MULTIPARAMETER PERSISTENT HOMOLOGY GENERIC STRUCTURES AND QUANTUM COMPUTING AMELIE SCHREIBER

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MULTIPARAMETER PERSISTENT HOMOLOGY: GENERIC STRUCTURES AND

QUANTUM COMPUTING

AMELIE SCHREIBER

Abstract. The following article is an application of commutative algebra to the study of multi-

parameter persistent homology in topological data analysis. In particular, the theory of ﬁnite free

resolutions of modules over polynomial rings is applied to multiparameter persistent modules.

The generic structure of such resolutions and the classifying spaces involved are studied using

results spanning several decades of research in commutative algebra, beginning with the study of

generic structural properties of free resolutions popularized by Buchsbaum and Eisenbud. Many

explicit computations are presented using the computer algebra package Macaulay2, along with

the code used for computations. This paper serves as a collection of theoretical results from

commutative algebra which will be necessary as a foundation in the future use of computational

methods using Gr¨

obner bases, standard monomial theories, Young tableaux, Schur functors and

Schur polynomials, and the classical representation theory and invariant theory involved in linear

algebraic group actions. The methods used are in general characteristic free and are designed

to work over the ring of integers in order to be useful for applications and computations in data

science. As an applications we explain how one could apply 2-parameter persistent homology to

study time-varying interactions graphs associated to quadratic Hamiltonians such as those in the

Ising model or Kitaev’s torus code and other surface codes.

Contents

1. Historical Background and Motivations 2

2. Young Tableaux and Schur Functors 4

3. Multiparameter Persistence Modules 5

4. Presentation Spaces, Rank Invariants, and Fitting Ideals 9

5. Generic Free Resolutions 19

6. Buchsbaum-Eisenbud Multipliers 20

7. Varieties of Complexes of Persistence Modules 22

8. Bases of Coordinate Rings via Young Tableaux 25

9. Examples 29

10. Applications: Ising Model Hamiltonian 30

11. Appendix: Macaulay2 Code 31

References 39

Date: October 21, 2022.

2020 Mathematics Subject Classiﬁcation. Primary 13P25 13P10 13P20 Secondary 05E40 13A30 .

Key words and phrases. persistent homology, multiparameter persistence, topological data analysis, generic free

resolutions, module varieties, quadratic Hamiltonians.

arXiv:2210.11433v1 [math.RT] 20 Oct 2022

2 A. SCHREIBER

1. Historical Background and Motivations

1.1. Introduction. The theory of multiparameter persistence arises in many contexts in com-

putational topology and data science. Often a ﬁnite point cloud P⊂Rdis obtained from some

observational data, and one would like to understand the signiﬁcant features of the data cloud.

It is typical that the data set will have some ”noise”, and such noise may need to be ﬁltered out.

It is in general a very diﬃcult problem to analyze a data set and determine what subset of the

data is noise and what is a signiﬁcant feature of the data. In this case, one can use topological

data analysis and persistent homology. This method has proven very robust and eﬀective in

applications, but is a relatively new ﬁeld.

The main object of study in persistent homology are the ”persistent modules”. It was es-

tablished by Carlsson and Zomorodian in [CZ1,CZ2] that multiparameter persistent modules

correspond to modules over a polynomial ring K[x1, ..., xn], nbeing the number of parameters.

In [CZ1] it is stated that certain coeﬃcient ﬁelds (or rings) Kmay be more desirable in cer-

tain cases than others. They state having a theory and methods for understanding persistence

modules over polynomial rings with arbitrary coeﬃcients is important for applications since

continuous invariants may be more diﬃcult computationally and in some applications it may be

desirable to have coeﬃcients in ﬁnite ﬁelds, ﬁelds of nonzero characteristic, or the integers Z.

This paper is primarily a survey, much like that of [HOST], which we will make some con-

nections to. Results and methods from commutative algebra and algebraic geometry are framed

in the setting of multiparameter persistent homology and topological data analysis. The appli-

cations come from research initiated in the early 1970s by Buchsbaum and Eisenbud on the

structure of free resolutions of modules over commutative rings, especially the theory of so-

called generic free resolutions.

In this paper, we will establish a structure theory for the free resolutions of multiparameter

persistent modules, which are ﬁnitely generated modules over the polynomial ring K[x1, ..., xn],

where Kmay be any ﬁeld, or the ring of integers Z. The study of free resolutions of modules,

especially that of the generic structure of resolutions over Noetherian rings has been extensively

studied for decades (see for example [Ho] and [BE1,BE2,BE3]). The quest for a complete and

practical understanding of the generic structure of free resolutions and the related representation

theory of algebraic groups acting on rings is an endeavor which is still an active area of research

(see for example [B], [W] ). In the following paper, many of these techniques and results are

used to study multiparameter persistent homology. Ample references to past research on the

subject are given.

1.2. Remarks on Background Material and Assumptions in the Paper. Throughout, the

ring R=K[x1, ..., xn] will always be the polynomial ring with coeﬃcients in an arbitrary ﬁeld

or in the integers. The results which follow are almost entirely independent of the characteristic

of the base ﬁeld (or ring), and one may assume we are working over Zunless speciﬁcally stated

otherwise. This is done for computational purposes, as computing over the ﬁeld Cor even Qor

its algebraic closure Qcan be computationally expensive, and can give rise to large errors when

computing with large data sets. For example, it is shown in §1.4.1 in [C] that even computations

with Hilbert matrices of relatively small size lead to large errors rather quickly. In certain

speciﬁc cases, we may need to assume the algebraic closure of the base ﬁeld. We will state this

assumption when it is necessary, but for the majority of the results not even this assumption is

required.

There will be applications of the representation theory of algebraic groups, especially of

GL(n) and SL(n), the general and special linear groups. Because much of what follows is done

MULTIPARAMETER PERSISTENT HOMOLOGY: GENERIC STRUCTURES AND QUANTUM COMPUTING 3

in the generality of coeﬃcients over Z, we will need a representation theory of these groups

which works in this generality. This will require notions such as schemes and functors of points,

which we will review the basics of. For the details and background we refer the reader to [EH]

and [M].

We will need this, for example, when constructing universal objects such as the Grassman-

nian functor. The Grassmannian over a ﬁeld is a classical construction, but if one wishes to

work over arbitrary base rings, for example Z, or a polynomial ring K[x1, ..., xn], one needs

something more general than the classical construction. In the case of the Grassmannian, this

construction is not hard. It turns out if one constructs the Grassmannian GrZ(r,k) over Z, then

for any other base scheme S=Spec(A), for Aany commutative ring, we have

GrS(r,k)=GrZ(r,k)×S.

More generally, if we have any homomorphism of aﬃne schemes φ:T→Swe have

GrT(r,k)=GrS(r,k)×ST

is given by a ﬁbre product. We will also talk about the universal complexes, which is a universal

object for complexes of free modules over a commutative ring in the same way GrZ(r,k) is for

Grassmannians. From this comes the study of generic free resolutions, which are in some sense

”the most general” free resolutions in a way we will make precise later on. They give a kind of

classifying space for free resolutions of a certain speciﬁed type.

At this point, we will also need the notion of Schur functors and the combinatorics of Young

tableaux in order to describe the irreducible representations of these linearly reductive groups.

We will review this material as well, but refer the reader to [E], [F], [FH], [DEP], [DS], [T1]. We

will look at some speciﬁc results on free resolutions where it is assumed that the base ﬁeld has

characteristic zero. So, for example, one may assume we are working over the p-adic numbers

Qm. As seen in [C], this can still be useful from a computer science perspective. We will also

state which speciﬁc cases require this particular assumption.

1.3. Organization of the Paper. In Sections 2and 3we review the theory of Schur functors

and Young tableaux, and recall basic information on multiparameter persistence homology and

related commutative algebra and algebraic geometry. Our ﬁrst task once this is done will be

to develop a generic structure theory in order to extend the theory of parametrizing spaces of

persistence modules started in [CZ1], where the so-called relation families are studied. As it

turns out, these are simply presentation spaces of multiparameter persistence modules. In this

case the theory of determinantal ideals and determinantal varieties are of great utility. We will

present the theory of the so-called Fitting invariants of persistence modules and use them to

study presentations of persistence modules. This is the content of Section 4. Next in Section

5we will deﬁne generic free resolutions of persistence modules and we will review Bruns’

”generic exactiﬁcation” of a complex of modules. Once the generic structure is described, we

introduce the Buchsbaum-Eisenbud multipliers and use these to state some deeper results about

generic free resolutions of persistence modules in Section 6. We then come to the varieties of

complexes in Section 7, where we study a scheme which gives geometric properties of families

of free resolutions of persistent modules. In Section 8we then give a Gr¨

obner basis of the

coordinate rings of varieties of complexes, and we develop a standard monomial theory using

the combinatorics of Young tableaux. This section in particular provides methods which are

particularly useful tools standard in computational algebraic geometry. Throughout we provide

many explicit examples, many of which were computed using Macaulay2. We provide the code

for the computations carried out in the examples in the appendix at the end.

4 A. SCHREIBER

2. Young Tableaux and Schur Functors

Apartition λ`m, of some nonnegative integer mis a sequence of nonincreasing numbers

λ=(λ1, ..., λs) such that Piλi=m. We deﬁne a Young diagram corresponding to a partition

λ=(λ1, ..., λs)`m, as the diagram with λiboxes in the ith row. We identify partitions with their

Young diagrams and speak of the two interchangeably. For a free module Eover a commutative

ring Kwith ordered basis {e1, ..., en}, we associate a ﬁlling of the diagram λby integers {1, ..., n}

to an element in the module

λ1

^E⊗ · · · ⊗

λs

as follows: If in row iand box jof λwe have the integers t(i,j) for j=1, ..., λi, we associate

the element

et(1,1) ∧ · · · ∧ et(1,λ1)⊗ · · · ⊗ et(s,1) ∧ · · · ∧ et(s,λs).

We deﬁne such a ﬁlling to be a tableaux. A tableaux is standard if its rows are strictly increas-

ing, and its columns are nondecreasing.

Fix a free module Eof rank nover a commutative ring K. Let λ=(λ1, ..., λs)`mbe a

partition. We deﬁne the module

LλE=

λ1

^⊗ · · · ⊗

λs

^E/R(λ, E)

where the submodule R(λ, E) is the sum of all submodules of the form

λ1

^E⊗ · · · ⊗

λa−1

^E⊗Ra,a+1E⊗

λa+2

^E⊗ · · · ⊗

λs

for 1 ≤a≤s−1. Here Ra,a+1Eis the submodule spanned by the images of the maps

θ(λ, a,u,v,E)

VuE⊗Vλa−u+λa+1−vE⊗VvE

1⊗∆⊗1



VuE⊗Vλa−uE⊗Vλa+1−vEVvE

m12⊗m34



VλaE⊗Vλa+1E

Here, ∆is the diagonal embedding (or comultiplication), and m:VrE⊗VsE→Vr+sEis a

component of the multiplication map on the exterior algebra V•E. Combinatorially, we may

use the theory of Young tableaux to describe the modules LλE. It is well known the standard

tableaux form a basis of LλE. Further, the relations R(λ, E) may be visualized as follows. Let λ`

mbe a partition with at most nparts. To the map θ(λ, a,u,v,E) we associate a Young scheme.

The Young scheme is deﬁned as being a diagram of shape λwith empty boxes everywhere

except the ath and a+1st rows. In row athere are uempty boxes followed by λa−uﬁlled

boxes. In row a+1 there are λa+1−vﬁlled boxes followed by vempty boxes. We restrict to

u+v< λa+1as in the deﬁnition of LλE. Let Uj=et(j,1) ∧ · · · ∧ et(j,λ j),V1=ex1∧ · · · ∧ exu,

V2=ey1∧ · · · ∧ eyλa−u+λv−v, and V3=ez1∧ · · · ∧ ezv. Then the image of a typical element

U1⊗ · · · ⊗ Ua−1⊗V1⊗V2⊗V3⊗Ua+2⊗ · · · ⊗ Us

is a sum of tableaux where we put x1, ..., xuin the empty uboxes in row a, and z1, ..., zvin the

empty vboxes of row a+1, and we shuﬄe the elements y1, ..., yλa−u+λa+1−vbetween the ﬁlled

MULTIPARAMETER PERSISTENT HOMOLOGY: GENERIC STRUCTURES AND QUANTUM COMPUTING 5

boxes in row aand a+1. The coeﬃcients of the tableaux in the summation are ±1 depending

on the sign of the permutations coming from the exterior diagonals.

Remark 2.1. From now on, we will let LλFdenote the Schur functor corresponding to the

diagram λ, and we will let S(λ−k)F=LλF⊗(Vdim FF∗)⊗k, where λ−kis obtained by subtracting

the integer kfrom every entry of the vector λ.

3. Multiparameter Persistence Modules

3.1. Homological Persistence. The theory of multiparameter persistence arises in many con-

texts in computational topology and data science. Often a point ﬁnite cloud P⊂Rdis obtained

from some observational data, and one would like to understand the signiﬁcant features of the

data cloud. It is typical that the data set will have some ”noise”, and such noise may need to

be ﬁltered out. It is in general a very diﬃcult problem to analyse a data set and determine what

subset of the data is noise and what is a signiﬁcant feature of the data. In this case, one can use

topological data analysis and persistence homology. This method has proven very robust and

eﬀective in applications.

Deﬁnition 3.1. Amultiﬁltered space X, is a topological space with a family of subspaces

{Xv⊆}v∈Nn, with inclusion maps Xu→Xvwhenever u≤vin the standard partial order on Nn.

We require the following diagram to commute:

Xu//



Xv1



Xv2//Xw

whenever u≤v1,v2≤w.

Deﬁnition 3.2. Let Kbe a ﬁeld. A persistence module M, is a family of K-modules {Mu}u∈Nn

with maps

φu,v:Mu→Mv

for all u≤vsuch that φv,w◦φu,v=φu,wfor all u≤v≤w. Let Mbe a persistence module and let

R=K[x1, ..., xn]. Deﬁne

α(M)=M

where the K-module structure is the direct sum structure, and xu−v:Mu→Mvis φu,vwhen

u≤vin Nn. This gives an equivalence of categories between the category of ﬁnite persistence

modules over K, and the category of Nn-graded ﬁnitely generated modules over R.

Now, the ﬁrst two terms of a minimal free resolution of a persistence module Mare unique

up to isomorphism of free chain complexes. This is important for classiﬁcation of isomorphism

classes of multigraded persistence modules.

3.2. Free Resolutions. The structure of free resolutions of modules over commutative rings is

one of the most fundamental objects of study in commutative algebra. If one wants to under-

stand a ring well, one needs to understand the modules over that ring. If one has some structure

theorems about what free (or projective/injective) resolutions of the modules ”look like”, then

one understands the modules quite well. In some very classical results which still have some sig-

niﬁcant implications which are being studied in commutative algebra, Buchsbaum and Eisenbud

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MULTIPARAMETERPERSISTENTHOMOLOGY:GENERICSTRUCTURESANDQUANTUMCOMPUTINGAMELIESCHREIBERAbstract.Thefollowingarticleisanapplicationofcommutativealgebratothestudyofmulti-parameterpersistenthomologyintopologicaldataanalysis.Inparticular,thetheoryofnitefreeresolutionsofmodulesoverpolynomialringsisappliedt...

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MULTIPARAMETER PERSISTENT HOMOLOGY GENERIC STRUCTURES AND QUANTUM COMPUTING AMELIE SCHREIBER

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