The Representation Theorem of Persistent Homology Revisited and Generalized
René Corbet Michael Kerber
math.AT math.RT 06F25, 16D90, 16W50, 55U99, 68W30





The Representation Theorem of Persistence Revisited and Generalized
    René CorbetGraz University of Technology, corbet@tugraz.at       
Michael KerberGraz University of Technology, kerber@tugraz.at 
    December 30, 2023
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The Representation Theorem by Zomorodian and Carlsson has been the
starting point of the study of persistent homology under the lens
of representation theory. In this work, we give a more accurate
statement of the original theorem and provide a complete and self-contained
proof. Furthermore, we generalize the statement from the case of linear
sequences of R-modules to R-modules indexed over more ge­neral
monoids. This generalization subsumes the Representation Theorem of
multidimensional persistence as a special case. 





§ INTRODUCTION



Persistent homology, introduced by Edelsbrunner et al. <cit.>,
is a multi-scale extension of classical homology theory. The idea
is to track how homological features appear and disappear in a shape
when the scale parameter is increasing. This data can be summarized
by a barcode where each bar corresponds to a homology class
that appears in the process and represents the range of scales where
the class is present. The usefulness of this paradigm in the context
of real-world data sets has led to the term topological data
analysis; see the surveys <cit.>
and textbooks <cit.> for various use
cases.

A strong point of persistent homology is that it can be defined and
motivated both in geometric and in algebraic terms. For the latter, the main
object are persistence modules. In the simplest case, such
a persistence module consists of a sequence of R-modules indexed
over ℕ and module homomorphisms connecting consecutive
modules, as in the following diagram:


    M_0[r]^φ_0    M_1[r]^φ_1   …[r]^φ_i-1    M_i[r]^φ_i    M_i+1[r]^φ_i+1   …


A persistence module as above is of finitely generated type
if each M_i is finitely generated and there is an m∈ℕ
such that φ_i is an isomorphism for all i≥ m. Under
this condition, Zomorodian and Carlsson <cit.> observed that
a persistence module can be expressed as single module over the polynomial
ring R[t]:

[Theorem 3.1 in <cit.>] Let R be a commutative
ring with unity. The category of persistence modules of finitely generated
type[In <cit.>, the term “finite type” is used instead, but
we renamed it here as we will define another finiteness condition
later.] over R is equivalent to the category of finitely generated graded
modules over R[t]. 

The importance of this equivalence stems from the case most important
for applications, namely if R is a field. In this case, graded
R[t]-modules, and hence also persistence modules of finitely generated
type, permit a decomposition


    (⊕_i=1^nΣ^α_iR[t])⊕(⊕_j=1^mΣ^β_jR[t]/(t^n_j)))


where Σ^· denotes a shift in the grading. The integers
α_i,β_j,n_j give rise to the aforementioned barcode
of the persistence module; see <cit.> for details. Subsequent
work studied the property of more general persistence modules, for
instance, for modules indexed over any subset of ℝ (and
not necessarily of finite type) <cit.>
and for the case that the M_i and φ_i are replaced
with any objects and morphisms in a target category <cit.>.



Given the importance of the ZC-Representation Theorem, it is remarkable
that a comprehensive proof seems not to be present in the literature. 
In <cit.>,
the authors assign an R[t]-module to a persistence module of finite
type and simply state: 

The proof is the Artin-Rees theory in commutative algebra (Eisenbud,
1995). 

In Zomorodian's textbook <cit.>, the same statement is
accompanied with this proof (where α is the assignment mentioned
above): 

It is clear that α is functorial. We only need to construct
a functor β that carries finitely generated non-negatively
graded k[t]-modules [sic] to persistence modules of finite[ly generated]
type. But this is readily done by sending the graded module M=⊕_i=0^∞M_i
to the persistence module {M_i,φ_i}_i∈ℕ
where φ_i:M^i→ M^i+1 is multiplication by
t. It is clear that αβ and βα are canonically
isomorphic to the corresponding identity functors on both sides. This
proof is the Artin-Rees theory in commutative algebra (Eisenbud, 1995). 


While that proof strategy works for the most important case of fields, 
it fails for “sufficiently” bad
choices of R, as the following example shows:


Let R=ℤ[x_1,x_2,…] and consider the graded R[t]
module M:=⊕_i∈ℕM_i with M_i=R/<x_1,…,x_i>
where multiplication by t corresponds to the map M_i→ M_i+1
that assigns p mod x_i to a polynomial p. M is generated
by { 1}. However, the persistence module β(M)
as in Zomorodian's proof is not of finitely generated type, because
no inclusion M_i→ M_i+1 is an isomorphism.



This counterexample raises the question: what are the requirements
on the ring R to make the claimed correspondence valid? In the
light of the cited Artin-Rees theory, it appears natural to require
R to be a Noetherian ring (that is, every ascending chain of ideals
becomes stationary), because the theory is formulated for such rings
only; see <cit.>. Indeed, as carefully exposed in the master's
thesis of the first author <cit.>, the above proof
strategy works under the additional assumption of R being Noetherian. 
We sketch the proof in Appendix <ref>.



  
Our contributions.

As our first result, we prove a generalized version of the ZC-Representation
Theorem. In short, we show that the original statement becomes valid
without additional assumptions on R if “finitely generated type”
is replaced with “finitely presented type” (that is, in particular,
every M_i must be finitely presented). 
Furthermore, we remove the requirement of R being commutative
and arrive at the following result.

Let R be a ring with unity. The category of persistence
modules of finitely presented type over R is isomorphic to the
category of finitely presented graded modules over R[t]. 

The example from above does not violate the statement of this theorem
because the module M is not finitely presented. Also, the statement
implies the ZC-Representation Theorem for commutative Noetherian rings,
because if R is commutative with unity and Noetherian, finitely
generated modules are finitely presented.

Our proof follows the same path as sketched by Zomorodian, using the
functors α and β to define a (straight-forward) correspondence
between persistence mo­dules and graded R[t]-modules. The technical
difficulty lies in showing that these functors are well-defined if
restricted to subclasses of finitely presented type. It is worth to
remark that our proof is elementary and self-contained and does not
require Artin-Rees theory at any point. We think that the ZC-Representation
Theorem is of such outstanding importance in the theory of persistent
homology that it deserves a complete proof in the literature.




As our second result, we give a Representation Theorem for a more
general class of persistence modules. We work over an arbitrary ring
R with unity and generalize the indexing set of persistence modules
to a monoid (G,⋆).[Recall that a monoid is almost a group,
except that elements might not have inverses.] 
We consider a subclass which we call “good” monoids in this work
(see Section <ref> for the definition and a discussion of 
related concepts). Among them is the case ℕ^k 
corresponding to multidimensional persistence modules,
but also other monoids such as (_≥ 0,+), (∩(0,1],·) and 
the non-commutative word monoid as illustrated in Figure <ref>. 
It is not difficult to show that such generalized persistence modules can be 
isomorphically described as a single module over the monoid ring R[G].

Our second main result 
is that finitely presented graded modules over R[G] correspond
again to generalized persistence modules with a finiteness condition.
Specifically, finiteness means that there exists a finite set S
of indices (i.e., elements in the monoid) such that for each monoid
element g with associated R-module R_g, there exists an
s∈ S such that each map R_s→ R_g̃ is an 
isomorphism, whenever g̃ lies between s and g. 

For G=^k, we prove that 
this condition is equivalent to the property that all
sequences in our persistence module are of finite type 
(as a persistence module over ℕ), see Figure <ref>, 
but this equivalence fails for general (good) monoids. 
Particularly, our second main result implies the first one, because
for G=, the monoid ring R[] is precisely the polynomial ring R[t].



