal{C}$. Define $\widehat{\mathcal{C}}((u,t),(s,v))$ to be
the set of equivalence classes of this relation.

Clearly the assignment $\mathcal{C}\mapsto\widehat{\mathcal{C}}$ does not
define an endofunctor on the category of small categories, since it depends on
the choice of a generating digraph $\mathcal{G}$ for $\mathcal{C}$. We refer
to $\widehat{\mathcal{C}}$ as the _line category associated to $\mathcal{C}$
with respect to the generating digraph $\mathcal{G}$._

######  Definition 2.3.

Let $\phi,\beta\colon\widehat{\mathcal{G}}\to\mathcal{G}$ denote the _front_
and _back_ graph morphisms, defined by $\phi(u,v)=v$ and $\beta(u,v)=u$, for
any edge $(u,v)$ in $\widehat{\mathcal{G}}$, with the action on edges in
$\widehat{\mathcal{G}}$ given by $\phi(u,v,w)=(v,w)$ and $\beta(u,v,w)=(u,v)$
for any edge $(u,v,w)$ in $\widehat{\mathcal{G}}$.

The maps $\phi$ and $\beta$ induce functors

$\phi,\beta\colon P(\widehat{\mathcal{G}})\to P(\mathcal{G}).$

The following lemma shows that the front and back functors induce the
corresponding functors on the line category.

######  Lemma 2.4.

Let $\mathcal{C}$ be a small category generated by a digraph
$\mathcal{G}=(V,E)$. Let $\widehat{\mathcal{C}}$ denote the line category
associated to $\mathcal{C}$ with respect to $\mathcal{G}$. Then the front and
back functors $\phi$ and $\beta$ on path categories induce functors

$\phi,\beta\colon\widehat{\mathcal{C}}\to\mathcal{C}.$

###### Proof.

The category $\mathcal{C}$ is a quotient category of the path category
$P(\mathcal{G})$ by the relations among morphisms in $\mathcal{C}$. Thus it
suffices to show that the square of functors

$\textstyle{P(\widehat{\mathcal{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\pi}$$\textstyle{P(\mathcal{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\widehat{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\psi}}$$\textstyle{\mathcal{C}}$

where $\psi$ is either $\phi$ or $\beta$, commutes. We prove the statement for
$\phi$. The proof for $\beta$ is essentially the same. The objects in
$P(\widehat{\mathcal{G}})$ are the edges $(u,v)$ in $\mathcal{G}$, and
$\phi(u,v)=v$. Define $\bar{\phi}(u,v)=v$. The square commutes on objects by
definition. Let

$\displaystyle\alpha\overset{\text{{def}}}{=}$
$\displaystyle((u,t)\to(t,a_{1})\to\cdots\to(a_{m},s)\to(s,v)),\quad\text{and}\quad$
$\displaystyle\gamma\overset{\text{{def}}}{=}$
$\displaystyle((u,t)\to(t,b_{1})\to\cdots\to(b_{k},s)\to(s,v))$

be two morphisms in $P(\widehat{\mathcal{G}})$, such that
$\pi(\alpha)=\pi(\gamma)$. Then, by definition,

$t\to a_{1}\to\cdots\to a_{m}\to s,\quad\text{and}\quad t\to b_{1}\to\cdots\to
b_{k}\to s$

coincide in $\mathcal{C}$. Hence the compositions

$t\to a_{1}\to\cdots s\to v,\quad\text{and}\quad t\to b_{1}\to\cdots s\to v$

also coincide in $\mathcal{C}$. But these are exactly the projections of
$\phi(\alpha)$ and $\phi(\gamma)$ into $\mathcal{C}$. This shows that
$\bar{\phi}$ is well defined and that the square commutes on morphisms, and
the proof is complete. ∎

A family of examples that are particularly relevant to this article arises
from considering posets and their associated Hasse diagrams.

######  Example 2.5.

Let $\mathcal{P}$ be a poset, let $\mathcal{H}_{\mathcal{P}}$ be its Hasse
diagram and let $\widehat{\mathcal{H}}_{\mathcal{P}}$ be the line digraph
associated to $\mathcal{H}_{\mathcal{P}}$. Then
$\widehat{\mathcal{H}}_{\mathcal{P}}$ is the Hasse diagram of a poset
$\widehat{\mathcal{P}}$ that is unique up to isomorphism.

To see this, note that any finite poset $\mathcal{P}$ can be reconstructed
uniquely up to isomorphism from its associated Hasse diagram, which by
definition is an acyclic and transitively reduced digraph. Thus it suffices to
show that if $\mathcal{H}=(V,E)$ is a finite transitively reduced acyclic
digraph, then $\widehat{\mathcal{H}}$ is also acyclic and transitively
reduced.

It is immediate from the definition that the existence of a cycle in
$\widehat{\mathcal{H}}$ implies that $\mathcal{H}$ itself contains a cycle.
Let $(u,v,w)$ be an edge in $\widehat{\mathcal{H}}$, and assume that it can be
decomposed as $(u,v,z)\cdot(v,z,w)$. Then $\mathcal{H}$ contains the
composable sequence

${u}$${v}$${z}$${w}$$\scriptstyle{(u,v)}$$\scriptstyle{(v,z)}$$\scriptstyle{(z,w)}$

Hence either $(u,v)$ is not composable with $(z,w)$ or $v=z$. The first option
contradicts the assumption that $(u,v,w)$ is an edge in
$\widehat{\mathcal{H}}$ and the second option contradicts the assumption that
$(v,z,w)$ is an edge in $\widehat{\mathcal{H}}$, or alternatively acyclicity
of $\mathcal{H}$ due to the self-loop $(v,v)$. Thus $(u,v,w)$ is
indecomposable, and so $\widehat{\mathcal{H}}$ is transitively reduced.

If $\mathcal{P}$ is a poset, then the poset $\widehat{\mathcal{P}}$ associated
to $\widehat{\mathcal{H}}_{\mathcal{P}}$ has as objects the edges $(u,v)$ in
$\mathcal{H}_{P}$ and $(u,v)\leq(z,w)$ if there is a sequence of directed
edges, for $n\geq 2$,

$(u,v)=(a_{0},a_{1})\xrightarrow{(a_{0},a_{1},a_{2})}(a_{1},a_{2})\xrightarrow{(a_{1},a_{2},a_{3})}(a_{2},a_{3})\to\cdots\xrightarrow{(a_{n-2},a_{n-1},a_{n})}(a_{n-1},a_{n})=(z,w).$

Let $\mathcal{S}$ be a category, and let $\mathcal{C}$ be a small category,
with a generating digraph $\mathcal{G}$ and an associated line digraph
$\widehat{\mathcal{G}}$. Let $\widehat{\mathcal{C}}$ be the associated line
category with respect to $\mathcal{G}$. The functors
$\phi,\beta\colon\widehat{\mathcal{C}}\to\mathcal{C}$ induce _restriction
functors_

$\phi^{*},\beta^{*}\colon\mathcal{S}^{\mathcal{C}}\to\mathcal{S}^{\widehat{\mathcal{C}}},$

where $\mathcal{S}^{\mathcal{C}}$ and $\mathcal{S}^{\widehat{\mathcal{C}}}$
are the functor categories from $\mathcal{C}$ and $\widehat{\mathcal{C}}$ to
$\mathcal{S}$, respectively. We are now ready to define the gradient.

######  Definition 2.6.

Fix a small category $\mathcal{C}$ and an associated line category
$\widehat{\mathcal{C}}$ with respect to some generating digraph $\mathcal{G}$.
Let $\mathcal{S}$ be an additive category. Let
$\EuR{Gr}(\mathcal{S}^{\mathcal{C}})$ denote the Grothendieck group of
isomorphism classes of functors $\mathcal{C}\to\mathcal{S}$, with the
operation $[F]+[G]=[F\sqcup G]$, where $(F\sqcup G)(c)=F(c)\sqcup G(c)$, and
where $\sqcup$ denote the coproduct in $\mathcal{S}$. Define the _gradient_

$\nabla\colon\EuR{Gr}(\mathcal{S}^{\mathcal{C}})\to\EuR{Gr}(\mathcal{S}^{\widehat{\mathcal{C}}})$

by

$\nabla[F]=[\phi^{*}(F)]-[\beta^{*}(F)]$

for each functor $F\in\mathcal{S}^{\mathcal{C}}$. Extend $\nabla$ by
additivity to the whole Grothendieck group.

Next we show that the standard definition of a gradient for a digraph with
vertex and edge weight functions taking values in the group of integers is a
particular case of our setup.

######  Example 2.7.

Let $\mathcal{G}=(V,E)$ be a digraph with line digraph
$\widehat{\mathcal{G}}$. Then $\phi$ and $\beta$ induce functors

$\phi^{*},\beta^{*}\colon\mathcal{S}^{P(\mathcal{G})}\to\mathcal{S}^{P(\widehat{\mathcal{G}})}.$

Consider $V$ and $E$ as discrete categories, and let $V\to P(\mathcal{G})$ and
$E\to P(\widehat{\mathcal{G}})$ be the obvious inclusions. Restriction of
$\phi^{*}$ and $\beta^{*}$ gives functors

$\phi^{*},\beta^{*}\colon\mathcal{S}^{V}\to\mathcal{S}^{E}.$

Now, assume that $\mathcal{S}={\mathrm{Vect}_{k}}$, the category of vector
spaces over a field $k$, with the coproduct $\sqcup$ given by the direct sum.
Since vector spaces are determined up to isomorphism by their dimension,
taking Grothendieck groups, we have

$\EuR{Gr}(\mathcal{S}^{V})=\mathbb{Z}^{V}\cong\bigoplus_{v\in
V}\mathbb{Z},\quad\text{and}\quad\EuR{Gr}(\mathcal{S}^{E})=\mathbb{Z}^{E}\cong\bigoplus_{e\in
E}\mathbb{Z}.$

Thus the gradient is given by

$\nabla[f](u,v)=\dim_{k}(f(v))-\dim_{k}(f(u)),$

where $[f]\in\EuR{Gr}(\mathcal{S}^{V})$.

Let $\varphi\colon V\to\mathbb{Z}$ be an arbitrary function that takes non-
negative values. Considering $V$ again as a discrete category, let $f\colon
V\to{\mathrm{Vect}_{k}}$ be the functor defined by $f(v)=k^{\varphi(v)}$. Then

$\nabla[f](u,v)=\nabla(\varphi)(u,v),$

where the left hand side is the gradient of the element
$[f]\in\EuR{Gr}(\mathcal{S}^{V})$, and the right hand side is the graph
theoretic definition for the gradient of the function $\varphi$, as in (2).

Next we define the divergence operators in our context. For this we require
that the category $\mathcal{S}$ is additive and bicomplete, namely that all
small limits and colimits exist in $\mathcal{S}$. In that case the functors
$\phi^{*}$ and $\beta^{*}$ have left and right adjoints given by the left and
right Kan extensions. Notice that because of pre-additivity of $\mathcal{S}$
finite products and coproducts coincide in $\mathcal{S}$.

######  Definition 2.8.

Fix a small category $\mathcal{C}$ and an associated line category
$\widehat{\mathcal{C}}$ with respect to some generating digraph $\mathcal{G}$.
Let $\mathcal{S}$ be a small bicomplete additive category. Define the _left
divergence_ and the _right divergence_ (with respect to $\mathcal{G}$)

$\nabla^{*},\nabla_{*}\colon\EuR{Gr}(\mathcal{S}^{\widehat{\mathcal{C}}})\to\EuR{Gr}(\mathcal{S}^{\mathcal{C}})$

by

$\nabla^{*}[T]=[L_{\phi}(T)]-[L_{\beta}(T)],\quad\text{and}\quad\nabla_{*}[T]=[R_{\phi}(T)]-[R_{\beta}(T)]$

for each functor $T\colon\widehat{\mathcal{C}}\to\mathcal{S}$. Extend by
additivity to the whole Grothendieck group.

Since limits commute with limits and colimits with colimits, and since finite
products and coproducts coincide in an additive category, both operators are
clearly well defined group homomorphisms.

We now show how the standard divergence operator for digraphs, with vertex and
edge weight functions taking values in the integers, is a particular example
of the general setup of Definition 2.8.

######  Example 2.9.

Let $\mathcal{G}=(V,E)$ be a digraph with line digraph
$\widehat{\mathcal{G}}$. Let $\mathcal{S}$ be a category satisfying the
requirement of Definition 2.8. Consider $V$ and $E$ as discrete subcategories
of $P(\mathcal{G})$ and $P(\widehat{\mathcal{G}})$ respectively, as in Example
2.7. Restriction of $\phi^{*}$ and $\beta^{*}$ gives functors

$\phi^{*},\beta^{*}\colon\mathcal{S}^{V}\to\mathcal{S}^{E}.$

Since $V$ and $E$ are discrete, we have for $f\in\mathcal{S}^{E}$

$L_{\phi}(f)(v)\overset{\text{{def}}}{=}\mathop{\textup{colim}}\limits_{\phi\downarrow
v}f=\coprod_{(u,v)\in E}f(u,v),\quad\text{and}\quad
L_{\beta}(f)(v)\overset{\text{{def}}}{=}\mathop{\textup{colim}}\limits_{\beta\downarrow
v}f=\coprod_{(v,w)\in E}f(v,w).$

Similarly

$R_{\phi}(f)(v)\overset{\text{{def}}}{=}\mathop{\textup{lim}}\limits_{v\downarrow\phi}f=\prod_{(u,v)\in
E}f(u,v),\quad\text{and}\quad
R_{\beta}(f)(v)\overset{\text{{def}}}{=}\mathop{\textup{lim}}\limits_{v\downarrow\beta}f=\prod_{(v,w)\in
E}f(v,w).$

Restrict attention to the case where $\mathcal{S}={\mathrm{Vect}_{k}}$ and
$\mathcal{G}$ is finite (this is essential because ${\mathrm{Vect}_{k}}$ is
only finitely bicomplete). Then, finite products (cartesian products) and
finite coproducts (direct sums) are isomorphic. Hence, the right and left Kan
extensions coincide. As in Example 2.7, we have

$\EuR{Gr}(\mathcal{S}^{V})=\mathbb{Z}^{V}\cong\bigoplus_{v\in
V}\mathbb{Z},\quad\text{and}\quad\EuR{Gr}(\mathcal{S}^{E})=\mathbb{Z}^{E}\cong\bigoplus_{e\in
E}\mathbb{Z}.$

Hence

$\nabla^{*}[f](v)=\nabla_{*}[f](v)=\sum_{(u,v)\in
E}\dim_{k}f(u,v)-\sum_{(v,w)\in E}\dim_{k}f(v,w)$

for $f\in\mathcal{S}^{E}$.

Let $\gamma\colon E\to\mathbb{Z}$ be an arbitrary function that takes non-
negative values. Considering $E$ as a discrete category, let $g\colon
E\to{\mathrm{Vect}_{k}}$ denote the functor defined by $g(e)=k^{\gamma(e)}$.
Then

$\nabla^{*}[g]=\nabla_{*}[g]=\nabla^{*}(\gamma)$

where the left and centre in the equation are the left and right divergence of
$[g]$ in $\EuR{Gr}(\mathcal{S}^{V})$, and the right hand side is the graph
theoretic divergence of $\gamma$, as in (2).

##  3\. Bilinear pairings on $\EuR{Gr}(\mathcal{S}^{\mathcal{C}})$

In this section we define two bilinear forms on
$\EuR{Gr}(\mathcal{S}^{\mathcal{C}})$, where $\mathcal{S}={\mathrm{Vect}_{k}}$
and $\mathcal{C}$ is a finite acyclic category (i.e. a category generated by a
finite acyclic digraph). Thus we use the notation $\EuR{Gr}(k\mathcal{C})$ for
$\EuR{Gr}(\mathcal{S}^{\mathcal{C}})$.

######  Definition 3.1.

Let $\mathcal{G}$ be a finite acyclic digraph that generates a category
$\mathcal{C}$. Define the _Hom pairing_

$\langle-,-\rangle_{\mathcal{C}}\colon\EuR{Gr}(k\mathcal{C})\times\EuR{Gr}(k\mathcal{C})\to\mathbb{Z}$

by

$\langle[M],[N]\rangle_{\mathcal{C}}\overset{\text{{def}}}{=}\dim_{k}\left(\mathrm{Hom}_{k\mathcal{C}}(M,N)\right),$

where $M$ and $N$ represent their isomorphism classes in
$\EuR{Gr}(k\mathcal{C})$. Extend the definition to the full Grothendieck group
by additivity.

Notice that acyclicity and finiteness guarantees that the category
$\mathcal{C}$ is finite, and so the Hom objects are guaranteed to be finite
dimensional. The Hom pairing is clearly bilinear and can be defined by analogy
also on $\EuR{Gr}(k\widehat{\mathcal{C}})$. Since the left and right Kan
extensions are left and right adjoints to the restrictions $\phi^{*}$ and
$\beta^{*}$ we have

(3)
$\langle\nabla^{*}[X],[Y]\rangle_{\mathcal{C}}=\langle[X],\nabla[Y]\rangle_{\widehat{\mathcal{C}}},\quad\text{and}\quad\langle\nabla[X],[Y]\rangle_{\widehat{\mathcal{C}}}=\langle[X],\nabla_{*}[Y]\rangle_{\mathcal{C}}$

for $[X]\in\EuR{Gr}(k{\widehat{\mathcal{C}}})$ and
$[Y]\in\EuR{Gr}(k{\mathcal{C}})$. Notice that the Hom pairing is not generally
symmetric; however, see Example 3.6 below.

Next we define another useful pairing on $\EuR{Gr}(k\mathcal{C})$.

######  Definition 3.2.

Let $\mathcal{G}$ be a finite acyclic digraph that generates a category
$\mathcal{C}$. Define the _Euler pairing_

$\chi_{\mathcal{C}}\colon\EuR{Gr}(k\mathcal{C})\times\EuR{Gr}(k\mathcal{C})\to\mathbb{Z}$

by

$\chi_{\mathcal{C}}([M],[N])\overset{\text{{def}}}{=}\chi(\mathrm{Ext}_{k\mathcal{C}}^{*}(M,N))=\sum_{n\geq
0}(-1)^{n}\dim_{k}(\mathrm{Ext}^{n}_{k\mathcal{C}}(M,N)),$

where $M$ and $N$ represent their isomorphism classes in
$\EuR{Gr}(k\mathcal{C})$, and extend to the Grothendieck group by additivity.

To show that this is well defined, we must argue that the $\mathrm{Ext}^{n}$
groups vanish for $n$ sufficiently large. This is implied by Lemma 3.3 below.

######  Lemma 3.3.

Let $\mathcal{G}$ be a finite acyclic digraph that generates a category
$\mathcal{C}$. Then the category algebra $k\mathcal{C}$ has finite projective
dimension.

###### Proof.

Since we assume that $\mathcal{G}$ is finite and acyclic, composition length
in $\mathcal{C}$ is bounded. For each object $x\in\mathcal{C}$, let $F_{x}\in
k\mathcal{C}\mbox{-}\EuR{mod}$ denote the projective module defined by

$F_{x}=k\mathcal{C}(x,-).$

Let $M$ be a finitely generated $k\mathcal{C}$-module. For each object
$x\in\mathcal{C}$, choose a basis $\\{v_{1},\ldots,v_{n_{x}}\\}$ for $M(x)$.
Let

$q_{x,i}\colon F_{x}\to M$

denote the transformation determined by taking $1_{x}\in\mathcal{C}(x,x)$, to
$v_{x,i}$. Thus one obtains a surjection of $k\mathcal{C}$-modules

$q_{M}\colon\bigoplus_{x\in\mathcal{C}}F_{x}^{d_{x}}\to M,$

where $d_{x}=\dim_{k}(M(x))$ and $F_{x}^{d_{x}}=\bigoplus_{d_{x}}F_{x}$. Let
$M_{1}$ denote the $\operatorname{Ker}\nolimits(q_{M})$. If $x$ is a minimal
object in the sense that $\mathcal{C}(-,x)=\emptyset$, then $M_{1}(x)=0$. Thus
$M_{1}$ vanishes on all minimal objects in $\mathcal{C}$. Notice that if
$d_{x}=0$, then no copy of $F_{x}$ is included in the cover. Next, construct a
projective cover of $M_{1}$ in a similar fashion, avoiding all minimal objects
in $\mathcal{C}$, and all those of graph distance $1$ from a minimal object,
for which $M_{1}$ vanishes. Let $M_{2}$ be the kernel of this cover. Then
$M_{2}$ vanishes of all minimal objects and all objects of graph distance $1$
from a minimal object. By induction on the length of the maximal chains in
$\mathcal{C}$, we obtain a finite projective resolution for $M$ of length
bounded above by the length of a maximal chain in $\mathcal{C}$. ∎

The pairing $\chi_{\mathcal{C}}$ is clearly bilinear, but in general it is
neither symmetric, nor is it true that

(4)
$\chi_{\mathcal{C}}(\nabla^{*}[X],[Y])=\chi_{\widehat{\mathcal{C}}}([X],\nabla[Y]),\quad\text{and}\quad\chi_{\mathcal{C}}([X],\nabla_{*}[Y])=\chi_{\widehat{\mathcal{C}}}(\nabla[X],[Y]).$

We now consider a special case where the generating digraph for $\mathcal{C}$
is a directed tree $\mathcal{T}$. In that case the path algebra $k\mathcal{T}$
of $\mathcal{T}$, considered as an acyclic quiver, and the category algebra
$k\mathcal{C}$ coincide. Path algebras over acyclic quivers are hereditary,
namely submodules of projective modules are projective. Hence for any $M,N\in
k\mathcal{C}$, one has $\mathrm{Ext}_{k\mathcal{C}}^{i}(M,N)=0$ for $i>1$.
Thus in this case

$\chi_{\mathcal{C}}([M],[N])=\dim_{k}\mathrm{Hom}_{k\mathcal{C}}(M,N)-\dim_{k}\mathrm{Ext}^{1}_{k\mathcal{C}}(M,N).$

In particular, if $\mathcal{C}$ is generated by a finite directed tree, then
the Euler pairing is rather easy to compute, as it depends only on the
dimension of point modules. To make this statement precise, we need the
following definition.

######  Definition 3.4.

Let $\mathcal{Q}=(V,E)$ be a quiver. Let $\alpha,\beta\in\mathbb{Z}^{V}$ be
integer valued functions on $V$. Define a function
$\chi_{\mathcal{Q}}\colon\mathbb{Z}^{V}\times\mathbb{Z}^{V}\to\mathbb{Z}$ by

$\chi_{\mathcal{Q}}(\alpha,\beta)\overset{\text{{def}}}{=}\sum_{x\in
V}\alpha(x)\beta(x)-\sum_{e\in E}\alpha(s(e))\beta(t(e)).$

The function $\chi_{\mathcal{Q}}$ is called the _Euler form for
$\mathcal{Q}$_.

######  Lemma 3.5 ([13, Proposition 2.5.2]).

Let $\mathcal{C}$ be a category generated by a finite tree $\mathcal{T}$. Then
for any $M,N\in k\mathcal{C}\mbox{-}\EuR{mod}$

$\chi_{\mathcal{C}}([M],[N])=\chi_{\mathcal{T}}(\mathbf{dim}_{k}[M],\mathbf{dim}_{k}[N]).$

Notice that a category generated by a tree is in fact a poset. Also, the lemma
shows that symmetry almost always fails, even under rather favourable
circumstances, because the second term in the definition of the Euler form is
not symmetric.

Looking back into graph theory, one considers the vertices $V$ and the edges
$E$ as discrete categories. The next example shows that in that case Hom
pairing and the Euler pairing coincide.

######  Example 3.6.

In the setup of Example 2.9 the sets $V$ and $E$ are considered as discrete
categories. Hence $kV$ and $kE$ are simply the $k$-vector spaces generated by
$V$ and $E$ respectively, with trivial products, and modules over them are
again just finite dimensional vector spaces associated to each vertex or edge.
Thus, all modules are projective, and so the respective $\mathrm{Ext}^{n}$
groups vanish for positive $n$. Hence, the corresponding pairings now take the
form

$\displaystyle\chi_{V}([f],[g])=\dim_{k}(\mathrm{Hom}_{kV}(f,g))=$
$\displaystyle\sum_{v\in
V}\dim_{k}(f(v))\dim_{k}(g(v))=\langle[f],[g]\rangle_{V},\quad\text{and}\quad$
$\displaystyle\chi_{E}([h],[s])=\dim_{k}(\mathrm{Hom}_{kE}(h,s))=$
$\displaystyle\sum_{e\in
E}\dim_{k}(h(e))\dim_{k}(s(e))=\langle[h],[s]\rangle_{E}$

Notice that the pairings in this case are commutative. Also, by Example 2.9
the divergence operators $\nabla^{*}$ and $\nabla_{*}$ coincide in this case.
In particular, the adjointness relations (3) trivially hold.

The following two lemmas are standard homological algebra, which we record
here for reference.

######  Lemma 3.7.

Let $\mathcal{A},\mathcal{B}$ be small abelian categories and assume that
$\mathcal{A}$ has enough projectives. Let $F\colon\mathcal{A}\to\mathcal{B}$
be a functor with a right adjoint $G\colon\mathcal{B}\to\mathcal{A}$. Then the
following statements are equivalent.

  1. (1)

$F$ sends projective objects in $\mathcal{A}$ to projective objects in
$\mathcal{B}$.

  2. (2)

$G$ is an exact functor.

Dually, if $\mathcal{B}$ has enough injectives, then the following statements
are equivalent.

  1. (3)

$G$ sends injective objects in $\mathcal{B}$ to injective objects in
$\mathcal{A}$.

  2. (4)

$F$ is an exact functor.

###### Proof.

We prove the equivalence of (1) and (2). The equivalence of (3) and (4)
follows by analogy. Let $P\in\mathcal{A}$ be a projective object. Then
$F(P)\in\mathcal{B}$ is projective if and only if the functor
$\mathrm{Hom}_{\mathcal{B}}(F(P),-)\cong\mathrm{Hom}_{\mathcal{A}}(P,G(-))$ is
exact. Thus, if $G$ is exact then $F(P)$ is projective. This shows that (2)
implies (1).

Conversely, since $\mathcal{A}$ is assumed to have enough projectives, every
object $X\in\mathcal{A}$ admits an epimorphism $P\to X$, where $P$ is
projective in $\mathcal{A}$. Let

$0\to B\xrightarrow{\alpha}C\xrightarrow{\beta}D\to 0$

be an exact sequence in $\mathcal{B}$. Since $G$ is a right adjoint, it is
left exact, so it suffices to show that

$G(C)\xrightarrow{G(\beta)}G(D)\to 0$

is exact. Let $\varphi\colon P\to G(D)$ be an epimorphism in $\mathcal{A}$,
where $P$ is projective. By (1), $F$ sends projective objects to projective
objects. Thus

$\mathrm{Hom}_{\mathcal{B}}(F(P),C)\xrightarrow{\beta_{*}}\mathrm{Hom}_{\mathcal{B}}(F(P),D)\to
0$

is exact, and by adjointness

$\mathrm{Hom}_{\mathcal{A}}(P,G(C))\xrightarrow{G(\beta)_{*}}\mathrm{Hom}_{\mathcal{A}}(P,G(D))\to
0$

is exact. Hence there exists $\psi\in\mathrm{Hom}_{\mathcal{A}}(P,G(C))$ such
that $G(\beta)_{*}(\psi)=G(\beta)\circ\psi=\varphi$. Since $\varphi$ is an
epimorphism, so is $G(\beta)$, as claimed. ∎

######  Lemma 3.8.

Let $\gamma\colon\mathcal{C}\to\mathcal{D}$ be a functor between small
categories. Let $\mathcal{A}$ be a bicomplete abelian category, and let
$\mathcal{A}^{\mathcal{C}}$ and $\mathcal{A}^{\mathcal{D}}$ denote the
respective functor categories. Let
$\gamma^{*}\colon\mathcal{A}^{\mathcal{D}}\to\mathcal{A}^{\mathcal{C}}$ denote
the restriction, and let $L_{\gamma}$ and $R_{\gamma}$ denote its left and
right Kan extensions. Then for any $M\in\mathcal{A}^{\mathcal{C}}$ and
$N\in\mathcal{A}^{\mathcal{D}}$,

$\mathrm{Ext}^{*}_{\mathcal{A}^{\mathcal{D}}}(L_{\gamma}(M),N)\cong\mathrm{Ext}^{*}_{\mathcal{A}^{\mathcal{C}}}(M,\gamma^{*}N)$

if $\mathcal{A}$ has enough projectives and either of the following two
equivalent conditions holds.

  1. (1)

$\gamma^{*}$ sends injective modules to injective modules.

  2. (2)

$L_{\gamma}$ is an exact functor.

Similarly, if $\mathcal{A}$ has enough injectives then

$\mathrm{Ext}^{*}_{\mathcal{A}^{\mathcal{D}}}(N,R_{\gamma}(M))\cong\mathrm{Ext}^{*}_{\mathcal{A}^{\mathcal{C}}}(\gamma^{*}N,M)$

if either of the following equivalent conditions holds.

  1. (3)

$\gamma^{*}$ sends projective modules to projective modules.

  2. (4)

$R_{\gamma}$ is an exact functor.

###### Proof.

We prove the first statement. The second follows by analogy. Let $N\to I^{*}$
be an injective resolution of $N$ in $\mathcal{A}^{\mathcal{D}}$. Since
$\gamma^{*}$ is exact and sends injective modules to injective modules,
$\gamma^{*}N\to\gamma^{*}(I^{*})$ is an injective resolution of $\gamma^{*}N$
in $\mathcal{A}^{\mathcal{C}}$. Then

$\mathrm{Ext}^{*}_{\mathcal{A}^{\mathcal{C}}}(M,\gamma^{*}N)=H^{*}(\mathrm{Hom}_{\mathcal{A}^{\mathcal{C}}}(M,\gamma^{*}(I^{*}))\cong
H^{*}(\mathrm{Hom}_{\mathcal{A}^{\mathcal{D}}}(L_{\gamma}(M),I^{*}))=\mathrm{Ext}_{\mathcal{A}^{\mathcal{D}}}^{*}(L_{\gamma}(M),N),$

as claimed. The equivalence of Conditions (1) and (2) follows from Lemma 3.7.
∎

######  Lemma 3.9.

Let $\mathcal{P}$ be a finite poset and assume that
$\mathcal{H}_{\mathcal{P}}$ is a tree. Let $\widehat{\mathcal{P}}$ be the
associated line poset (i.e., the line category associated to $\mathcal{P}$),
and let $\phi,\beta\colon\widehat{\mathcal{P}}\to\mathcal{P}$ be the front and
back functors. Then $\phi^{*}\colon k\mathcal{P}\mbox{-}\EuR{mod}\to
k\widehat{\mathcal{P}}\mbox{-}\EuR{mod}$ sends injective modules to injective
modules and $\beta^{*}\colon k\mathcal{P}\mbox{-}\EuR{mod}\to
k\widehat{\mathcal{P}}\mbox{-}\EuR{mod}$ sends projective modules to
projective modules.

###### Proof.

We prove that $\phi^{*}$ sends injective modules to injective modules. The
corresponding statement for $\beta^{*}$ is similar. By [13, Lemma 2.2.3] every
indecomposable injective $k\mathcal{P}$-module has the form $G_{v}$, for some
$v\in\mathcal{P}$, where for any $u\in\mathcal{P}$,

$G_{v}(u)\overset{\text{{def}}}{=}k\operatorname{Mor}_{\mathcal{P}}(u,v)=\begin{cases}k&u\leq
v\\\ 0&\text{otherwise}\end{cases}$

with $G_{v}(x)\to G_{v}(y)$ the identity morphism, for each relation $x\leq y$
in $\mathcal{P}_{\leq v}$. Thus to prove the lemma, we must show that for any
module of the form $G_{v}\in k\mathcal{P}$, the corresponding module
$\phi^{*}G_{v}$ is a finite sum of indecomposable injective modules in
$k\widehat{\mathcal{P}}$.

Let $(u,w)\in\widehat{\mathcal{P}}$ be any object. Then

$\phi^{*}G_{v}(u,w)=G_{v}(w)=\begin{cases}k&w\leq v\\\
0&\text{otherwise}\end{cases}=\begin{cases}k&(u,w)\leq(a,v)\\\
0&\text{otherwise}\end{cases}$

where $a<v$ is any indecomposable relation in $\mathcal{P}$. Since
$\mathcal{P}$ is generated by a tree, for each such relation the sub-posets
$\widehat{\mathcal{P}}_{\leq(a,v)}$ are pairwise disjoint. Thus

$\phi^{*}G_{v}\cong\bigoplus_{(a,v)\in\widehat{\mathcal{P}}}G_{(a,v)}.$

This completes the proof. ∎

######  Example 3.10.

Let $\mathcal{P}$ be a finite poset and assume that
$\mathcal{H}_{\mathcal{P}}$ is a tree. By Lemmas 3.8 and 3.9 one has for $N\in
k\mathcal{P}\mbox{-}\EuR{mod}$ and $M\in
k\widehat{\mathcal{P}}\mbox{-}\EuR{mod}$

$\mathrm{Ext}^{*}_{k\mathcal{P}}(L_{\phi}(M),N)\cong\mathrm{Ext}^{*}_{k\widehat{\mathcal{P}}}(M,\phi^{*}N),\quad\text{and}\quad\mathrm{Ext}^{*}_{k\mathcal{C}}(N,R_{\beta}(M))\cong\mathrm{Ext}^{*}_{k\widehat{\mathcal{C}}}(\beta^{*}N,M).$

In general $\phi^{*}$ does not send projective $k\mathcal{P}$-modules to
projective modules, and $\beta^{*}$ does not send injective
$k\mathcal{P}$-modules to injective modules. See section 5 for further
discussion.

##  4\. The gradient of modules over posets

In this section we specialise the idea of a gradient, as defined in Section 2,
to persistence modules. Let $\mathcal{P}$ be a finite poset and let
$\mathcal{H}_{\mathcal{P}}$ denote its Hasse diagram. Thus
$\mathcal{H}_{\mathcal{P}}$ is a generating digraph for $\mathcal{P}$ in the
sense of Definition 2.1. Let $\widehat{\mathcal{H}}_{P}$ be the associated
line digraph, which by Example 2.5 is the Hasse diagram of a poset
$\widehat{\mathcal{P}}$. Let $k$ be a fixed field of coefficients. Throughout
we consider $k\mathcal{P}$\- and $k\widehat{\mathcal{P}}$-modules alternately
as modules over the category algebras, or as functors from the respective
categories to $k$-vector spaces.

Let $\phi,\beta\colon\widehat{\mathcal{P}}\to\mathcal{P}$ be the front and
back maps, and let

$\phi^{*},\beta^{*}\colon\EuR{Gr}(k\mathcal{P})\to\EuR{Gr}(k\widehat{\mathcal{P}})$

be the corresponding ring homomorphisms. The following is a particular case of
Definition 2.6.

######  Definition 4.1.

For any finite poset $\mathcal{P}$, define the gradient

$\nabla\colon\EuR{Gr}(k\mathcal{P})\to\EuR{Gr}(k\widehat{\mathcal{P}})$

by $\nabla\overset{\text{{def}}}{=}\phi^{*}-\beta^{*}$.

Figure 3 illustrates some small posets and the front and back modules in
Definition 4.1.

|  |  |
---|---|---|---
|  |  |
$\textstyle{{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x<y}$$\textstyle{{y}}$$\textstyle{{\phi^{*}M(x,y)\atop=M(y)}}$$\textstyle{{\beta^{*}M(x,y)\atop=M(x)}}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x<y}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y<z}$$\textstyle{z}$$\textstyle{{\phi^{*}M(x,y)\atop=M(y)}\quad\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(y<z)}$$\textstyle{\phi^{*}M(y,z)\atop=M(z)}$$\textstyle{{\beta^{*}M(x,y)\atop=M(x)}\quad\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(x<y)}$$\textstyle{\beta^{*}M(y,z)\atop=M(y)}$$\textstyle{z}$$\textstyle{{\phi^{*}M(y,z)\atop=M(z)}}$$\textstyle{{\beta^{*}M(y,z)\atop=M(y)}}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x<y}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y<z}$$\scriptstyle{y<w}$$\textstyle{{\phi^{*}M(x,y)\atop=M(y)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(y<z)}$$\scriptstyle{M(y<w)}$$\textstyle{{\beta^{*}M(x,y)\atop=M(x)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(x<y)}$$\scriptstyle{M(x<y)}$$\textstyle{w}$$\textstyle{{\phi^{*}M(y,w)\atop=M(w)}}$$\textstyle{\beta^{*}M(y,w)\atop=M(y)}$$\textstyle{y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\phi^{*}M(x,y)\atop=M(y)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(y<w)}$$\textstyle{\phi^{*}M(y,w)\atop=M(w)}$$\textstyle{{\beta^{*}M(x,y)\atop=M(x)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(x<y)}$$\textstyle{\beta^{*}M(y,w)\atop=M(y)}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{w}$$\textstyle{z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\phi^{*}M(x,z)\atop=M(z)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(z<w)}$$\textstyle{\phi^{*}M(z,w)\atop=M(w)}$$\textstyle{{\beta^{*}M(x,z)\atop=M(x)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(x<z)}$$\textstyle{{\beta^{*}M(z,w)\atop=M(z)}}$
Figure 3. Left: Four sample posets given by their respective Hasse diagrams.
Centre and Right: For an arbitrary persistence module $M$ on $\mathcal{P}$,
the corresponding modules $\phi^{*}M$ and $\beta^{*}M$ on
$\widehat{\mathcal{P}}$. The gradient is given by formal difference
$[\phi^{*}M]-[\beta^{*}M]$ in $\EuR{Gr}(k\widehat{\mathcal{P}})$

We next study some basic properties of the gradient.

######  Theorem 4.2.

Let $\mathcal{P}$ be a finite poset. Then the gradient operator
$\nabla=\nabla_{\mathcal{P}}\colon\EuR{Gr}(k\mathcal{P})\to\EuR{Gr}(k\widehat{\mathcal{P}})$
satisfies the following properties:

  1. a)

$\nabla$ is a well defined group homomorphism.

  2. b)

If $[X]\in\EuR{Gr}(k\mathcal{P})$ is locally constant then $\nabla[X]=0$.

  3. c)

$\nabla$ satisfies a Leibniz type rule, i.e. for all
$[X],[Y]\in\EuR{Gr}(k\mathcal{P})$, the identity

$\nabla([X]\cdot[Y])=\nabla[X]\cdot\phi^{*}[Y]+\beta^{*}[X]\cdot\nabla[Y]$

holds in $\EuR{Gr}(k\widehat{\mathcal{P}})$.

Furthermore, $\nabla$ is natural with respect to restrictions to sub-posets,
namely, if $\iota\colon\mathcal{Q}\xrightarrow{\subseteq}\mathcal{P}$ is a
sub-poset, then
$\iota^{*}\circ\nabla_{\mathcal{P}}=\nabla_{\mathcal{Q}}\circ\iota^{*}$.

###### Proof.

Since
$\phi^{*},\beta^{*}\colon\EuR{Gr}(k\mathcal{P})\to\EuR{Gr}(k\widehat{\mathcal{P}})$
are ring homomorphisms (see Remark 1.14), they are in particular homomorphisms
of abelian groups, and hence so is their difference. This proves Part a.

Let $M\in k\mathcal{P}\mbox{-}\EuR{mod}$ be locally constant. Define a natural
isomorphism $\mu\colon\beta^{*}M\to\phi^{*}M$, by sending an object $(x,y)$ in
$\widehat{\mathcal{P}}$ to the morphism

$\mu_{(x,y)}\colon\beta^{*}M(x,y)=M(x)\xrightarrow{M(x,y)}M(y)=\phi^{*}M(x,y).$

Naturality is clear. It follows that $\nabla[M]=0$. By definition, an element
of $[X]\in\EuR{Gr}(k\mathcal{P})$ is locally constant if it is represented as
a difference of locally constant modules. By additivity $\nabla$ vanishes on
any locally constant virtual module in $\EuR{Gr}(k\mathcal{P})$. This proves
Part b.

Finally, since $\phi^{*}$ and $\beta^{*}$ are ring homomorphisms, one has for
$M,N\in k\mathcal{P}\mbox{-}\EuR{mod}$,

$\displaystyle\nabla[M]\cdot\phi^{*}[N]+\beta^{*}[M]\cdot\nabla[N]$
$\displaystyle=(\phi^{*}[M]-\beta^{*}[M])\cdot\phi^{*}[N]+\beta^{*}[M]\cdot(\phi^{*}[N]-\beta^{*}[N])$
$\displaystyle=\phi^{*}[M]\cdot\phi^{*}[N]-\beta^{*}[M]\cdot\beta^{*}[N]$
$\displaystyle=\nabla([M]\cdot[N])$

as claimed in Part c. For virtual modules $[X]=[M]-[N]$, extend the statement
by linearity of $\phi^{*}$, $\beta^{*}$ and $\nabla$.

The last statement follows from commutativity of the square

$\textstyle{k\mathcal{P}\mbox{-}\EuR{mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}}$$\scriptstyle{\alpha^{*}}$$\textstyle{k\mathcal{Q}\mbox{-}\EuR{mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{*}}$$\textstyle{k\widehat{\mathcal{P}}\mbox{-}\EuR{mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}}$$\textstyle{k\widehat{\mathcal{Q}}\mbox{-}\EuR{mod}}$

where $\alpha$ is either $\phi$ or $\beta$. ∎

An obvious question is what can be said about a module with a vanishing
gradient. To answer this question we need a preparatory lemma.

######  Lemma 4.3.

Let $\mathcal{P}$ be a line connected finite poset. Then its Hasse diagram
$\mathcal{H}_{\mathcal{P}}$ has a line connected maximal tree.

###### Proof.

The Hasse diagram $\mathcal{H}_{\mathcal{P}}$ is a connected digraph (a line
connected digraph is in particular connected), and by definition it is acyclic
and transitively reduced. Let $\widehat{\mathcal{H}}_{\mathcal{P}}$ denote its
line digraph, as usual. If $\mathcal{H}_{\mathcal{P}}$ is not a tree, then
there are two objects $x<y\in\mathcal{P}$ and at least two distinct paths in
$\mathcal{H}_{\mathcal{P}}$ from $x$ to $y$. We proceed by showing that one
can disconnect one of the paths from $x$ to $y$ in $\mathcal{H}_{\mathcal{P}}$
with the resulting digraph remaining line connected. The claim then follows by
induction on the number of multiple paths between pairs of points in
$\mathcal{H}_{\mathcal{P}}$.

Let $x<y\in\mathcal{P}$ be two distinct vertices, and assume that there are
more than one directed path from $x$ to $y$. Denote the collection of all
directed paths from $x$ to $y$ by $l_{1},\ldots,l_{n}$, for $n>1$. We may
assume that in all $l_{j}$, except possibly one of them (since double edges
are not allowed in a poset), there is a vertex $x<a_{i}<y$, such that either
$x<a_{i}$ or $a_{i}<y$ is indecomposable in $\mathcal{P}$. We may assume
without loss of generality that the $l_{j}$ have no common vertices except $x$
and $y$.

We consider four possibilities.

(1) $x$ is not minimal and $y$ is maximal: Then there exists some relation
$x_{0}<x$ in $\mathcal{P}$, and removing a single edge $x<a_{j}$ from all
$l_{j}$ except one of them, disconnects all the $l_{j}$ except the one that
was not modified. Since this leaves exactly one path from $x$ to $y$, the
resulting digraph is still line connected.

(2) $y$ is not maximal and $x$ is minimal: Then there exists a relation
$y<y_{0}$ in $\mathcal{P}$, and removing a single edge $a_{j}<y$ from each
$l_{j}$ except one of them, disconnects all the $l_{j}$ except the one that
was not modified. Again the resulting digraph is line connected.

(3) $x$ is not minimal and $y$ is not maximal: Then remove a single
indecomposable edge from each $l_{j}$ except one of them, as in (1) and (2).
In that case as well the resulting digraph is line connected.

(4) $x$ is minimal and $y$ is maximal: In that case the associated line
digraph splits into $n$ connected components, each containing the line digraph
of $l_{j}$ for a unique $1\leq j\leq n$. This contradicts the assumption that
$\mathcal{P}$ is line connected.

The proof is now complete by inductively performing this procedure for any set
of multiple paths between pairs of vertices in $\mathcal{H}_{\mathcal{P}}$. ∎

Let $\mathcal{P}$ be a finite line connected poset, and let $\mathcal{T}$ be a
line connected sub-tree of $\mathcal{H}_{\mathcal{P}}$. Let
$\mathcal{P}_{\mathcal{T}}\subseteq\mathcal{P}$ be the sub-poset generated by
$\mathcal{T}$, and let $\iota_{\mathcal{T}}$ denote the inclusion functor.
Functoriality of the line digraph construction gives an inclusion
$\widehat{\iota}_{\mathcal{T}}\colon\widehat{\mathcal{P}_{\mathcal{T}}}\to\widehat{\mathcal{P}}$.
Furthermore, one easily verifies commutativity of the square

$\textstyle{\widehat{\mathcal{P}_{\mathcal{T}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\iota}_{\mathcal{T}}}$$\scriptstyle{\alpha}$$\textstyle{\widehat{\mathcal{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathcal{P}_{\mathcal{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\mathcal{T}}}$$\textstyle{\mathcal{P}}$

where $\alpha$ is either the front functor $\phi$ or the back functor $\beta$.
This square induces homomorphisms on the respective Grothendieck rings, and
hence commutative square of group homomorphisms

$\textstyle{\EuR{Gr}(k\mathcal{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\mathcal{T}}^{*}}$$\scriptstyle{\nabla}$$\textstyle{\EuR{Gr}(k\mathcal{P}_{\mathcal{T}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla}$$\textstyle{\EuR{Gr}(k\widehat{\mathcal{P}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\iota}_{\mathcal{T}}^{*}}$$\textstyle{\EuR{Gr}(k\widehat{\mathcal{P}_{\mathcal{T}}})}$

######  Definition 4.4.

Let $\mathcal{P}$ be a finite line connected poset, and let $\mathcal{T}$ be a
line connected sub-tree of $\mathcal{H}_{\mathcal{P}}$. We say that an element
$[X]\in\EuR{Gr}(k\mathcal{P})$ has a _vanishing gradient on $\mathcal{T}$_ if

$\nabla(\iota_{\mathcal{T}}^{*}[X])=\widehat{\iota}_{\mathcal{T}}^{*}(\nabla[X])=0.$

Notice that for a module $M\in k\mathcal{P}\mbox{-}\EuR{mod}$,
$\iota_{\mathcal{T}}^{*}[M]$ is just the restriction of $M$ to the sub-poset
$\mathcal{P}_{\mathcal{T}}$, which justifies Definition 4.4. Notice also that
if $\nabla[X]=0$ in $\EuR{Gr}(k\widehat{\mathcal{P}})$, then
$\nabla(\iota^{*}_{\mathcal{T}}[X])=0$ as well, namely the restriction of $X$
to $\mathcal{P}_{\mathcal{T}}$ has a vanishing gradient on $\mathcal{T}$.

The following theorem is stated with respect to a maximal tree in
$\mathcal{H}_{\mathcal{P}}$, but one may easily obtain a restricted analog for
any sub-tree of the Hasse diagram.

######  Theorem 4.5.

Let $\mathcal{P}$ be a finite line connected poset, and let $\mathcal{T}$ be a
line connected maximal tree for its Hasse diagram $\mathcal{H}_{\mathcal{P}}$.
Let $\mathcal{P}_{\mathcal{T}}\subseteq\mathcal{P}$ denote the sub-poset
generated by $\mathcal{T}$. Let $M\in k\mathcal{P}\mbox{-}\EuR{mod}$ be a
module with a vanishing gradient on $\mathcal{T}$. Let $M_{\mathcal{T}}$
denote the restriction of $M$ to $\mathcal{P}_{\mathcal{T}}$, so
$\iota_{\mathcal{T}}^{*}[M]=[M_{\mathcal{T}}]\in\EuR{Gr}(k\mathcal{P}_{\mathcal{T}})$.
Then the following statements hold.

  1. (1)

For any pair of objects $u,v\in\mathcal{P}$, there is an isomorphism
$\alpha_{u,v}\colon M(u)\to M(v)$, such that $\alpha_{u,u}=1_{M(u)}$ and
$\alpha_{v,w}\circ\alpha_{u,v}=\alpha_{u,w}$.

  2. (2)

For every pair of indecomposable morphisms $u\leq w$ and $s\leq t$ in
$\mathcal{P}_{\mathcal{T}}$,

$\alpha_{w,t}\circ M(u\leq w)=M(s\leq t)\circ\alpha_{u,s}.$

  3. (3)

$M_{\mathcal{T}}$ is locally constant if and only if $M(u\leq v)$ is an
isomorphism for some indecomposable relation $u\leq v$ in
$\mathcal{P}_{\mathcal{T}}$.

Furthermore, if $M\in k\mathcal{P}\mbox{-}\EuR{mod}$ is a module such that for
any indecomposable relation $u\leq v$ in $\mathcal{P}_{\mathcal{T}}$ there
exists an isomorphism $\alpha_{u,v}\colon M(u)\to M(v)$ that satisfy
Statements (1) and (2), then $M$ has a vanishing gradient on $\mathcal{T}$.

###### Proof.

By hypothesis $\nabla[M_{\mathcal{T}}]=0$. Hence there is a natural
isomorphism
$\alpha\colon\beta^{*}M_{\mathcal{T}}\xrightarrow{}\phi^{*}M_{\mathcal{T}}$.
If $(u,v)$ is an edge in $\mathcal{T}$, and hence a vertex in
$\widehat{\mathcal{P}}_{\mathcal{T}}$, then we have

$\alpha_{u,v}\colon\beta^{*}M_{\mathcal{T}}(u,v)=M(u)\to
M(v)=\phi^{*}M_{\mathcal{T}}(u,v),$

that is, $\alpha_{u,v}$ is the evaluation of the natural isomorphism $\alpha$
on the vertex $(u,v)$. Define

$\alpha_{v,u}\overset{\text{{def}}}{=}\alpha_{u,v}^{-1}.$

Since $\mathcal{P}_{\mathcal{T}}$ is a poset and $u\lneq v$, the opposite
inequality does not hold, so $(v,u)$ is not a vertex in
$\widehat{\mathcal{P}_{\mathcal{T}}}$. Thus, $\alpha_{v,u}$ is well defined.
Notice that the vertices of $\mathcal{P}_{\mathcal{T}}$ coincide with those of
$\mathcal{P}$.

Let $v_{0}$ be any minimal object in $\mathcal{P}_{\mathcal{T}}$. Then
$\alpha_{v_{0},u}$ is defined and is an isomorphism for each indecomposable
relation $v_{0}<u$ and, by induction on the length of a chain starting at
$v_{0}$, $\alpha_{u,v}$ is defined and is an isomorphism for any
$u,v\in\mathcal{P}_{\mathcal{T}_{\geq v_{0}}}$. Since
$\mathcal{P}_{\mathcal{T}}$ has only finitely many minimal objects, it follows
that $\alpha_{u,v}$ is an isomorphism on any sub-poset of the form
$\mathcal{P}_{\mathcal{T}_{\geq v_{0}}}$ where $u_{0}$ is minimal.

Let $u_{0},u_{0}^{\prime}$ be two distinct minimal objects in
$\mathcal{P}_{\mathcal{T}}$. Since $\mathcal{P}_{\mathcal{T}}$ is line
connected by construction, it is in particular connected. Hence there are
minimal objects $u_{0}=v_{1},v_{2},\ldots,v_{k}=u_{0}^{\prime}$, such that for
each $1\leq i\leq k-1$, the intersection of sub-posets
$\mathcal{P}_{\mathcal{T}_{\geq v_{i}}}\cap\mathcal{P}_{\mathcal{T}_{\geq
v_{i+1}}}$ is nonempty. Let $a_{i}\in\mathcal{P}_{\mathcal{T}_{\geq
v_{i}}}\cap\mathcal{P}_{\mathcal{T}_{\geq v_{i+1}}}$ be any object. Thus one
has an isomorphism

$\alpha_{a_{i},v_{i+1}}\circ\alpha_{v_{i},a_{i}}=\alpha_{v_{i+1},a_{i}}^{-1}\circ\alpha_{v_{i},a_{i}}\colon
M(v_{i})\to M(v_{i+1}).$

Since between any two objects in $\mathcal{P}_{\mathcal{T}}$ there is by
assumption at most one path in $\mathcal{T}$, compositions of isomorphisms of
the form $\alpha_{x,y}$, where $(x,y)$ is an object in
$\widehat{\mathcal{P}}_{\mathcal{T}}$, and the inverses of such isomorphisms
define $\alpha_{u,v}$ for any pair of objects
$u,v\in\mathcal{P}_{\mathcal{T}}$. Since $\mathcal{P}$ and
$\mathcal{P}_{\mathcal{T}}$ coincide on objects, this proves Part (1).

Let $(u_{1},u_{2},u_{3})$ be an edge in $\widehat{\mathcal{P}}_{\mathcal{T}}$.
Then we have a commutative square

$\textstyle{M(u_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\beta^{*}M_{\mathcal{T}}(u_{1},u_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\alpha_{u_{1},u_{2}}}$$\scriptstyle{M_{\mathcal{T}}(u_{1}\leq
u_{2})}$$\textstyle{\beta^{*}M_{\mathcal{T}}(u_{2},u_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{u_{2},u_{3}}}$$\scriptstyle{\cong}$$\textstyle{M(u_{2})}$$\textstyle{M(u_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi^{*}M_{\mathcal{T}}(u_{1},u_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M_{\mathcal{T}}(u_{2}\leq
u_{3})}$$\textstyle{\phi^{*}M_{\mathcal{T}}(u_{2},u_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(u_{3})}$

Thus $M(u_{2}\leq u_{3})=\alpha_{u_{2},u_{3}}\circ M(u_{1}\leq
u_{2})\circ\alpha_{u_{2},u_{1}}$. By induction, if $u_{1}\leq
u_{2}\leq\cdots\leq u_{n}$ is any chain of indecomposable relations in
$\mathcal{P}_{\mathcal{T}}$, then

$M(u_{n-1}\leq u_{n})=\alpha_{u_{2},u_{n}}\circ M(u_{1}\leq
u_{2})\circ\alpha_{u_{n-1},u_{1}}.$

Let $u\leq v$ and $a\leq b$ be two indecomposable relations in
$\mathcal{P}_{\mathcal{T}}$. Let $u_{0},a_{0}$ be two minimal objects in
$\mathcal{P}_{\mathcal{T}}$, such that $u_{0}\leq u$ and $a_{0}\leq a$. Let

$u_{0}\leq u_{1}\leq\cdots\leq u_{n}\leq u\leq v,\quad\text{and}\quad
a_{0}\leq a_{1}\leq\cdots\leq a_{k}\leq a\leq b$

be chains of indecomposable relations in $\mathcal{P}_{\mathcal{T}}$. Then

$M(u\leq v)=\alpha_{u_{1},v}\circ M(u_{0}\leq
u_{1})\circ\alpha_{u,u_{0}},\quad\text{and}\quad M(a\leq
b)=\alpha_{a_{1},b}\circ M(a_{0}\leq a_{1})\circ\alpha_{a,a_{0}}.$

Thus it suffices to prove that

$M(u_{0}\leq u_{1})=\alpha_{a_{1},u_{1}}\circ M(a_{0}\leq
a_{1})\circ\alpha_{u_{0},a_{0}}.$

Notice that $(u_{0},u_{1})$ and $(a_{0},a_{1})$ are minimal objects in
$\widehat{\mathcal{P}}_{\mathcal{T}}$. Since
$\widehat{\mathcal{P}}_{\mathcal{T}}$ is connected, there is a sequence of
minimal objects in $\widehat{\mathcal{P}}_{\mathcal{T}}$

$(u_{0},u_{1})=(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{r},y_{r})=(a_{0},a_{1})$

such that
$\widehat{\mathcal{P}}_{\mathcal{T}_{\geq(x_{i},y_{i})}}\cap\widehat{\mathcal{P}}_{\mathcal{T}_{\geq(x_{i+1},y_{i+1})}}$
is nonempty for each $1\leq i\leq r-1$. Let $(s_{i},t_{i})$ be an object in
the intersection, so that $(s_{i},t_{i})\geq(x_{i},y_{i}),(x_{i+1},y_{i+1})$.
It follows that

$\alpha_{y_{i},t_{i}}\circ M(x_{i}\leq
y_{i})\circ\alpha_{s_{i},x_{i}}=M(s_{i}\leq t_{i})=\alpha_{y_{i+1},t_{i}}\circ
M(x_{i+1}\leq y_{i+1})\circ\alpha_{s_{i},x_{i+1}}.$

Thus

$M(x_{i+1}\leq y_{i+1})=\alpha_{t_{i},y_{i+1}}\alpha_{y_{i},t_{i}}\circ
M(x_{i}\leq
y_{i})\circ\alpha_{s_{i},x_{i}}\alpha_{x_{i+1},s_{i}}=\alpha_{y_{i},y_{i+1}}\circ
M(x_{i}\leq y_{i})\circ\alpha_{x_{i+1},x_{i}}$

for each $1\leq i\leq r-1$. Part (2) follows by induction on $r$.

Part (3) follows at once from (2).

Finally, assume that $M\in k\mathcal{P}\mbox{-}\EuR{mod}$ satisfies Statements
(1) and (2). Let $\alpha$ be a family of morphisms for which these statements
hold. By (1), for each object $(u,v)\in\widehat{\mathcal{P}_{\mathcal{T}}}$ we
have an isomorphism

$\alpha_{u,v}\colon M(u)=\beta^{*}M(u,v)\to\phi^{*}M(u,v)=M(v),$

If $(u,v)\leq(x,y)$ in $\widehat{\mathcal{P}_{\mathcal{T}}}$, then by
Condition (2)

$\alpha_{v,y}\circ M(u<v)=M(x<y)\circ\alpha_{u,x}.$

This shows that $\alpha$ restricts to a natural isomorphism
$\beta^{*}M\to\phi^{*}M$, and hence $[M]$ has a vanishing gradient on
$\mathcal{T}$. ∎

######  Remark 4.6.

Theorem 4.5 shows that the vanishing of the gradient on a sub-poset generated
by a tree gives a lot of information about the module in question. It is
instructive however to note that if the poset in question consists of a single
pair of comparable objects, then any module on such poset, where the point
modules are isomorphic, will have a vanishing gradient. This of course does
not contradict the conclusion of the theorem. We leave the details to the
reader.

######  Definition 4.7.

With the notation of Theorem 4.5, define $\operatorname{Ker}\nolimits
M,\operatorname{Im}M\in k\mathcal{P}\mbox{-}\EuR{mod}$ as follows.

  1. (1)

$\operatorname{Ker}\nolimits
M(u)\overset{\text{{def}}}{=}\begin{cases}\operatorname{Ker}\nolimits(M(u<v))&u<v\;\text{indecomposable},\;u\;\text{not
maximal}\\\ 0&u\quad\text{maximal}\end{cases}.$

Define $\operatorname{Ker}\nolimits M(x<y)\overset{\text{{def}}}{=}0$ for any
non-identity relation in $\mathcal{P}$.

  2. (2)

$\operatorname{Im}M(u)\overset{\text{{def}}}{=}\begin{cases}\operatorname{Im}(M(w<u))&\textup{for
any indecomposable}\;w<u,\;\;u\;\textup{not minimal}\\\
M(u)/\operatorname{Ker}\nolimits(M(u<v))&\textup{for any
indecomposable}\;u<v,\;u\;\textup{minimal}\end{cases}$

Define $\operatorname{Im}M(u<v)$ to be the restriction of $M(u<v)$ to
$\operatorname{Im}M(u)$.

######  Corollary 4.8.

With the notation and hypotheses of Theorem 4.5, the modules
$\operatorname{Ker}\nolimits M$ and $\operatorname{Im}M$ are well defined
$k\mathcal{P}_{\mathcal{T}}$-modules. Furthermore,
$\operatorname{Ker}\nolimits M(u)\cong\operatorname{Ker}\nolimits M(v)$ for
all non-maximal $u,v\in\mathcal{P}$, and
$\operatorname{Im}M(u)\cong\operatorname{Im}M(v)$ for all $u,v\in\mathcal{P}$.

###### Proof.

Let $u<v$ and $u<v^{\prime}$ be indecomposable morphisms in $\mathcal{P}$.
Then by Theorem 4.5(2),

$\alpha_{v,v^{\prime}}\circ M(u<v)=M(u<v^{\prime}).$

Since $\alpha_{v,v^{\prime}}$ is an isomorphism,
$\operatorname{Ker}\nolimits(M(u<v))=\operatorname{Ker}\nolimits(M(u<v^{\prime}))$.
This shows that $\operatorname{Ker}\nolimits M$ is well defined.

Let $w<u$ and $w^{\prime}<u$ be two indecomposable morphisms in $\mathcal{P}$.
Then, again by 4.5(2),

$M(w<u)=M(w^{\prime}<u)\circ\alpha_{w,w^{\prime}}.$

Thus $\operatorname{Im}M$ is well defined on non-minimal objects. If $u$ is
minimal, then $\operatorname{Im}M(u)$ is also well defined since
$\operatorname{Ker}\nolimits(M(u<v))$ does not depend on the choice of $v$.
The definition on morphisms is clear.

Let $u,s\in\mathcal{P}$ be non-maximal, and let $u<w$ and $s<t$ be
indecomposable morphisms in $\mathcal{P}$. Let
$x\in\operatorname{Ker}\nolimits(M(u<w))$, and let $y=\alpha_{u,s}(x)$. Then
by Theorem 4.5(2),

(5) $0=M(u<w)(x)=\alpha_{t,w}\circ
M(s<t)\circ\alpha_{u,s}(x)=\alpha_{t,w}\circ M(s<t)(y).$

Consider the following diagram with exact rows. Equation 5 implies that the
left square commutes, and hence that so does the right square.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}\nolimits
M(u)=\operatorname{Ker}\nolimits(M(u<w))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(u)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{u,s}}$$\scriptstyle{\cong}$$\textstyle{M(u)/\operatorname{Ker}\nolimits(M(u<w))\simeq\mathrm{Im}M(w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}\nolimits
M(s)=\operatorname{Ker}\nolimits(M(s<t))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(s)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(s)/\operatorname{Ker}\nolimits(M(s<t))\simeq\mathrm{Im}M(t)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

for all $u,s\in\mathcal{P}$. Hence $\alpha_{u,s}$ restricted to
$\operatorname{Ker}\nolimits M(u)$ is a monomorphism and the induced map on
$\operatorname{Im}M(w)$ is an epimorphism. The same argument using
$\alpha_{s,u}=\alpha_{u,s}^{-1}$ and finite dimensionality of all point
modules completes the proof that $\operatorname{Ker}\nolimits
M(u)\cong\operatorname{Ker}\nolimits M(s)$ for all non-maximal
$u,s\in\mathcal{P}$. Notice also that at the same time we have shown that
$\operatorname{Im}M(w)\cong\operatorname{Im}M(t)$ for all $w,t\in\mathcal{P}$
that are not minimal. Finally, notice that if $u$ is minimal, then

$\operatorname{Im}M(u)\overset{\text{{def}}}{=}M(u)/\operatorname{Ker}\nolimits(M(u<v))=\operatorname{Im}(M(u<v))\overset{\text{{def}}}{=}\operatorname{Im}M(v).$

Thus $\operatorname{Im}M(u)\cong\operatorname{Im}M(v)$ for all
$u,v\in\mathcal{P}$. ∎

######  Corollary 4.9.

Assume the notation and hypotheses of Theorem 4.5. Fix a non-maximal object
$x_{0}\in\mathcal{P}$. Then there is an endomorphism
$\gamma_{0}\in\mathrm{End}(M(x_{0}))$, such that for any $y<y^{\prime}$ in
$\mathcal{P}_{\mathcal{T}}$,

$M(y<y^{\prime})=\alpha_{x_{0},y^{\prime}}\circ\gamma_{0}^{k}\circ\alpha_{y,x_{0}},$

where $k$ is the graph distance from $y$ to $y^{\prime}$ in $\mathcal{T}$.

###### Proof.

Let $\alpha\colon\beta^{*}M_{\mathcal{T}}\to\phi^{*}M_{\mathcal{T}}$ be a
natural isomorphism. Let $x_{0}<x_{1}$ be an indecomposable relation in
$\mathcal{P}_{\mathcal{T}}$, and set

$\gamma_{0}=\alpha_{x_{1},x_{0}}\circ
M(x_{0}<x_{1})\in\mathrm{End}(M(x_{0})).$

Let $y<y^{\prime}$ in $\mathcal{P}_{\mathcal{T}}$ be any pair of comparable
objects, and let

$y=z_{1}<z_{2}<\cdots<z_{k}=y^{\prime}$

be a decomposition of the relation $y<y^{\prime}$ into a sequence of
indecomposable relations in $\mathcal{P}_{\mathcal{T}}$. By Theorem 4.5(2),
for each $1\leq i\leq k-1$,

$M(z_{i},z_{i+1})=\alpha_{x_{1},z_{i+1}}\circ
M(x_{0}<x_{1})\circ\alpha_{z_{i},x_{0}}.$

Hence

$\displaystyle M(y<y^{\prime})$
$\displaystyle=M(z_{k-1}<z_{k})\circ\cdots\circ M(z_{1}<z_{2})$
$\displaystyle=\alpha_{x_{1},z_{k}}\circ
M(x_{0}<x_{1})\circ\alpha_{z_{k-1},x_{0}}\circ\alpha_{x_{1},z_{k-1}}\circ
M(x_{0}<x_{1})\circ\alpha_{z_{k-2},x_{0}}\circ\cdots\circ$
$\displaystyle\alpha_{x_{1},z_{2}}\circ
M(x_{0}<x_{1})\circ\alpha_{z_{1},x_{0}}$
$\displaystyle=\alpha_{x_{1},z_{k}}\circ
M(x_{0}<x_{1})\circ\alpha_{x_{1},x_{0}}\circ
M(x_{0}<x_{1})\circ\cdots\circ\alpha_{x_{1},x_{0}}\circ
M(x_{0}<x_{1})\circ\alpha_{z_{1},x_{0}}$
$\displaystyle=\alpha_{x_{1},y^{\prime}}\circ\alpha_{x_{0},x_{1}}\circ\alpha_{x_{1},x_{0}}\circ
M(x_{0}<x_{1})\circ\gamma_{0}^{k-1}\circ\alpha_{y,x_{0}}$
$\displaystyle=\alpha_{x_{0},y^{\prime}}\circ\gamma_{0}^{k}\circ\alpha_{y,x_{0}},$

as claimed. ∎

The next obvious question is what can be said about a virtual module
$[X]=[M]-[N]$ with a vanishing gradient on a tree, where $M,N\in
k\mathcal{P}\mbox{-}\EuR{mod}$. By linearity, $\nabla[X]=0$ if and only if

$\phi^{*}[M]-\beta^{*}[M]=\nabla[M]=\nabla[N]=\phi^{*}[N]-\beta^{*}[N],$

or equivalently if and only if there is a natural isomorphism

(6) $\phi^{*}M\oplus\beta^{*}N\cong\phi^{*}N\oplus\beta^{*}M.$

One may expect that $M$ and $N$ in that case would differ by locally constant
modules, namely that $M\oplus C\cong N\oplus D$ for some locally constant
modules $C,D\in k\mathcal{P}\mbox{-}\EuR{mod}$. This is not true in general as
Example 4.10 demonstrates.

######  Example 4.10.

Let $\mathcal{P}$ be any finite poset. Consider the constant module
$\underline{k}\in k\mathcal{P}\mbox{-}\EuR{mod}$ that assigns $k$ to each
object and the identity to each morphism. On the other hand let
$\underline{k}_{0}\in k\mathcal{P}\mbox{-}\EuR{mod}$ denote the module that
assigns $k$ to each object and the zero homomorphism to each morphism. It is
immediate that $\nabla[\underline{k}]=\nabla[\underline{k}_{0}]=0$. Let
$[X]\in\EuR{Gr}(k\mathcal{P})$ be any element and set
$[Y]=[X]+[\underline{k}]$ and $[Y_{0}]=[X]+[\underline{k}_{0}]$. Then
$\nabla[Y]=\nabla[Y_{0}]$, but the difference

$[Y]-[Y_{0}]=[\underline{k}]-[\underline{k}_{0}]$

is not locally constant.

Next we consider the relationship between the gradient and the rank invariant.

######  Definition 4.11.

Let $M\in k\mathcal{P}\mbox{-}\EuR{mod}$ be a persistence module. Define the
_rank invariant_

$\operatorname{rk}\nolimits[M]\colon\operatorname{Mor}(\mathcal{P})\to\mathbb{N}$

by $\operatorname{rk}\nolimits[M](x\leq
y)\overset{\text{{def}}}{=}\operatorname{rk}\nolimits M(x\leq y)$ for each
relation $x\leq y$ in $\mathcal{P}$. For a virtual module $[X]=[M]-[N]$ in
$\EuR{Gr}(k\mathcal{P})$, where $M,N\in k\mathcal{P}\mbox{-}\EuR{mod}$, define
the rank invariant

$\operatorname{rk}\nolimits[X]\colon\operatorname{Mor}(\mathcal{P})\to\mathbb{Z}$

by
$\operatorname{rk}\nolimits[X]\overset{\text{{def}}}{=}\operatorname{rk}\nolimits[M]-\operatorname{rk}\nolimits[N]$.

Notice that $\operatorname{rk}\nolimits[M](x\leq x)=\dim_{k}M(x)$ for each
object $x\in\mathcal{P}$. Notice also that the rank invariant can be regarded
as a group homomorphism

$\operatorname{rk}\nolimits\colon\EuR{Gr}(k\mathcal{P})\to\mathbb{Z}^{\operatorname{Mor}(\mathcal{P})},$

where the codomain is the abelian group of all functions from
$\operatorname{Mor}(\mathcal{P})$ to $\mathbb{Z}$, or in other words the free
abelian group generated by all morphisms in $\mathcal{P}$.

######  Remark 4.12.

The way we define the rank invariant here includes the rank of a module (or a
virtual module) on identity morphisms, namely it incorporates the dimensions
of the point modules $M(v)$ for all objects $v$ in $\mathcal{P}$. Notice also
that for $M\in k\mathcal{P}\mbox{-}\EuR{mod}$,
$[M]\in\operatorname{Ker}\nolimits(\operatorname{rk}\nolimits)$ if and only if
$M=0$, but as Example 4.10 shows, this is not the case for virtual modules.

Next, we turn to investigate virtual modules with a vanishing gradient.

######  Theorem 4.13.

Let $\mathcal{P}$ be a finite poset. Let
$[X]=[M]-[N]\in\EuR{Gr}(k\mathcal{P})$ be an element with $M,N\in
k\mathcal{P}\mbox{-}\EuR{mod}$. Then

  1. (1)

Let $(u_{0},u_{1})<(v_{0},v_{1})$ be a pair of comparable objects in
$\widehat{\mathcal{P}}$. Assume that $\nabla[X]=0$. Then
$\operatorname{rk}\nolimits[X](u_{0}<v_{0})=\operatorname{rk}\nolimits[X](u_{1}<v_{1})$.

Assume in addition that $\mathcal{P}$ is line connected, let $\mathcal{T}$ be
a line connected maximal tree for $\mathcal{H}_{\mathcal{P}}$, and let
$\mathcal{P}_{\mathcal{T}}\subseteq\mathcal{P}$ be the sub-poset generated by
$\mathcal{T}$. If $[X]$ has a vanishing gradient on $\mathcal{T}$, then
$\operatorname{rk}\nolimits[X]$ has the following properties:

  1. (2)

It is constant on all identity morphisms in $\mathcal{P}$.

  2. (3)

For any pair of indecomposable relations $u_{0}<u_{1}$, and $v_{0}<v_{1}$ in
$\mathcal{P}_{\mathcal{T}}$, one has
$\operatorname{rk}\nolimits[X](u_{0}<u_{1})=\operatorname{rk}\nolimits[X](v_{0}<v_{1})$.

###### Proof.

Since $\nabla[X]=0$, we have a natural isomorphism
$\eta\colon\phi^{*}M\oplus\beta^{*}N\xrightarrow{\cong}\phi^{*}N\oplus\beta^{*}M$
(See Equation (6)). Let $(u_{0},u_{1})<(v_{0},v_{1})$ be a pair of comparable
objects in $\widehat{\mathcal{P}}$. Then one has a commutative square

$\textstyle{M(u_{1})\oplus
N(u_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(u_{1}<v_{1})\oplus
N(u_{0}<v_{0})}$$\scriptstyle{\cong}$$\scriptstyle{\eta}$$\textstyle{M(v_{1})\oplus
N(v_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eta}$$\textstyle{N(u_{1})\oplus
M(u_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N(u_{1}<v_{1})\oplus
M(u_{0}<v_{0})}$$\textstyle{N(v_{1})\oplus M(v_{0})}$

It follows that $\operatorname{rk}\nolimits
M(u_{0}<v_{0})+\operatorname{rk}\nolimits
N(u_{1}<v_{1})=\operatorname{rk}\nolimits
M(u_{1}<v_{1})+\operatorname{rk}\nolimits N(u_{0}<v_{0})$, and hence that
$\operatorname{rk}\nolimits[X](u_{0}<v_{0})=\operatorname{rk}\nolimits[X](u_{1}<v_{1})$.
This proves Part (1).

Throughout the rest of the proof assume that $\mathcal{P}$ is line connected.
Fix a line connected maximal tree $\mathcal{T}$ for
$\mathcal{H}_{\mathcal{P}}$. Since $[X]$ has a vanishing gradient on
$\mathcal{T}$, we have a natural isomorphism
$\phi^{*}M_{\mathcal{T}}\oplus\beta^{*}N_{\mathcal{T}}\cong\phi^{*}N_{\mathcal{T}}\oplus\beta^{*}M_{\mathcal{T}}$.
Thus, for any object $(x,y)\in\widehat{\mathcal{P}}_{\mathcal{T}}$ one has
$M(x)\oplus N(y)\cong N(x)\oplus M(y)$. Hence

$\operatorname{rk}\nolimits[X](x\leq
x)=\dim_{k}M(x)-\dim_{k}N(x)=\dim_{k}M(y)-\dim_{k}N(y)=\operatorname{rk}\nolimits[X](y\leq
y).$

Since every $x\in\mathcal{P}_{\mathcal{T}}$ is either the source coordinate or
the target coordinate in an object of $\widehat{\mathcal{P}}_{\mathcal{T}}$,
and since $\widehat{\mathcal{P}}_{\mathcal{T}}$ is connected, Part (2)
follows.

Let $(u_{0},u_{1})$ be a minimal object in
$\widehat{\mathcal{P}}_{\mathcal{T}}$. By Part (1) and induction,
$\operatorname{rk}\nolimits[X](u_{r}<u_{r+1})=\operatorname{rk}\nolimits[X](u_{0}<u_{1})$
for any object $(u_{r},u_{r+1})\in\widehat{\mathcal{P}}_{\mathcal{T}}$ that is
the target of a directed path from $(u_{0},u_{1})$ in
$\widehat{\mathcal{P}}_{\mathcal{T}}$. Let $(u_{0},u_{1})$ and $(v_{0},v_{1})$
be minimal objects in $\widehat{\mathcal{P}}_{\mathcal{T}}$. Since
$\widehat{\mathcal{P}}_{\mathcal{T}}$ is connected, there are minimal objects

$(u_{0},u_{1})=(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{k},y_{k})=(v_{0},v_{1})$

such that for each $1\leq i\leq k-1$ the intersection of sub-posets
$\widehat{\mathcal{P}}_{\mathcal{T}_{\geq(x_{i},y_{i})}}\cap\widehat{\mathcal{P}}_{\mathcal{T}_{\geq(x_{i+1},y_{i+1})}}$
is nonempty. Let $(a_{i},b_{i})$ be any object in the intersection. By the
argument above,

$\operatorname{rk}\nolimits[X](x_{i}<y_{i})=\operatorname{rk}\nolimits[X](a_{i}<b_{i})=\operatorname{rk}\nolimits[X](x_{i+1}<y_{i+1})$

and since this holds for all $i$, we have
$\operatorname{rk}\nolimits[X](u_{0}<u_{1})=\operatorname{rk}\nolimits[X](v_{0},v_{1})$.
It follows that $\operatorname{rk}\nolimits[X]$ is constant on all
indecomposable relations in $\mathcal{P}_{\mathcal{T}}$, as claimed in Part
(3). ∎

The following example shows that the statements of Theorem 4.13 are best
possible in the sense that two modules with equal gradients may have different
rank invariants when evaluated on morphisms of length greater than 1.

######  Example 4.14.

Let $\mathcal{P}$ denote the poset $[2]$ with objects $0,1,2$ and the ordinary
order relation. Define $M,N\in[2]\mbox{-}\EuR{mod}$ as follows.

$M\colon\quad k^{2}\xrightarrow{(1,0)}k^{2}\xrightarrow{(0,1)}k^{2},\qquad
N\colon\quad k^{2}\xrightarrow{(1,0)}k^{2}\xrightarrow{(1,0)}k^{2}.$

It is immediate that $\phi^{*}M\cong\beta^{*}M$ and
$\phi^{*}N\cong\beta^{*}N$, so $\nabla[M]=\nabla[N]=0$. But
$\operatorname{rk}\nolimits M(0<2)=0$ while
$\operatorname{rk}\nolimits[N](0<2)=1$.

Above we considered the kernel and image modules associated to a module $M\in
k\mathcal{P}\mbox{-}\EuR{mod}$ in the case where $M$ has a vanishing gradient
on some tree $\mathcal{T}\subseteq\mathcal{H}_{\mathcal{P}}$ (See Definition
4.7 and Corollary 4.8). Next, we consider a kernel and a cokernel modules in
$k\widehat{\mathcal{P}}\mbox{-}\EuR{mod}$, associated to any module $M\in
k\mathcal{P}\mbox{-}\EuR{mod}$.

######  Lemma 4.15.

Let $M\in k\mathcal{P}\mbox{-}\EuR{mod}$ be any module. Let $K_{M},C_{M}\in
k\widehat{\mathcal{P}}\mbox{-}\EuR{mod}$ denote the modules defined on objects
by

$K_{M}(x,y)\overset{\text{{def}}}{=}\operatorname{Ker}\nolimits(M(x,y)\colon
M(x)\to M(y))\quad\text{and}\quad
C_{M}(x,y)\overset{\text{{def}}}{=}\operatorname{coKer}\nolimits(M(x,y)\colon
M(x)\to M(y))$

with the natural induced maps on morphisms. Then $K_{M}$ and $C_{M}$ are
virtually trivial.

###### Proof.

For any objects $(x,y)$ and $(y,z)$ in $\widehat{\mathcal{P}}$ one has the
following diagram of vector spaces with exact rows.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{M}(x,y)\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K_{M}((x,y)<(y,z))}$$\textstyle{M(x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(x,y)}$$\scriptstyle{M(x,y)}$$\textstyle{M(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(y,z)}$$\textstyle{C_{M}(x,y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C_{M}((x,y)<(y,z))}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{M}(y,z)\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(y,z)}$$\textstyle{M(z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{M}(y,z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

A straightforward diagram chase now shows that $K_{M}((x,y)<(y,z))$ and
$C_{M}((x,y)<(y,z))$ are the zero homomorphism. ∎

######  Corollary 4.16.

Let $\mathcal{P}$ be a finite line connected poset, let $\mathcal{T}$be a line
connected maximal tree for $\mathcal{H}_{\mathcal{P}}$, and let
$\mathcal{P}_{\mathcal{T}}\subseteq\mathcal{P}$ be the sub-poset generated by
$\mathcal{T}$. Let $[X]=[M]-[N]\in\EuR{Gr}(k\mathcal{P})$ be an element of
vanishing gradient on $\mathcal{T}$, with $M,N\in
k\mathcal{P}\mbox{-}\EuR{mod}$. Then there is an integer $D$, such that
$[K_{M}]=[K_{N}]+[\underline{D}]$ and $[C_{M}]=[C_{N}]+[\underline{D}]$ in
$\EuR{Gr}(k\widehat{\mathcal{P}}_{\mathcal{T}})$, where $[\underline{D}]$
denotes the virtually trivial module that associates $k^{D}$ with every object
of $\widehat{\mathcal{P}}_{\mathcal{T}}$.

###### Proof.

By Lemma 4.15, $K_{M}$ and $K_{N}$, as well as $C_{M}$ and $C_{N}$ are
virtually trivial, and hence their isomorphism type is determined by their
values on objects by Corollary 1.6. Thus, it suffices to prove that the
appropriate ranks coincide on all objects of $\widehat{\mathcal{P}}_{T}$.

By Theorem 4.13,
$\operatorname{rk}\nolimits[X]=\operatorname{rk}\nolimits[M]-\operatorname{rk}\nolimits[N]$
is constant on all identity morphisms and all indecomposable morphisms $x<y$
in $\mathcal{P}_{\mathcal{T}}$. Thus we may write
$\dim_{k}M(x)-\dim_{k}N(x)=K$ for all $x\in\mathcal{P}$, and
$\operatorname{rk}\nolimits[M](x<y)-\operatorname{rk}\nolimits[N](x<y)=T$ for
all indecomposable relations $x<y$ in $\mathcal{T}$, where $K$ and $T$ are
some fixed non-negative integers. Set $D=K-T$. Thus for an object
$(x,y)\in\widehat{\mathcal{P}}_{\mathcal{T}}$,

$\displaystyle\dim_{k}K_{M}(x,y)$
$\displaystyle=\operatorname{rk}\nolimits[M](x\leq
x)-\operatorname{rk}\nolimits[M](x<y)$
$\displaystyle=\operatorname{rk}\nolimits[N](x\leq
x)-\operatorname{rk}\nolimits[N](x<y)+K-T$
$\displaystyle=\dim_{k}K_{N}(x,y)+D.$

Since both $K_{M}$ and $K_{N}$ are virtually trivial, it follows that
$[K_{M}]=[K_{N}]+[\underline{D}]$. This proves the statement for $K_{M}$ and
$K_{N}$.

Similarly,

$\displaystyle\dim_{k}C_{M}(x,y)$
$\displaystyle=\operatorname{rk}\nolimits[M](y\leq
y)-\operatorname{rk}\nolimits[M](x<y)$
$\displaystyle=\operatorname{rk}\nolimits[N](y\leq
y)-\operatorname{rk}\nolimits[N](x<y)+K-T$
$\displaystyle=\dim_{k}C_{N}(x,y)+D.$

By the same argument as above $[C_{M}]=[C_{N}]+[\underline{D}]$, as claimed. ∎

Recall that for any ring $A$ one has a ring homomorphism

$\Upsilon_{A}\colon\EuR{Gr}(A)\to\EuR{Gr}_{e}(A),$

where $\EuR{Gr}_{e}(A)$ is the quotient of $\EuR{Gr}(A)$ by the relation
$[K]-[M]+[C]$, if $M$ is an extension of $C$ by $K$ (See Definition 1.11).
Since $\phi^{*}$ and $\beta^{*}$ are exact functors, they induce group
homomorphisms

$\phi_{e}^{*},\beta_{e}^{*}\colon\EuR{Gr}_{e}(k\mathcal{P})\to\EuR{Gr}_{e}(k\widehat{\mathcal{P}}),$

such that
$\Upsilon_{k\widehat{\mathcal{P}}}\circ\phi^{*}=\phi_{e}^{*}\circ\Upsilon_{k\mathcal{P}}$,
and similarly for $\beta^{*}$. Thus, one may define

$\nabla_{e}\colon\EuR{Gr}_{e}(k\mathcal{P})\to\EuR{Gr}_{e}(k\widehat{\mathcal{P}})$

by $\nabla_{e}\overset{\text{{def}}}{=}\phi^{*}_{e}-\beta^{*}_{e}$. The
following proposition shows that a significant amount of information about
modules is lost upon passing to the reduced Grothendieck groups.

######  Corollary 4.17.

For any $M\in k\mathcal{P}\mbox{-}\EuR{mod}$, one has

$\nabla_{e}[M]=[C_{M}]-[K_{M}]$

in $\EuR{Gr}_{e}(k\widehat{\mathcal{P}})$.

###### Proof.

For each $M\in k\mathcal{P}\mbox{-}\EuR{mod}$ let $I_{M}\in
k\widehat{\mathcal{P}}\mbox{-}\EuR{mod}$ denote the image functor, i.e.
$I_{M}(u,v)\overset{\text{{def}}}{=}\operatorname{Im}(M(u<v))$. Then we obtain
two short exact sequences in $k\widehat{\mathcal{P}}\mbox{-}\EuR{mod}$

$0\to K_{M}\to\beta^{*}M\to I_{M}\to 0\quad\text{and}\quad 0\to
I_{M}\to\phi^{*}M\to C_{M}\to 0.$

It follows that in $\EuR{Gr}_{e}(k\widehat{\mathcal{P}})$,

$[\beta_{e}^{*}M]=[K_{M}]+[I_{M}]\quad\text{and}\quad[\phi_{e}^{*}M]=[I_{M}]+[C_{M}],$

so
$\nabla_{e}[M]\overset{\text{{def}}}{=}[\phi_{e}^{*}M]-[\beta_{e}^{*}M]=[C_{M}]-[K_{M}]$.
∎

######  Remark 4.18.

The functor $K_{M}\in k\widehat{\mathcal{P}}\mbox{-}\EuR{mod}$ evaluated at an
object $(x,y)$ returns the subspace of elements that are “present” at $x$, but
do not “survive” to $y$. Thus for a fixed $x\in\mathcal{P}$, the intersection
$\bigcap_{(x,y)\in\widehat{\mathcal{P}}}K_{M}(x,y)$ is the subspace of $M(x)$
of all elements that “die” at $x$.

Similarly, the functor $C_{M}\in k\widehat{\mathcal{P}}\mbox{-}\EuR{mod}$,
evaluated at an object $(x,y)$ returns the quotient of $M(y)$ by the image of
$M(x<y)$. For a fixed object $y$, consider the system of all homomorphisms
$M(y)\to C_{M}(x,y)$ for all $(x,y)\in\widehat{\mathcal{P}}$. The coequaliser
of this system can be thought of as the space representing all elements in
$M(y)$ that are “born” at $y$. The coequaliser is easily seen to be the
quotient of $M(y)$ by the image of the composite

$\bigoplus_{(x,y)}M(x)\xrightarrow{\oplus
M(x<y)}\bigoplus_{(x,y)}M(y)\xrightarrow{\sum}M(y),$

where $\Sigma$ is the map given by summing coordinates.

Thus if $\mathcal{P}$ is line connected and $M,N\in
k\mathcal{P}\mbox{-}\EuR{mod}$ are modules with equal gradients, then for any
line connected maximal tree $\mathcal{T}$, spaces of “births” and “deaths” of
$M$ and $N$ restricted to $\mathcal{P}_{\mathcal{T}}$ object-wise coincide.

We end this section with an example of modules with non-isomorphic gradients
and the same rank invariant.

######  Example 4.19.

Let $\mathcal{P}$ be the poset with objects $\emptyset,a,b,c,d,m,\infty$, and
relations

$\emptyset<a,b,c,d<m<\infty.$ (7)
$\textstyle{\mathcal{H}_{\mathcal{P}}}$$\textstyle{\infty}$$\textstyle{m\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{d\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\emptyset\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\widehat{\mathcal{H}}_{\mathcal{P}}}$$\textstyle{(m,\infty)}$$\textstyle{(a,m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(b,m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(c,m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(d,m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\emptyset,a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\emptyset,b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\emptyset,c)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\emptyset,d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(8)
---
$\textstyle{X}$$\textstyle{0}$$\textstyle{k^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{*}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{*}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{*}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{*}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\beta(X)}$$\textstyle{k^{2}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{*}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{*}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{*}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{*}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\phi(X)}$$\textstyle{0}$$\textstyle{k^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{*}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{*}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{*}}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{*}}$

Diagram (7) shows the poset $\mathcal{P}$ and its associated line digraph.
Consider $k\mathcal{P}$-modules $X$ that take the value $k$ on $a,b,c,d$, the
value $k^{2}$ on $m$, and $0$ on $\emptyset$ and $\infty$. Furthermore, assume
that for each of the morphisms $\alpha,\beta,\gamma,\delta$, the induced map
under $X$ is an injection, and such that the images of each pair of
homomorphisms form a basis for $X(m)=k^{2}$. Clearly all such modules have
exactly the same rank invariant.

Let $M,N\in k\mathcal{P}\mbox{-}\EuR{mod}$ be modules satisfying these
requirements. Let $X,Y\in k\widehat{\mathcal{P}}\mbox{-}\EuR{mod}$ denote the
modules $\phi^{*}(M)\oplus\beta^{*}(N)$ and $\phi^{*}(N)\oplus\beta^{*}(M)$
respectively. Then $\nabla[M]=\nabla[N]$ if and only if $X$ and $Y$ are
isomorphic. Let $X_{0}$ and $Y_{0}$ denote the restrictions of $X$ and $Y$ to
the full sub-poset of $\widehat{\mathcal{P}}$ consisting of the objects
$(a,m)$, $(b,m)$, $(c,m)$, $(d,m)$ and $(m,\infty)$. Denote by
$\alpha_{*},\beta_{*},\ldots$ the homomorphisms $N(\alpha),N(\beta),\ldots$
and by $\alpha^{\prime}_{*},\beta^{\prime}_{*},\ldots$ the homomorphisms
$M(\alpha),M(\beta),\ldots$. Assume that there is an isomorphism $\Theta\colon
X\to Y$ and let $\Theta_{0}\colon X_{0}\to Y_{0}$ be the restriction of
$\theta$:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.92082pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 46.68123pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{k^{2}}$}}}}}}}{\hbox{\kern 104.68329pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-14.92082pt\raise-39.16554pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{k^{2}\oplus
k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.40639pt\raise-13.86194pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.79028pt\hbox{$\scriptstyle{0\oplus\alpha_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 49.71664pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
38.92082pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{k^{2}\oplus
k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
32.9378pt\raise-19.58278pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{0\oplus\beta_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 53.84164pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
92.76247pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{k^{2}\oplus
k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
62.98593pt\raise-25.51888pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.57501pt\hbox{$\scriptstyle{0\oplus\gamma_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 57.96371pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
146.60411pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{k^{2}\oplus
k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
101.29842pt\raise-13.68695pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.96527pt\hbox{$\scriptstyle{0\oplus\delta_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
61.00208pt\raise-2.59425pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}\ignorespaces\Longrightarrow\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
16.0319pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 48.9034pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{k^{2}}$}}}}}}}{\hbox{\kern 109.12762pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-16.0319pt\raise-39.16554pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{k\oplus
k^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
10.59593pt\raise-13.86194pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.79028pt\hbox{$\scriptstyle{\alpha^{\prime}_{*}\oplus 0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 51.76889pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
40.0319pt\raise-39.16554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{k\oplus
k^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
36.05998pt\raise-19.58278pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30556pt\hbox{$\scriptstyle{\beta^{\prime}_{*}\oplus 0}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 56.06381pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise