Betti Cones of Stanley-Reisner Ideals

David Carey

A thesis presented for the degree of

Doctor of Philosophy

School of Mathematics and Statistics

University of Sheffield

August 2023

CHAPTER: ACKNOWLEDGEMENTS

I would like to thank a number of people for their help during the course of my PhD.

Firstly, I am grateful to the Engineering and Physical Sciences Research Council for supporting my work.

During the course of my PhD I was fortunate enough to attend the virtual Research Encounters in Algebraic and Combinatorial Topics conference in 2021; the Graduate Meeting on Combinatorial Commutative Algebra at the University of Minnesota in 2022; and the 4th Graduate Meeting on Applied Algebra and Combinatorics at KTH in 2023. I would like to thank the coordinators of all of these conferences, both for the hard work they put into organising the events, and for giving me the opportunity to present my research there. I am also very grateful to the many mathematicians I met across these events who took the time to tell me about their own work (or other work that interested them), for providing countless fascinating discussions about combinatorics and algebra. In particular I am indebted to Alex Lazar for an invaluable conversation about Cohen-Macaulay complexes.

I would also like to thank my parents for providing a comfortable space to finish writing this thesis (and for keeping me supplied with cups of tea and great meals throughout); my brother for his assistance with some of my coding problems; Joseph Martin, Lewis Combes and David Williams for attending a mock run-through of my first ever academic talk, and offering some extremely helpful feedback on how to improve it; the Soho Scribblers for providing a regular, relaxed and friendly environment to write whenever I find myself down in London; and all of the friends and family who have helped to keep me grounded and boost my morale when the workload was most stressful.

Lastly, and most importantly, I would like to thank my supervisor Moty Katzman for his endless supply of support, guidance and patience throughout all aspects of this project. I am very lucky to have had him as my supervisor.

CHAPTER: ABSTRACT

The aim of this thesis is to investigate the Betti diagrams of squarefree monomial ideals in polynomial rings. Betti diagrams encode information about the minimal free graded resolutions of these ideals, and are therefore important algebraic invariants.

Computing resolutions is a difficult task in general, but in our case there are tools we can use to simplify it. Most immediately, the Stanley-Reisner Correspondence assigns a unique simplicial complex to every squarefree monomial ideal, and Hochster’s Formula ([25] Theorem 5.1) allows us to compute the Betti diagrams of these ideals from combinatorial properties of their corresponding complexes. This reduces the algebraic problem of computing resolutions to the (often easier) combinatorial problem of computing homologies. As such, most of our work is combinatorial in nature.

The other key tool we use in studying these diagrams is Boij-Söderberg Theory. This theory views Betti diagrams as vectors in a rational vector space, and investigates them by considering the convex cone they generate. This technique has proven very instructive: it has allowed us to classify all Betti diagrams up to integer multiplication. This thesis applies the theory more narrowly, to the cones generated by diagrams of squarefree monomial ideals.

In Chapter <ref> we introduce all of these concepts, along with some preliminary results in both algebra and combinatorics we will need going forward. Chapter <ref> presents the dimensions of our cones, along with the vector spaces they span.

Chapters <ref> and <ref> are devoted to the pure Betti diagrams in these cones (motivated in part by [1] and [3]), and the combinatorial properties of their associated complexes. Finally Chapter <ref> builds on these results to prove a partial analogue of the first Boij-Söderberg conjecture for squarefree monomial ideals, by detailing an algorithm for generating pure Betti diagrams of squarefree monomial ideals of any degree type.

CHAPTER: INTRODUCTION

Throughout this thesis we fix a field $\KK$ of arbitrary characteristic and a positive integer $n$, and we set $R$ to be the polynomial ring $\KK[x_1,\dots, x_n]$. Our aim is to investigate both the squarefree monomial ideals of $R$ and simplicial complexes on the vertex set $[n]=\{1,\dots,n\}$, which are related via the Stanley-Reisner Correspondence. More specifically we wish to study an important algebraic invariant of squarefree monomial ideals: their Betti diagrams.

Betti diagrams are an invariant of any graded $R$-module $M$; they are defined as the matrix of exponents $\beta_{i,d}$ in a minimal free graded resolution of $M$,

$$0\rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{p,d}} \rightarrow ... \rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{1,d}} \rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{0,d}}\rightarrow M \rightarrow 0.$$

and they encode a lot of important information about their respective modules: in particular, the Krull-dimension, projective dimension, Hilbert polynomial, and Castelnuovo-Mumford regularity of a module can all be computed from its Betii diagram.

Unfortunately there does not, in general, exist an efficient way of computing Betti diagrams for arbitrary modules. However, in the case of monomial ideals, there are often combinatorial properties of the ideal which we can exploit to help us compute its Betti diagram.

More specifically, in the case of squarefree monomial ideals, the Stanley-Reisner correspondence assigns the ideal a unique simplicial complex; and a seminal theorem of Hochster's ([25] Theorem 5.1) reframes the Betti numbers of the ideal in terms of homological data from its corresponding complex. Hochster's Formula has proven to be an incredibly powerful tool in the study of Betti diagrams. It has been successfully used to compute diagrams of many families of squarefree monomial ideals – in particular, the edge ideals of certain families of graphs, such as in [34] Theorem 2.3.3 (complements of cyclic graphs) or [27] Theorem 5.3.8 (complete multipartite graphs) – and to find combinatorial bounds for other algebraic invariants (such as Theorem 6.7 in [22], which gives a bound on the regularity of an edge ideal in terms of matchings in its corresponding graph).

Note that every Betti diagram of a monomial ideal can be realised as the Betti diagram of a squarefree monomial ideal, via the process of polarisation (see [31] Definition 2.1). Thus, restricting our attention to the squarefree case does not in fact require us to neglect any Betti diagrams of arbitrary monomial ideals.

This thesis is particularly focussed on those complexes whose Stanley-Reisner ideals have pure Betti diagrams – these are Betti diagrams arising from pure resolutions of the form

$$0\rightarrow R(-c_p)^{\beta_{p,c_p}} \rightarrow ... \rightarrow R(-c_1)^{\beta_{1,c_1}} \rightarrow R(-c_0)^{\beta_{0,c_0}}\rightarrow I \rightarrow 0.$$

Put simply, a Betti diagram is pure if it has only a single nonzero entry in each column. Stanley-Reisner ideals with pure resolutions have been investigated extensively in the literature. Most notably, [3], [18] and [1] all include a number of results in this area. Between them they classify all the Stanley-Reisner ideals with pure resolutions corresponding to $1$-dimensional complexes (Proposition 2.2 of [3]); Cohen-Macaulay posets (Theorem 3.6 of [3]); and clique complexes of graphs (Theorem 1 in [18] and Theorem 2.1 in [1]). In the case of clique complexes, [15] builds on Theorem 1 of [18] to show that any 2-linear diagrams arising from a clique complex has a unique corresponding threshold graph.

Our specific interest is in the possible shapes of pure diagrams of Stanley-Reisner ideals – that is, the positions of their nonzero entries. This is equivalent to the notion of shift type given in [1] (page 1203), defined as the sequence $(c_p,\dots,c_0)$ of shifts in the resolution. The authors of this paper give a number of examples of, and restrictions on, the possible shift types of pure resolutions of Stanley-Reisner ideals (see e.g. Theorem 3.1 and Propositions 3.2 and 3.3).

One reason pure diagrams have enjoyed particular attention in the literature comes from Boij-Söderberg Theory, which is a central motivation for our work. This theory (originally expounded in [5]) envisages all Betti diagrams of $R$-modules as vectors lying in the infinite-dimensional rational vector space $\Vn=\bigoplus_{d\in \ZZ} \QQ^{n+1}$, and studies the cone generated by the positive rational rays spanned by these diagrams. The central insight of Boij and Söderberg was that every ray in this cone can be decomposed into the sum of rays corresponding to pure diagrams of Cohen-Macaulay modules (and these pure rays can be easily computed). This result amounts to a complete classification of all Betti diagrams of $R$-modules up to multiplication by a positive rational.

In the case of monomial ideals, there has been considerable interest in obtaining combinatorial descriptions of the Boij-Söderberg decomposition described above. This avenue has proven very fruitful in certain cases (see e.g. [33] Theorem 3.6, which looks at Ferrers ideals, or [16] Theorem 3.2, which looks at Borel ideals). However, there is a significant obstacle to this goal in general: while we know that there exists a pure Betti diagram of any given shift type (this is the first Boij-Söderberg conjecture, [17] Theorem 1.9), this pure diagram need not be the diagram of a monomial ideal. Hence, the decomposition of the diagram of a monomial ideal $\beta(I)$ will often contain pure diagrams which do not correspond to monomial ideals themselves (and thus do not lend themselves readily to combinatorial manipulation).

This raises a natural question: to what extent does the first Boij-Söderberg conjecture apply to the cone generated by diagrams of squarefree monomial ideals? Or put more simply, for which shift types does there exist a squarefree monomial ideal with a pure resolution of that shift type? This question is the central motivation for our study of pure Betti diagrams of Stanley-Reisner ideals in Chapters <ref>, <ref> and <ref>.

Another way to navigate the obstacle above would be to obtain a decomposition of $\beta(I)$ into the rational sum of diagrams of other squarefree monomial ideals. This would require a description of the extremal rays of the cone generated by Betti diagrams of squarefree monomial ideals. We study this cone in Chapter <ref>, along with three notable subcones. While a complete description of these cones may currently be out of reach, we present some important properties, which may be of use in future classifications: in particular, we present and prove formulae for their dimensions, and describe the minimal subspaces of $\Vn$ that they span.

For the avoidance of ambiguity we now present a brief overview of our substantial new results, broken down by chapter. It is worth noting that none of these results depend upon the characteristic of $\KK$.

§ CHAPTER <REF>: BACKGROUND MATERIAL

Our first chapter is a survey of relevant algebraic and combinatorial background material from the literature, to provide context and motivation for our main results, as well as a number of key tools that will help us in proving them. This includes a brief overview of topics surrounding Betti diagrams, Boij-Söderberg Theory, Stanley-Reisner and edge ideals, and Hochster's Formula. As such, none of the results in this chapter are new (with the possible exception of Lemma <ref> – a result detailing a condition under which a face may be deleted from a complex while preserving homotopy – which we have been unable to find in the literature).

§ CHAPTER <REF>: DIMENSIONS OF BETTI CONES

This chapter details our work on the dimensions of Betti cones generated exclusively by diagrams of squarefree monomial ideals. The chapter is built around the proofs of four key theorems.

* Theorem <ref> on the dimension of the Betti cone $\Dn$ generated by the diagrams of all squarefree monomial ideals in $R$.

* Theorem <ref> on the dimension of the Betti cone $\Cn$ generated by the diagrams of edge ideals in $R$.

* Theorem <ref> on the dimension of the Betti cone $\Dnh$ generated by the diagrams of squarefree monomial ideals in $R$ of a given height $h$.

* Theorem <ref> on the dimension of the Betti cone $\Cnh$ generated by the diagrams of edge ideals in $R$ of a given height $h$.

While proving Theorems <ref> and <ref> we also demonstrate how they can be altered slightly to compute the dimensions of the cones $\Dntilde$ and $\Dnhtilde$, generated by diagrams squarefree monomial ideals of degree at least $2$ (equivalently, these are the diagrams corresponding to simplicial complexes with no missing vertices).

Our proofs for each of these theorems involve constructing the minimal vector spaces containing the cone. We construct these spaces in each case by compiling a list of restrictions that the diagrams in the cone must satisfy, most of which can be found in the literature. Hence, the first part of each of our proofs (the `upper bound' section) consists predominantly of a compilation of existing results. The new part of each theorem, and the bulk of each proof (the `lower bound' section), is that the vector spaces carved out by these restrictions are the minimal spaces containing their respective cones. This demonstrates that (up to linear combination) these restrictions represent the only linear relations satisfied by every diagram in the cone, and in that sense they are best possible.

While the auxiliary results in this chapter are presented primarily in pursuit of proving our four main theorems, the following may be of interest in their own right.

* Proposition <ref> shows that the regularity of a Stanley-Reisner ideal of height $h$ is at most $n-h+1$.Diane expects this is known already, and in retrospect I'm sure she's right! In fact, none of these four results were difficult to prove, so presumably all four of them are in fact known already. Maybe do a quick literature search to see if you can find any of the four of them, and consider deleting this - though also Diane says I can ignore this comment so it's not urgent

* Proposition <ref> shows that the regularity of an edge ideal of height $h$ is at most $\min \{h,n-h\}+1$.

* Lemma <ref> and Corollaries <ref> and <ref> detail the homology and Betti diagrams of various complexes obtained by the starring operation laid out in [30] (page 3).

* Proposition <ref> shows how the Betti diagram of a Stanley-Reisner ideal is altered by adding a number of isolated vertices to its corresponding complex.

§ CHAPTER <REF>: PR COMPLEXES AND DEGREE TYPES

This chapter begins our investigation into pure resolutions of squarefree monomial ideals by introducing the family of PR complexes – simplicial complexes $\Delta$ whose dual Stanley-Reisner ideals $\Idstar$ have pure Betti diagrams. We present an entirely combinatorial description of these complexes (Corollary <ref>) which follows from the Alexander Dual variant of Hochster's Formula, ADHF (Proposition 1 in [10]). While this description is an elementary consequence of ADHF, we have not yet come across a treatment of these complexes in the literature.

A central result of Eagon and Reiner in [10] (Theorem 3) is that the dual Stanley-Reisner ideal $I_\Delta^*$ of a complex $\Delta$ has a linear resolution if and only if the Stanley-Reisner ring $\KK[\Delta]$ is Cohen-Macaulay. Thus PR complexes are a generalisation of the family of Cohen-Macaulay complexes. Combinatorially speaking, Cohen-Macaulay complexes are defined as complexes whose links have homology only at the highest possible degree (this is Reisner's criterion, in e.g. [32] Theorem 5.53). The PR condition is a relaxation of the Cohen-Macaulay one, requiring only that the homology of each non-acyclic link is a function of the size of its corresponding face. Cohen-Macaulay complexes are the subject of a large amount of literature, ranging from largely algebraic investigations (see e.g. [26]) to almost entirely combinatorial ones (see e.g. [9]).

Of particular note in this chapter is the concept of maximal intersections in Section <ref>, particularly Proposition <ref>, which details the conditions under which the second column of the Betti diagram $\beta(\Idstar)$ is pure; and the link poset (Definition <ref>), which is a poset structure that can be imposed on the links of a simplicial complex to help us identify whether its dual Stanley-Reisner ideal admits a pure resolution.

§ CHAPTER <REF>: SOME FAMILIES OF PR COMPLEXES

This chapter presents a number of highly symmetric families of PR complexes, and proves that they satisfy the PR property. These include the following.

* The family of cycle complexes $\fC_{a,b}$ given in Definition <ref>; Theorem <ref> states that the complex $\fC_{a,b}$ is PR with degree type $(a,b)$.

* The family of intersection complexes $\calI(\bm)$ given in Definition <ref>; Theorems <ref> and <ref> state that these complexes are PR, and detail their degree types and Betti diagrams.

* The family of partition complexes $\calP(a,p,m)$ given in Definition <ref>; Theorem <ref> states that the complex $\calP(a,p,m)$ is PR with degree types of the form $(\overbrace{1,\dots,1,\underbrace{a,1,\dots,1}_{m}}^{p})$.

§ CHAPTER <REF>: PURE RESOLUTIONS OF ANY DEGREE TYPE

This chapter is devoted to the proof of our most significant result, which is the following.

* Theorem <ref> states that there exist squarefree monomial ideals with pure resolutions of any given degree type.

We define the degree type of a pure resolution explicitly in Chapter <ref> – the terminology is borrowed from [1] (page 1203), although it should be noted that our definition differs slightly from theirs. In essence, the degree type contains all the information of the shift type $(c_p,\dots,c_0)$ except for the value of the initial shift $c_0$. Thus, Theorem <ref> can be seen as a slightly weaker variant of the first Boij-Söderberg conjecture, for the cone generated by diagrams of squarefree monomial ideals.

In proving this result, we construct a number of operations on simplicial complexes which preserve the purity of the corresponding Betti diagrams, while altering the diagram's degree type.

* The operations $\{\phi_i : i\in \ZZ^+\}$ in Definition <ref> map PR complexes of degree type $(d_p,\dots,d_i,1,\dots,1)$ to PR complexes of degree type $(d_p,\dots,d_{i+1},d_i+1,1,\dots,1)$ (this is Theorem <ref>)

* The operations $f^\lambda$ and $f^{\free}$ in Definitions <ref> and <ref> have similar PR preserving properties, as shown in Corollaries <ref> and <ref>.

Our work in this chapter also involves a brief investigation in to the barycentric subdivision $\BDelta$ of arbitrary simplicial complexes $\Delta$. There are a number of results in the literature on the combinatorial effects of the barycentric subdivision process (for instance, [7] contains a number of results on its effects on $f$- and $h$-vectors, such as Theorems 2.2 and 3.1). The central results of Section <ref>, which we believe to be new, demonstrate the effects of the process on links, homologies of links, and both the PR and Cohen-Macaulay properties.

* Proposition <ref> is a description of the links in $\BDelta$ in terms of links in $\Delta$.

* Corollary <ref> is a description of the homologies of these links in terms of the homologies of the links in $\Delta$.

* Proposition <ref> is an equivalence statement about the conditions under which the barycentric subdivision of $\Delta$ is PR (in particular, this occurs if and only if $\Delta$ is Cohen-Macaulay).

§ CHAPTER <REF>: FUTURE DIRECTIONS

This final chapter is a brief account of possible future research directions that could be explored based on our main results. As such, it does not contain any substantial new results, but it presents a number of open questions and conjectures.

CHAPTER: BACKGROUND MATERIAL

We begin by presenting some preliminary results from the literature which we will need going forward.

Our most central tool, the Stanley-Reisner correspondence, provides a bridge between the fields of algebra and combinatorics; as such, this introductory chapter will be split into two sections: Results from Algebra and Results from Combinatorics. The former of these will include a brief exposition of Betti diagrams, before moving on to Boij-Söderberg theory, and detailing how this theory allows us to classify all Betti diagrams up to multiplication by an integer. The latter will describe the Stanley-Reisner Correspondence, as well as presenting some important preliminary results concerning simplicial complexes, and graph theory.

The latter section also reviews some key results from simplicial homology, but does not contain an overview of the topic; an overview can be found in [23] Chapter 2.1. All of the homology we consider in this thesis will be reduced homology over our arbitrary field $\KK$, and we use the notation $\Hred_i(\Delta)$ to denote the $\nth{i}$ reduced homology group of $\Delta$ with coefficients in $\KK$. None of our results depend upon the characteristic of $\KK$.

Wherever possible we define all notation we use; any exceptions to this can be found in the List of Symbols and Notation section at the end of the thesis.

§ RESULTS FROM ALGEBRA

§.§ Free Resolutions and Betti Diagrams

We fix a finitely generated graded $R$-module $M$ (this thesis is only concerned with graded modules, so from now on we will often simply use the term `module' to mean `graded module'). Consider a minimal free resolution of $M$, given by

\begin{equation}\label{Equation: Free Resolution}
0 \rightarrow R^{\beta_p}\rightarrow \dots \rightarrow R^{\beta_0} \rightarrow M \rightarrow 0 \, .
\end{equation}

The length $p$ of this resolution is equal to the projective dimension of $M$, $\pdim M$. By Hilbert's Syzygy Theorem ([11] Corollary 19.7), we have $\pdim M\leq n$.

We call the ranks $\beta_i$ of the free modules in this resolution the total Betti numbers of $M$, and often write them as $\beta_i(M)$.

Note that while, in general, $M$ will admit many distinct free resolutions, the Betti numbers of $M$ are an invariant. They also encode a lot of information about $M$. In particular, $\beta_0$ is the minimal size of a generating set for $M$, because the image of any basis of $R^{\beta_0}$ under the map $R^{\beta_0}\rightarrow M$ in (<ref>) generates $M$. Similarly, $\beta_1$ is equal to the minimal number of generators for the first syzygy module of $M$; and more generally, $\beta_i$ is equal to the minimal number of generators of the $\nth{i}$ syzygy module.

Informally, we can think of this as follows: $\beta_0$ tells us the number of generators of the module, $\beta_1$ tells us the number of relations between those generators, $\beta_2$ tells us the number of relations between those relations, and so on.

One significant limitation of total Betti numbers is that they contain no information about the grading of $M$. Ideally, we would like an invariant that encodes not just the number of generators of $M$ and its syzygy modules, but also the degrees of those generators.

To construct such an invariant we now consider a minimal graded free resolution of $M$, given by

\begin{equation}\label{Equation: Graded Free Resolution}
0\rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{p,d}} \rightarrow ... \rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{1,d}} \rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{0,d}}\rightarrow M \rightarrow 0
\end{equation}

where $R(-d)$ denotes the graded $R$-module with grading given by $[R(-d)]_a = R_{a-d}$ for $a\in \ZZ$. For each $i$, the integers $d$ for which $\beta_{i,d}\neq 0$ are called shifts (at homological degree $i$). The resolution (<ref>) is graded in the sense that the maps in the resolution send elements of degree $d$ in one module to elements of degree $d$ in the next. Note that the nonzero entries in the matrices representing these maps all have positive degree.

We call the exponents $\beta_{i,d}$ in (<ref>) the graded Betti numbers of $M$ (or in this thesis, usually simply the Betti numbers of $M$). We often write them as $\beta_{i,d}(M)$.

Suppose $n=4$. The squarefree monomial ideal $I$ of $R$ generated by $x_1x_4$ and $x_1x_2x_3$ admits the following minimal graded free resolution.

$$\begin{tikzcd}[ampersand replacement=\&]
0 \rar \& R(-4) \rar{ \begin{pmatrix}
x_2x_3 \\ -x_4
\end{pmatrix} }\&[1em] R(-2)\oplus R(-3) \rar{ \begin{pmatrix}
x_1x_4 & x_1x_2x_3
\end{pmatrix} } \&[3em] I\rar \& 0.
\end{tikzcd}$$

The nonzero Betti numbers of $I$ are $\beta_{0,2}(I)=\beta_{0,3}(I)=\beta_{1,4}(I)=1$.

Just as the total Betti number $\beta_0(M)$ is equal to the minimum number of generators of $M$, the graded Betti number $\beta_{0,d}(M)$ is equal to the minimum number of generators of $M$ of degree $d$. Similarly $\beta_{i,d}(M)$ is equal to the minimum number of generators of the $\nth{i}$ syzygy module of $M$ of degree $d$.

The total Betti numbers and graded Betti numbers of $M$ are related via the equations

$$\beta_i = \sum_{d\in \ZZ} \beta_{i,d} \text{ for each } 0\leq i \leq n\, .$$

The Betti diagram of $M$, denoted $\beta(M)$, is a matrix containing the graded Betti numbers of $M$. In order to reduce the number of rows of this matrix, the standard notational convention (which we adopt for this thesis) is to write $\beta(M)$ as a matrix $(a_{ij})$ with $a_{ij}=\beta_{j,i+j}$, as shown.

\vdots& \vdots& &\vdots \\
\beta_{0,0} & \beta_{1,1} & \dots & \beta_{n,n} \\
\beta_{0,1} & \beta_{1,2} & \dots & \beta_{n,n+1}\\
\vdots & \vdots & & \vdots\\
\end{bmatrix}$$

Note that while this matrix is infinite, only finitely many of the entries are nonzero.

We also occasionally write Betti diagrams in table notation as follows.
Suppose $\beta$ is a Betti diagram of an $R$-module. We can write $\beta$ as a table in the following way.

$$\begin{array}{c | cccc}
& 0 & 1 & \dots & n\\
\hline
\vdots&\vdots&\vdots&\vdots&\vdots
\end{array}$$

The table $\begin{array}{c | ccc}
& 0 & 1 & 2\\
\hline
4 & 8 & 9 & 1\\
5 & . & . & 1\\
\end{array}$ denotes a Betti diagram $\beta$ for which $\beta_{0,4}=8$, $\beta_{1,5}=9$, $\beta_{2,6}=\beta_{2,7}=1$, and all other Betti numbers are zero.

Just as the projective dimension of a module can be seen as the “length” of its Betti diagram, the following property can be seen as the Betti diagram's “width”.

Let $M$ be an $R$-module with Betti diagram $\beta(M)$. The (Castelnuovo-Mumford) regularity of $M$ is given by $\reg(M)=\max\{d-i: \beta_{i,d}(M)\neq 0\}$.

We have a number of important constraints on the Betti numbers of $R$-modules. The following is perhaps one of the most famous, and the most important for our work going forward. It holds for all $R$-modules of a given codimension $h$ (for a proof, see e.g. [17] Section 1.3).

Let $M$ be a graded $R$-module of codimension $h$, and let $\beta=\beta(M)$. We have

\begin{equation*}
\sum_{i,d}(-1)^i d^j \beta_{i,d} = 0 \text{ for each } 0\leq j\leq h-1.
\end{equation*}

For convenience, for each $0\leq j\leq h-1$, we use the notation $\HK_j(\beta)$ to represent the expression $\sum_{i,d}(-1)^i d^j \beta_{i,d}$. In this notation, the Herzog-Kühl equations for modules of codimension $h$ can be rephrased as the statement $\HK_j(\beta)=0$ for each $0\leq j\leq h-1$.

All of the $R$-modules we care about in this thesis are ideals, so we end this section with two key observations about the Betti diagrams of ideals.

Let $I$ be an ideal in $R$ of height $h$. This means that the quotient $R/I$ has codimension $h$, and therefore, setting $M=R/I$ in Lemma <ref> gives us $\HK_j(\beta(R/I))=0$ for each $0\leq j \leq h-1$. We can use this to find similar linear dependency relations for the diagram $\beta(I)$.

Specifically, if

$$0\rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{p,d}} \rightarrow ... \rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{1,d}} \rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{0,d}}\rightarrow I \rightarrow 0$$

is a minimal graded free resolution of $I$, then

$$0\rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{p,d}} \rightarrow ... \rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{1,d}} \rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{0,d}}\rightarrow R(0) \rightarrow R/I \rightarrow 0$$

is a minimal graded free resolution of $R/I$.

This means that the Betti diagrams of $R$ and $R/I$ are directly related via

\begin{align*}
\beta_{i,d}(R/I)=\begin{cases}
\beta_{i-1,d}(I) &\text{ if } i > 0\\
1  &\text{ if } (i,d)=(0,0) \\
0 & \text{ otherwise.}
\end{cases}
\end{align*}

Hence, for any $j\geq 0$, we have

\begin{align*}
\HK_j(\beta(R/I)) &= \sum_{i,d}(-1)^i d^j \beta_{i,d}(R/I)\\
&= 0^j + \sum_{i,d}(-1)^i d^j \beta_{i-1,d}(I)\\
&= 0^j - \sum_{i,d}(-1)^i d^j \beta_{i,d}(I)\\
1 - \HK_j(\beta(I)) & \text{ if } j=0\\
- \HK_j(\beta(I))  & \text{ otherwise.}
\end{cases}
\end{align*}

Thus, for any ideal $I$ of height $h$, the Herzog-Kühl equations tell us that $\HK_0(\beta(I))=1$ and $\HK_j(\beta(I))=0$ for each $1\leq j\leq h-1$.

Suppose $I$ is an ideal in $R$, and let

$$F_\bullet: 0\rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{p,d}} \rightarrow ... \rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{1,d}} \rightarrow \bigoplus_{d\in \ZZ} R(-d)^{\beta_{0,d}}\rightarrow I \rightarrow 0$$

be a minimal free graded resolution for $I$ as an $R$-module.

Let $S$ denote the polynomial ring $R[x_{n+1}]$, and consider the ideal $IS$ in $S$. Because $S$ is a free $R$-module, it is also faithfully flat, and hence
$$F_\bullet\otimes_R S: 0\rightarrow \bigoplus_{d\in \ZZ} S(-d)^{\beta_{p,d}} \rightarrow ... \rightarrow \bigoplus_{d\in \ZZ} S(-d)^{\beta_{1,d}} \rightarrow \bigoplus_{d\in \ZZ} S(-d)^{\beta_{0,d}}\rightarrow IS \rightarrow 0$$
is a minimal free graded resolution for $IS$ as an $S$-module.

In particular, any generating set for $I$ as an $R$-ideal is also a generating set for $IS$ as an $S$-ideal. Hence the Betti diagram of $I$ is dependent only on its generators, and not on the number of variables in its ambient polynomial ring.

§.§ Boij-Söderberg Theory and Betti Cones

One of the most significant advances in our understanding of Betti Diagrams came from a paper put on the Arxiv by Boij and Söderberg (and later published in 2008 as [5]), which laid out two important conjectures about the Betti diagrams of Cohen-Macaulay modules, both of which have since been proven. In this section we give a brief exposition of these two conjectures, and the surrounding theory; a more comprehensive overview can be found in [17] Section 1.

The aim of [5] was a classification result for Betti diagrams of $R$-modules; but instead of trying to classify all such Betti diagrams directly, the authors set themselves the task of classifying them up to multiplication by a positive rational (or equivalently, by a positive integer, because all of the diagrams have integer values). Their central insight was that any Betti diagram of an $R$-module may be viewed as a vector in the infinite-dimensional rational vector space $\Vn=\bigoplus_{d\in \ZZ} \QQ^{n+1}$, and thus their goal was to classify the positive rational rays spanned by the diagrams in this space.

Note in particular that for any two $R$-modules $M_1$ and $M_2$, and any two positive integers $q_1$ and $q_2$, we have $q_1 \beta(M_1)+q_2 \beta(M_2)=\beta(M_1^{q_1}\oplus M_2^{q_2})$. It follows that the set of positive rational rays spanned by the Betti diagrams in $\Vn$ form a cone. The task, therefore, was to describe the cone.

The key insight of Boij and Söderberg was that any Betti diagram of an $R$-module may be viewed as a vector in the infinite-dimensional rational vector space $\Vn=\bigoplus_{d\in \ZZ} \QQ^{n+1}$. Thus, instead of studying Betti diagrams themselves, we can broaden our attention to the rational cone generated by these diagrams.

Note in particular that for any two Betti diagrams $M_1$ and $M_2$, and any two integers $q_1$ and $q_2$, we have $q_1 \beta(M_1)+q_2 \beta(M_2)=\beta(M_1^{q_1}\oplus M_2^{q_2})$. It follows that every ray in this cone contains the Betti diagram of some $R$-module, and thus classifying the rays in this cone allows us to classify the Betti diagrams of all $R$-modules up to multiplication by a positive integer.

Boij and Söderberg initially restricted their attention to cones generated by diagrams of Cohen-Macaulay $R$-modules of a given codimension $h$, for some fixed $h\leq n$.  Recall that Cohen-Macualay modules are defined in terms of the Krull-dimension and depth of a module as follows (see [2] Section 2.1 for more details).

An $R$-module $M$ is Cohen-Macaulay if we have $\dim M = \depth M$.

We can find the projective dimensions of these modules from the Auslander-Buchsbaum formula for graded rings (see e.g. [11] Exercise 19.8), which is the following.

Let $S$ be a graded ring and let $M$ be a finitely generated graded $S$-module of finite projective dimension. We have
$$\pdim M = \depth S - \depth M.$$

Let $M$ be a finitely generated graded $R$-module. We have
$$\pdim M = n - \depth M.$$

The polynomial ring $R$ is Cohen-Macaulay with depth and Krull-dimension both equal to $n$, and all finitely generated $R$-modules have finite projective dimension.

In the case where $M$ is a Cohen-Macaulay $R$-module of codimension $h$ we have
$$\pdim M = n - \dim M=\codim M = h.$$
More generally, if $M$ is an arbitrary $R$-module we have $\depth M \leq \dim M$, and hence $\pdim M \geq \codim M$. Thus the diagrams of Cohen-Macaulay modules are the ones with minimal length for their given codimension.

The cone generated by all Cohen-Macaulay $R$-modules of codimension $h$ is infinite-dimensional, which makes it difficult to work with. For this reason, Boij and Söderberg restricted their attention further to finite dimensional subcones generated by Betti diagrams within a given window. We lay out how to build these cones below.

For the rest of this section we fix two strictly decreasing sequences of integrs $\ba =(a_h,\dots,a_0)$ and $\bfb=(b_h,\dots,b_0)$ in $\ZZ^{h+1}$, such that $a_i\leq b_i$ for each $0\leq i \leq h$. Note that this condition imposes a partial order on the sequences in $\ZZ^{h+1}$, which we will henceforth denote by $\ba \leq \bfb$.

We define

* $\II(\ba,\bfb)=\left\{(i,d)\in \{0,\dots,h\}\times \ZZ : a_i\leq d \leq b_i \right \}$.

* $\calV(\ba,\bfb)= \bigoplus_{(i,d)\in \II(\ba,\bfb)} \QQ \subset \Vn$.

* $\calC(\ba,\bfb)$ to be the cone of positive rational rays $t \cdot\beta$ for Betti diagrams $\beta$ of Cohen-Macaulay $R$-modules of codimension $h$ which lie inside the subspace $\calV(\ba,\bfb)$.

Boij and Söderberg's key aim was to find the extremal rays of the cone $\calC(\ba,\bfb)$. These are vectors $v_1,\dots, v_m$ in the cone such that

* Every vector in the cone can be written as a positive rational sum of the form $\lambda_1 v_1 + \dots \lambda_m v_m$.

* For each $1\leq i \leq m$ there do not exist linearly independent vectors $u$ and $w$ in the cone such that $v_i = u + w$.

It turns out that these extremal rays come from so-called pure Betti diagrams, defined as follows.

Let $M$ be an $R$-module. We say the Betti diagram $\beta(M)$ is pure if for every $0\leq i \leq n$ there is at most a single value of $c$ such that the diagram $\beta_{i,c}(M)$ is nonzero.

In this case, $M$ admits a graded resolution of the form
$$0\rightarrow R(-c_p)^{\beta_{p,c_p}} \rightarrow ... \rightarrow R(-c_1)^{\beta_{1,c_1}} \rightarrow R(-c_0)^{\beta_{0,c_0}}\rightarrow M.$$
We call such a resolution a pure resolution, and we say it has shift type $(c_p,..., c_0)$ (after the terminology of [1] page 1203).

Let $\bc=(c_h,\dots,c_0)$ be a strictly decreasing sequence such that $\ba \leq \bc \leq \bfb$. Any pure Betti diagram $\beta$ of shift type $\bc$ corresponding to a Cohen-Macaulay module of codimension $h$ must satisfy the Herzog-Kühl equations $\HK_0(\beta)=\dots = \HK_{h-1}(\beta)=0$. This gives us a system of $h$ linearly independent equations in the $h+1$ variables $\beta_{0,c_0},\dots,\beta_{h,c_h}$, and hence there is only a single ray of solutions in $\Vn$. We denote the smallest integer solution on this ray by $\pi(\bc)$, and we can compute it as follows (see [5], Section 2.1).

Let $\pi=\pi(\bc)\in \Vn$ be the smallest pure integer solution to the Herzog-Kühl equations $\HK_0(\beta)=\dots=\HK_{h-1}(\beta)=0$. We have
$$\pi_{i,c_i}=\Pi_{0\neq k \neq i} \frac{c_k - c_0}{c_k - c_i}.$$

To show that the rays spanned by these pure diagrams are the extremal rays of the cone $\calC(\ba,\bfb)$, we need to prove two things. First, we need to show that they actually lie in the cone: that is, for every decreasing sequence $\bc$ in $\ZZ^{h+1}$ with $\ba\leq \bc \leq \bfb$ the ray $t\cdot \pi(\bc)$ actually contains a Betti diagram of a Cohen-Macaulay module of codimension $h$ (this is not immediate from the above discussion; Proposition <ref> shows that any pure diagram of shift type $\bc$ corresponding to a Cohen-Macaulay module of height $h$ must lie on the ray $t\cdot \pi(\bc)$ if it exists, but does not guarantee the existence of such a diagram).

Second, we must show that the diagrams $\pi(\bc)$ satisfy the conditions of extremal rays. It is easy to see that $\pi(\bc)$ cannot be decomposed into the sum of two linearly independent vectors in the cone (indeed, if $\pi(\bc)=\beta^1+\beta^2$ then $\beta^1$ and $\beta^2$ must also lie on the ray $t\cdot \pi(\bc)$ by considering the position of zero entries), so what remains to show is that the diagram of any Cohen-Macaulay module of codimension $h$ can be decomposed into a rational sum of pure diagrams.

The two claims above are the Boij-Söderberg conjectures (Theorems 1.9 and 1.11 in [17]). The first conjecture was subsequently proven by Eisenbud, Fløystad and Weyman in [12],and the second by Eisenbud and Schreyer in [14].

For any strictly decreasing sequence of integers $\bc=(c_h,\dots,c_0)$, there exists a Cohen-Macaulay graded $R$-module $M$ of codimension $h$ with a pure resolution of shift type $\bc$.

Let $M$ be a Cohen-Macaulay graded $R$-module of codimension $h$ whose Betti diagram lies inside $\calV(\ba,\bfb)$. There exist positive rational numbers $q_i$ and a chain of decreasing sequences $\bc^1 < \dots < \bc^r$ in $\ZZ^{h+1}$ such that

$$\beta(M) = q_1 \pi(\bc^1)+\dots + q_r \pi (\bc^r).$$

Boij and Söderberg later extended this second result to the non-Cohen-Macaulay case ([6] Theorem 2). In order to state this revised result, we will use the notation $\ZZ^{\leq n+1}$ to denote the set of integer sequences of length at most $n+1$. We can extend our earlier partial order on $\ZZ^{h+1}$ to this set by stipulating that $\ba \leq \bfb$ for sequences $\ba=(a_{p_1},\dots,a_0)$ and $\bfb=(b_{p_2},\dots,b_0)$ in $\ZZ^{\leq n+1}$ whenever $p_1\geq p_2$ and $a_i\leq b_i$ for each $0\leq i \leq p_1$ .

Let $M$ be an arbitrary $R$-module. There exist positive rational numbers $q_i$ and a chain of decreasing sequences $\bc^1<\dots<\bc^r$ in $\ZZ^{\leq n+1}$ such that
$$\beta(M) = q_1 \pi(\bc^1)+\dots + q_r \pi (\bc^r).$$

Together, Theorem <ref> and Theorem <ref> provide a complete classification of all Betti diagrams of $R$-modules up to multiplication by a positive integer. Theorem <ref> also allows us to classify the vector space spanned by the cone $\calC(\ba,\bfb)$.

We define

* $\calV^{\HK }(\ba,\bfb)= \left\{ \beta \in \calV(\ba,\bfb) : \HK_0(\beta)=\dots=\HK_{h-1}(\beta)=0 \right\}$.

* $\WW(\ba,\bfb)$ to be the subspace of $\calV(\ba,\bfb)$ generated by the diagrams in $\calC(\ba,\bfb)$.

The former space $\calV^{\HK }(\ba,\bfb)$ has a basis consisting of pure diagrams $\pi(\bc)$ (this is Proposition 1.8 in [17]).

For any maximal chain $\ba = \bc^1 < \dots < \bc^r = \bfb$ in $\ZZ^{h+1}$, the diagrams $\pi(\bc^1),\dots, \pi(\bc^r)$ form a basis for $\calV^{\HK }(\ba,\bfb)$.

The latter space $\WW(\ba,\bfb)$ must be a subspace of $\calV^{\HK }(\ba,\bfb)$, because all of its generating Betti diagrams satisfy the Herzog-Kühl Equations. In fact, it is a consequence of the first Boij-Söderberg conjecture that these two spaces are equal (this is Corollary 1.10 in [17]).

The spaces $\WW(\ba,\bfb)$ and $\calV^{\HK}(\ba,\bfb)$ are equal.

By Theorem <ref>, the space $\WW(\ba,\bfb)$ contains all the diagrams $\pi(\bc)$ for strictly decreasing sequences $\bc$ in $\ZZ^{h+1}$ with $\ba \leq \bc \leq \bfb$, and these diagrams generate $\calV^{\HK}(\ba,\bfb)$ by Proposition <ref>

The key results of this thesis (Theorems <ref>, <ref>,  <ref> and <ref> im Chapter <ref>, and Theorem <ref> in Chapter <ref>) can be seen as analogues of either the first Boij-Söderberg conjecture or its above corollary, for Betti cones generated exclusively by diagrams of squarefree monomial ideals.

In order to study the diagrams of squarefree monomial ideals, we must now delve into the world of combinatorics.

§ RESULTS FROM COMBINATORICS

§.§ Simplicial Complexes and Stanley-Reisner Ideals

Simplicial complexes are important combinatorial and topological objects, defined as follows.

A simplicial complex $\Delta$ on vertex set $V$ is a set of subsets of $V$ such that for any $G \subseteq F \subseteq V$, if $F$ is in $\Delta$ then $G$ is also in $\Delta$.

We often denote the vertex set of a complex $\Delta$ as $V(\Delta)$.

For our purposes the vertex set $V$ will always be finite.

Let $\Delta$ be a simplicial complex on a vertex set $V$ of size $n$.

* We refer to the elements of $\Delta$ as simplices or faces.

* A facet of $\Delta$ is a face of $\Delta$ that is maximal with respect to inclusion.

* A minimal nonface of $\Delta$ is a subset of $V$ that is not a face of $\Delta$ and is minimal with this property with respect to inclusion.  Note that $\Delta$ need not contain every singleton subset of $V$. A singleton minimal nonface of $\Delta$ is called a missing vertex.

* A vertex in $V$ which is contained in only one facet of $\Delta$ is called a free vertex.

* The dimension of a face $\sigma$ in $\Delta$, denoted $\dim \sigma$, is $|\sigma|-1$. The dimension of $\Delta$, denoted $\dim \Delta$, is the dimension of its largest facet. If all facets of $\Delta$ have the same dimension, we say $\Delta$ is pure.

* The codimension of $\Delta$, denoted $\codim \Delta$, is $n-\dim \Delta -1$.

* For a face $\sigma\in \Delta$ we define the link of $\sigma$ in $\Delta$ to be the complex
$$\lkds=\{\tau \in \Delta : \sigma \cap \tau = \emptyset, \sigma \cup \tau \in \Delta\}.$$

* We define the Alexander Dual of $\Delta$ to be the complex
$$\Delta^* = \{F \subseteq V : V - F \notin \Delta\}.$$

* For two complexes $\Delta_1$ and $\Delta_2$ on disjoint vertex sets $V_1$ and $V_2$ we define the join of $\Delta_1$ and $\Delta_2$ on vertex set $V_1 \sqcup V_2$ as
$$ \Delta_1 \ast \Delta_2 = \{\sigma_1 \sqcup \sigma_1 : \sigma_1 \in \Delta_1, \sigma_2 \in \Delta_2 \}.$$

* The join of $\Delta$ with a single vertex is called the cone over $\Delta$. We denote it by  $C\Delta$.

* The join of $\Delta$ with two disjoint vertices is called the suspension of $\Delta$. We denote it by $S\Delta$.

We often write simplicial complexes in facet notation, as follows.

For subsets $F_1$,...,$F_m$ in a vertex set $V$, we use $\langle F_1,...,F_m\rangle$ to denote the smallest simplicial complex on $V$ containing $F_1$,...,$F_m$. Note that if the set $\{F_1,...,F_m\}$ is irredundant (i.e. no pair of subsets $F_i$ and $F_j$ satisfy $F_i \subseteq F_j$), then this is the same as the complex on $V$ with facets $F_1,...,F_m$.

Suppose $\sigma$ is a face in a simplicial complex $\Delta$, and $F_1,\dots,F_m$ are all the facets of $\Delta$ which contain $\sigma$. We have $\lkds = \langle F_1 -\sigma, \dots, F_m- \sigma \rangle$.

Suppose $\Delta$ is the complex  $\langle \{1,2,3\},\{3,4\} \rangle$ on vertex set $[4]$. We can draw $\Delta$ as follows.

[scale = 0.75]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]

(a)[point,label=left:$1$] at (0,1) ;
(b)[point,label=left:$2$] at (0,-1) ;
(c)[point,label=above:$3$] at (2,0) ;
(d)[point,label=above:$4$] at (4,0) ;

[on background layer]
[fill=gray] (a.center) – (b.center) – (c.center) – cycle;
[fill=gray] (c.center) – (d.center);

* This complex has dimension $2$, but it is not pure because it also has a facet of dimension $1$.

* The minimal nonfaces of $\Delta$ are $\{1,4\}$ and $\{2,4\}$, and hence the Alexander dual $\Delta^*$ has facets $[4]-\{1,4\}=\{2,3\}$ and $[4]-\{2,4\}=\{1,3\}$.

* The cone $C\Delta$ looks like the following.

$$\begin{tikzpicture}[scale=1][line join = round, line cap = round]

\coordinate [label=above:5] (5) at ({.5*sqrt(3)},{sqrt(2)},-.5);
\coordinate [label=left:1] (1) at ({-.5*sqrt(3)},0,-.5);
\coordinate [label=below:2] (2) at (0,0,1);
\coordinate [label=below:3] (3) at ({.5*sqrt(3)},0,-.5);
\coordinate [label=right:4] (4) at ({1.5*sqrt(3)},0,-1.3);

\begin{scope}
\draw (1)--(3);
\draw[fill=darkgray,fill opacity=.5] (2)--(1)--(5)--cycle;
\draw[fill=gray,fill opacity=.5] (3)--(2)--(5)--cycle;
\draw[fill=gray,fill opacity=.5] (3)--(4)--(5)--cycle;
\draw (2)--(1);
\draw (2)--(3);
\draw (3)--(5);
\draw (2)--(5);
\draw (1)--(5);
\draw (3)--(4);
\end{scope}
\end{tikzpicture}$$

[scale = 1][line join = round, line cap = round]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]

[label = left:$3$] (3) at (-.5*sqrt(3),0,-.5);
[label =below left:$2$] (2) at (.2,0,1) ;
[label =above:$1$](1) at (.5*sqrt(3),0,-.5);

[label = $4$] (a) at (2,0,-) ;

[label = $5$] (4) at (0,.8*sqrt(2),0);
[label =below:$6$] (5) at (0,-.8*sqrt(2),0);

[fill=gray,fill opacity=.5] (2)–(1)–(4)–cycle;
[fill=darkgray,fill opacity=.5] (3)–(2)–(4)–cycle;

[fill=gray,fill opacity=.5] (2)–(1)–(5)–cycle;
[fill=darkgray,fill opacity=.5] (3)–(2)–(5)–cycle;

[fill=gray, fill opacity=.5] (4) – (a) –(1);
[fill=gray, fill opacity =.5] (5) – (a) –(1);

We have the following result about Alexander Duals.

Let $\Delta$ be a simplicial complex. We have $(C\Delta)^*=C(\Delta^*)$ (i.e. the operations of coning commutes with taking Alexander Duals).

Let $V$ be the vertex set of $\Delta$ and $\tV=V\cup \{v\}$ be the vertex set of $C\Delta$ for some additional vertex $v$. A subset $F\subseteq \tV$ is a facet of $(C\Delta)^*$ if and only if $\tV-F$ is a minimal nonface of $C\Delta$. Note that the minimal nonfaces of $C\Delta$ are the same as the minimal nonfaces of $\Delta$, none of which contain $v$. Thus the facets of $(C\Delta)^*$ are precisely those sets of the form $F=G\cup \{v\}$ for some subset $G\subseteq V$ such that $V-G$ is a minimal nonface of $\Delta$. In other words, the facets of $(C\Delta)^*$ are the sets $G\cup \{v\}$ for which $G$ is a facet of $\Delta^*$. The result follows.

We now review some key subcomplex constructions.

Let $\Delta$ be a simplicial complex on vertex set $V$.

* For a subset $U\subset V$, we define the induced subcomplex of $\Delta$ on $U$ to be the complex
$$\Delta_U=\{\sigma \in \Delta : \sigma \subseteq U\}.$$

* For a face $g$ in $\Delta$ we define the deletion of $g$ from $\Delta$ to be the complex
$$\Delta-g = \{\sigma \in \Delta : \sigma \nsupseteq g \}.$$

We will sometimes denote the induced subcomplex $\Delta_U$ as $\Delta|_U$ to avoid ambiguity.

The complex $\Delta - g$ is the largest subcomplex of $\Delta$ that does not contain $g$. Note that, except in the case where $g$ is a vertex, this is larger than the induced subcomplex $\Delta_{V-g}$, because it contains all faces of $\Delta$ which intersect with $g$ strictly along its boundary. The following example illustrates the difference between the two constructions.

Let $\Delta$ be the complex

[scale = 0.75]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=3pt,minimum height=3pt]
(a)[point,label=above:$1$] at (0,3.5) ;
(b)[point,label=above:$2$] at (2,3) ;
(c)[point,label=above:$3$] at (4,3.5) ;
(d)[point,label=left:$4$] at (1.3,1.8) ;
(e)[point,label=right:$5$] at (2.7,1.8) ;
(f)[point,label=left:$6$] at (2,0) ;

[on background layer]
[fill=gray] (a.center) – (b.center) – (d.center) – cycle;
[fill=gray] (b.center) – (c.center) – (e.center) – cycle;
[fill=gray]   (d.center) – (e.center) – (f.center) – cycle;

on vertex set $[6]$, and let $g$ denote the edge {2,4}.

The complex $\Delta - g$ is equal to

[scale = 0.75]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=3pt,minimum height=3pt]
(a)[point,label=above:$1$] at (0,3.5) ;
(b)[point,label=above:$2$] at (2,3) ;
(c)[point,label=above:$3$] at (4,3.5) ;
(d)[point,label=left:$4$] at (1.3,1.8) ;
(e)[point,label=right:$5$] at (2.7,1.8) ;
(f)[point,label=left:$6$] at (2,0) ;

[on background layer]
(d.center) – (a.center) – (b.center);
[fill=gray] (b.center) – (c.center) – (e.center) – cycle;
[fill=gray]   (d.center) – (e.center) – (f.center) – cycle;

whereas the induced subcomplex $\Delta_{[6]-g}$ is equal to

[scale = 0.75]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=3pt,minimum height=3pt]
(a)[point,label=above:$1$] at (0,3.5) ;
(c)[point,label=above:$3$] at (4,3.5) ;
(e)[point,label=right:$5$] at (2.7,1.8) ;
(f)[point,label=left:$6$] at (2,0) ;

[on background layer]
(c.center) – (e.center);
(e.center) – (f.center);

We now review some examples of important complexes, which will be crucial to our work going forward.

There are two trivial simplicial complexes on any vertex set.

* The irrelevant complex is the set $\{\emptyset\}$.

* The void complex is the empty set $\emptyset$.

Note that the irrelevant complex and the void complex are not the same complex. The former has a single face, of dimension $-1$, while the latter has no faces at all. In particular, the former complex has $\nth[st]{(-1)}$ reduced homology, while the latter is acyclic.

Let $j\geq -1$ be an integer, and let $V$ be a vertex set of size $j+1$.

* The full simplex on vertex set $V$ is the complex whose faces are every subset of $V$. We often call $\Delta$ the $j$-simplex, and denote it by $\Delta^j$.

* The boundary of the simplex on vertex set $V$ is the complex whose faces are every proper subset of $V$. We often call $\Delta$ the boundary of the $j$-simplex, and denote it by $\partial \Delta^j$.

Technically speaking, the $j$-simplex and its boundary are actually isomorphsim classes of complexes rather than individual complexes, because they can be defined on any vertex set of size $j+1$. By convention, we will assume they are defined on vertex set $[j+1]$ unless otherwise stated.

Let $r$ be an integer greater than or equal to $-1$, and let $\Delta$ be any complex. We define the $r$-skeleton of $\Delta$ to be the complex

$$\Skel_r(\Delta) = \{\sigma \in \Delta : \dim \sigma \leq r\}$$

If $\Delta$ is a full simplex on vertex set $V$ we often denote this as $\Skel_r(V)$.

* The $j$-simplex $\Delta^j$ is equal to $\Skel_j([j+1])$.

* The boundary of the $j$-simplex $\partial \Delta^j$ is equal to $\Skel_{j-1}([j+1])$.

* For any complex $\Delta$ and any $r\geq \dim \Delta$ we have $\Skel_r(\Delta) = \Delta$.

* For any complex $\Delta$ we have $\Skel_{-1}(\Delta)$ is the irrelevant complex $\{\emptyset\}$ on $V(\Delta)$, and for $r<-1$ we have that $\Skel_{r}(\Delta)$ is the void complex $\emptyset$.

The $3$-simplex $\Delta^3$ on vertex set $[4]$ has the following nontrivial skeleton complexes.

[line join = round, line cap = round]
[label=above:4] (4) at (0,sqrt(2),0);
[label=left:3] (3) at (-.5*sqrt(3),0,-.5);
[label=below:2] (2) at (0,0,1);
[label=right:1] (1) at (.5*sqrt(3),0,-.5);

[fill=darkgray,fill opacity=1] (2)–(1)–(4)–cycle;
[fill=black,fill opacity=1] (3)–(2)–(4)–cycle;

[line join = round, line cap = round]

[label=above:4] (4) at (0,sqrt(2),0);
[label=left:3] (3) at (-.5*sqrt(3),0,-.5);
[label=below:2] (2) at (0,0,1);
[label=right:1] (1) at (.5*sqrt(3),0,-.5);

[fill=lightgray,fill opacity=.5] (2)–(1)–(4)–cycle;
[fill=gray,fill opacity=.5] (3)–(2)–(4)–cycle;

[line join = round, line cap = round]

[label=above:4] (4) at (0,sqrt(2),0);
[label=left:3] (3) at (-.5*sqrt(3),0,-.5);
[label=below:2] (2) at (0,0,1);
[label=right:1] (1) at (.5*sqrt(3),0,-.5);

[line join = round, line cap = round]

point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(4) [point,label=above:4] at (0,sqrt(2),0) ;
(3) [point,label=left:3] at (-.5*sqrt(3),0,-.5) ;
(2) [point,label=below:2] at (0,0,1) ;
(1) [point,label=right:1] at (.5*sqrt(3),0,-.5) ;

$\Skel_3(\Delta^3)=\Delta^3$    $\Skel_2(\Delta^3) = \partial \Delta^3$    $\Skel_1(\Delta^3)$     $\Skel_0(\Delta^3)$

The darker shading of the image of $\Skel_3(\Delta^3)$ indicates that this complex contains the $3$-dimensional face $\{1,2,3,4\}$, while the lighter shading for $\Skel_2(\Delta^3)$ indicates that this complex only contains the bounding $2$-simplices of the face $\{1,2,3,4\}$.

The $d$-dimensional cross polytope (also known as the $d$-dimensional orthoplex) is the simplicial complex $O^d$, defined recursively for $d\geq -1$ as follows:

\begin{align*}
O^{-1} &= \{\emptyset\}\\
O^{d+1}&= S(O^d).
\end{align*}

The cross polytopes of dimensions $-1$, $0$, $1$ and $2$ are shown below.

point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point] at (0,0) ;
(b)[point] at (1,0) ;

$O^{-1}$            $O^0$

(Irrelevant complex)             (Disjoint vertices)

point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point] at (1,0) ;
(b)[point] at (0,1) ;
(c)[point] at (1,2) ;
(d)[point] at (2,1) ;

(a.center) – (b.center) – (c.center) – (d.center) – cycle;

[line join=bevel,z=-5.5]
(A1) at (0,0,-1);
(A2) at (-1,0,0);
(A3) at (0,0,1);
(A4) at (1,0,0);
(B1) at (0,1,0);
(C1) at (0,-1,0);

(A1) – (A2) – (B1) – cycle;
(A4) – (A1) – (B1) – cycle;
(A1) – (A2) – (C1) – cycle;
(A4) – (A1) – (C1) – cycle;
[fill opacity=0.7,fill=lightgray!80!gray] (A2) – (A3) – (B1) – cycle;
[fill opacity=0.7,fill=gray!70!lightgray] (A3) – (A4) – (B1) – cycle;
[fill opacity=0.7,fill=darkgray!30!gray] (A2) – (A3) – (C1) – cycle;
[fill opacity=0.7,fill=darkgray!30!gray] (A3) – (A4) – (C1) – cycle;

$O^1$            $O^2$

(Square)             (Octahedron)

Having reviewed some key examples of simplicial complexes, we now move on to the Stanley-Reisner correspondence. For any simplicial complex on vertex set $[n]$, we may assign a unique corresponding squarefree monomial ideal.

Let $\Delta$ be a simplicial complex on vertex set $[n]$. We define the Stanley-Reisner ideal of $\Delta$ to be
$$I_\Delta=\langle \mathbf{x}^\sigma : \sigma \subseteq [n], \sigma \notin \Delta \rangle$$

where $\mathbf{x}^\sigma$ denotes the squarefree monomial $\prod_{i\in \sigma}x_i$ in $R$.

We define the Stanley-Reisner ring of $\Delta$ to be

We also extend this definition to complexes $\Delta$ on arbitrary vertex set $V$ of size $n$ by first choosing an ordering on $V$, which allows us to define an isomorphism from $\Delta$ to a complex on $[n]$. Technically this makes the Stanley-Reisner ideal dependent on the specific ordering we choose, but we still use the phrase `the' Stanley-Reisner ideal anyway, because all possible orderings give us the same ideal up to isomorphism.

For a simplicial complex $\Delta$ the minimal generators of $I_\Delta$ correspond to the minimal nonfaces of $\Delta$. In particular, note that every squarefree monomial ideal is uniquely determined by its minimal generators, and similarly every simplicial complex is uniquely determined by its minimal nonfaces. Thus, the assignment $\Delta \mapsto I_\Delta$ gives us a one-to-one correspondence from simplicial complexes on $[n]$ to squarefree monomial ideals in $R$ (this is Theorem 1.7 in [32]).

The assignment $\Delta \mapsto I_\Delta$ gives a bijection
$$\{\text{Simplicial complexes on } [n]\}\isomto \{\text{Squarefree monomials in } R\}.$$

This one-to-one correspondence allows us to reframe algebraic properties of Stanley-Reisner ideals and rings in terms of combinatorial properties of their corresponding complexes, and vice versa. For example, we can compute the Krull-dimension of the ring $\KK[\Delta]$ from the dimension of $\Delta$ (this is Theorem 5.1.4 in [2]).

For any simplicial complex $\Delta$, the Krull-dimension of the Stanley-Reisner ring $\dim \KK[\Delta]$ is given by

$$\dim \KK[\Delta] = \dim \Delta + 1.$$

For any simplicial complex $\Delta$ on vertex set $[n]$, we have

$$\height I_\Delta = n - \dim \Delta - 1 = \codim \Delta.$$

For any ideal $I$ in $R$ we have $\height I = n - \dim R/I$, so this follows directly from Lemma <ref>.

For our purposes the most important example of this combinatorial reframing is Hochster's Formula, which is the subject of the following section.

This one-to-one correspondence allows us to reframe algebraic properties of Stanley-Reisner ideals and rings in terms of combinatorial properties of their corresponding complexes, and vice versa. For example, we can compute the height of the ideal $I_\Delta$, and hence the Krull-dimension of the ring $\KK[\Delta]$, respectively from the codimension and dimension of $\Delta$.

For any simplicial complex $\Delta$ on vertex set $[n]$, we have

$$\height I_\Delta = n - \dim \Delta - 1 = \codim \Delta.$$

For a face $F$ of $\Delta$ of size $d+1$, we define $P_F$ to be the ideal in $R$ generated by the vertices $x_i$ where $i\in [n] - F$. This is a monomial prime ideal in $R$ of height $n-d-1$, and because every nonface of $\Delta$ shares at least one vertex with $[n]-F$ we must have $P_F \supseteq I_\Delta$.

Conversely, any monomial prime ideal $P$ in $R$ containing $I_\Delta$ is of the form $\langle x_{i_1},\dots,x_{i_h} \rangle$ for some $1\leq i_1 < \dots < i_h \leq n$ such that each nonface of $\Delta$ contains at least one of the vertices $x_{i_1},\dots,x_{i_h}$. Thus the set $F_P = [n]-\{x_{i_1},\dots,x_{i_h}\}$ is a face of $\Delta$.

This gives us a one-to-one correspondence between faces of $\Delta$ and monomial prime ideals in $R$ containing $I_\Delta$, which sends faces of dimension $d$ to ideals of height $n-d-1$. Under this correspondence, facets of maximum dimension are sent to monomial primes containing $I_\Delta$ of minimal height. Because $I_\Delta$ is a squarefree monomial ideal then all of the minimal primes of $I_\Delta$ are monomial (see [24] Corollary 1.3.6), and the result follows.

For any simplicial complex $\Delta$, we have

$$\dim \KK[\Delta] = \dim \Delta + 1.$$

For any ideal $I$ in $R$ we have $\dim R/I = n - \height I$, so this follows directly from Lemma <ref>.

For our purposes the most important example of this combinatorial reframing is Hochster's Formula, which is the subject of the following section.

§.§ Hochster's Formula and Simplicial Homology

As we have seen already, the Stanley-Reisner construction allows us to translate algebraic problems into combinatorial ones, and vice versa, by reframing the algebraic invariants of squarefree monomial ideals in terms of combinatorial invariants of their corresponding complexes.

In this section we present a crucial result of Hochster's, which reframes the Betti diagrams of Stanley-Reisner ideals in terms of simplicial homology. As mentioned already, all of our homology is over $\KK$, and we use the notation $\Hred_i(\Delta)$ to denote the $\nth{i}$ reduced homology group of $\Delta$ with coefficients in $\KK$.

Let $\Delta$ be a simplicial complex on a vertex set $V$ of size $n$. For any integers $0\leq i \leq n$ and $d$, we have

\begin{equation*}
\beta_{i,d}(I_\Delta) =\sum_{U\in {V \choose d}} \dim_\KK \Hred_{d-i-2}(\Delta_U)
\end{equation*}

where ${V \choose d}$ denotes the subsets of $V$ of size $d$.

For a proof of Hochster's Formula see [24], Theorem 8.1.1.

Hochster's Formula shows us that the Betti numbers of the form $\beta_{i,i+1}(I_\Delta)$ come from induced subcomplexes of $\Delta$ with $\nth[st]{(-1)}$ reduced homology. The only simplicial complex which has $\nth[st]{(-1)}$ reduced homology is the irrelevant complex $\{\emptyset\}$, and for any nonempty subset $U\subseteq V$, we have $\Delta_U=\{\emptyset\}$ if and only if $U$ consists solely of missing vertices of $\Delta$. In particular, if $\Delta$ has $m$ missing vertices, then for any $0\leq i \leq n-1$ we have

\begin{align*}
\beta_{i,i+1}(I_\Delta) &= \sum_{U\in {V \choose i+1}} \dim_\KK \Hred_{-1}(\Delta_U)\\
&= \#\left\{U\in {V \choose i+1} : \Delta_U = \{\emptyset\}\right\}\\
&= {m \choose i+1}\, .
\end{align*}

Sometimes it will be more useful for us to study squarefree monomial ideals using a dual Stanley-Reisner construction, which we lay out below.

Note that every simplicial complex $\Delta$ is equal to the Alexander dual of its own Alexander dual $\Delta^*$ (i.e. we have $(\Delta^*)^*=\Delta$). In particular, this means that every squarefree monomial ideal in $R$ can be viewed uniquely as the dual Stanley-Reisner ideal $I_{\Delta^*}$ of some complex $\Delta$.

Let $\Delta$ be a simplicial complex on vertex set $[n]$. The dual Stanley-Reisner ideal $\Idstar$ is given by
$$\Idstar=\langle \mathbf{x}^\sigma : \sigma \subseteq [n], [n]-\sigma \in \Delta \rangle$$
where $\mathbf{x}^\sigma$ denotes the squarefree monomial $\prod_{i\in \sigma}x_i$ in $R$.

By construction, the nonfaces of the complex $\Delta^*$ are the complements of the faces of $\Delta$. The result follows from Definition <ref>.

We sometimes denote the dual Stanley-Reisner ideal $I_{\Delta^*}$ of $\Delta$ by $I^*_{\Delta}$.

There is an Alexander dual variant of Hochster's Formula, due to Eagon and Reiner, which allows us to compute the Betti numbers of the dual Stanley-Reisner ideal $I_{\Delta^*}$ directly using combinatorial and homological data from the complex $\Delta$ (this is [10] Proposition 1, rephrased in terms of Betti numbers of Stanley-Reisner ideals as opposed to Betti polynomials of Stanley-Reisner rings; see this article for a proof).

Let $\Delta$ be a simplicial complex on a vertex set $V$ of size $n$. For any integers $0\leq i\leq n$ and $d$, we have

\begin{equation*}
\beta_{i,d}(\Idstar)=\sum_{\sigma\in \Delta, |\sigma|=n-d} \dim_\KK \Hred_{i-1}(\lkds)
\end{equation*}

Let $\Delta$ be a simplicial complex.

* We have $\lkds[\emptyset]=\Delta$, which tells us that  $\beta_{i,n}(\Idstar)=\Hred_{i-1}(\Delta)$ for any integer $1\leq i \leq n$.

* For any facet $F$ of $\Delta$ we have $\lkds[F]=\{\emptyset\}$, which has only $\nth[st]{(-1)}$ homology of dimension $1$. Thus for any integer $d$, the Betti number $\beta_{0,d}(\Idstar)$ is equal to the number of facets of $\Delta$ of size $n-d$. In particular, the minimum degree of a generator of $\Idstar$ is equal to the codimension of $\Delta$.

Note that the smallest shift $d$ for which $\beta_{i,d}(\Idstar)\neq 0$ is equal to $n-\dim \Delta - 1$. In other words, the degree of the dual Stanley-Reisner ideal $\Idstar$ is equal to the codimension of $\Delta$.

Both Hochster's Formula and its Alexander Dual variant will be fundamental to our work going forwards, and for this reason most of our proofs centre around the computation of simplicial homology. We devote the rest of this section to some important results which can simplify these computations.

Firstly, note that for any $j\geq -1$, the definition of the homology group $\Hred_j(\Delta)$ (as given in e.g. [23], pages 104-6) relies solely on the faces of $\Delta$ of dimension $j+1$ and $j$, and this data is contained in the skeleton complex $\Skel_r(\Delta)$ for any $r>j$. This gives us the following lemma.

Let $\Delta$ be a simplicial complex, and let $r>j\geq -1$ be two integers. We have $\Hred_j(\Delta)=\Hred_j(\Skel_r(\Delta))$.

Next we introduce two important results which allow us to compute the homology of certain complexes from the homologies of their subcomplexes. Both of these results have more general topological analogues, regarding the singular homology of any topological space; but we present only the cases where the space in question is a simplicial complex. For a proof of the Mayer-Vietoris Sequence see [23] pages 149-150; for a proof of the Künneth Formula see [23] page 276, Corollary 3B.7.

Let $\Delta$ be a simplicial complex and suppose we have $\Delta = A \cup B$ for two subcomplexes $A$ and $B$ with nonempty intersection. There is a long exact sequence

\begin{align*}
\dots \rightarrow \Hred_{r+1}(\Delta) \rightarrow \Hred_r(A\cap B)\rightarrow \Hred_r(A)\oplus \Hred_r(B) \rightarrow &\Hred_r(\Delta)\rightarrow \dots\\
&\dots \rightarrow \Hred_0(\Delta)\rightarrow 0.
\end{align*}

Let $\Delta$ be a simplicial complex and suppose we have $\Delta = A \ast B$ for two subcomplexes $A$ and $B$. For any integer $r\geq -1$ we have an isomorphism

\begin{equation*}
\Hred_{r+1}(\Delta)\cong \sum_{i+j=r}\Hred_i(A)\otimes\Hred_j(B).
\end{equation*}

In particular the Künneth Formula allows us to derive the homology of cones and suspensions.

Let $\Delta$ be a simplicial complex. The cone $C\Delta$ is acyclic.

The cone $C\Delta$ is the join of $\Delta$ with  a single vertex, and the complex consisting of a single vertex is acyclic.

Let $\Delta$ be a simplicial complex. For any integer $i\geq -1$ the suspension $S\Delta$ has homology $\Hred_i(S\Delta)=\Hred_{i-1}(\Delta)$.

The suspension $S\Delta$ is the join of $\Delta$ with the complex consisting of two disjoint vertices [scale = 0.2]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point] at (0,0) ;
(c)[point] at (1.5,0) ;
, and the homology of this complex is given by
$$\Hred_i( \begin{tikzpicture}[scale = 0.2]
\tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
\node (a)[point] at (0,0) {};
\node (c)[point] at (1.5,0) {};
\end{tikzpicture}) = \begin{cases}
\KK &\text{ if } i=0\\
0 &\text{otherwise.}
\end{cases}$$

Many of the results in this thesis are about the shapes of Betti diagrams of Stanley-Reisner ideals (i.e. the indices for which the Betti numbers are nonzero) rather than the values of specific Betti numbers. Thus we will often be more interested in the degrees at which a complex has nontrivial homology than in what those nontrivial homologies are. For this reason, we sometimes make use of the following notation.

Let $\Delta$ be a simplicial complex. We define the homology index set of $\Delta$ as $h(\Delta)=\{i\in \ZZ : \Hred_i(\Delta) \neq 0\}$.

We may add homology index sets $A$ and $B$ together using the rule $A+B=\{i+j: i\in A, j\in B\}$. If $A$ and $B$ are both singletons, then this is the same as adding the elements of those singletons together; and if either set is empty then the resulting sum is also empty.

Using this notation we have the following corollary to the Künneth Formula.

Let $\Delta_1,\dots,\Delta_m$ be simplicial complexes, and let $\circledast_{j=1}^m \Delta_j$ denote the join of all of them. We have that $h(\circledast_{j=1}^m \Delta_j)=\sum_{j=1}^m h(\Delta_j) + \{m-1\}$.

We prove this by induction on $m\geq 1$. For the base case $m=1$, there is nothing to prove. For the inductive step, assuming that $h(\circledast_{j=1}^m \Delta_j)=\sum_{j=1}^m h(\Delta_j) + \{m-1\}$, we have

\begin{align*}
h(\circledast_{j=1}^{m+1} \Delta_i) &= h((\circledast_{j=1}^m \Delta_j)\ast \Delta_{m+1}) &\\
&= h(\circledast_{j=1}^m \Delta_j)+ h(\Delta_{m+1}) +\{1\} &\text{by Prop. \ref{Proposition: Kunneth Formula}}\\
&= (\sum_{j=1}^m h(\Delta_j) + \{m-1\})+h(\Delta_{m+1}) +\{1\} &\text{by ind. hyp.}\\
&= \sum_{j=1}^{m+1} h(\Delta_j) + \{m\}.
\end{align*}

Another crucial tool for computing homology comes from the following result (see e.g. [23], Corollary 2.11).

If two topological spaces are homotopy equivalent, they are homology equivalent.

In particular, every deformation retraction is a homotopy equivalence, and for this reason we will often make use of deformation retractions when computing homology. As an example, this allows us to find the homology of cones and suspensions.

Let $\Delta$ be a simplicial complex. The cone $C\Delta$ is acyclic.

The cone $C\Delta$ is the join of $\Delta$ with a single vertex $v$, and it deformation retracts on to the vertex $v$, which is acyclic.

Let $\Delta$ be a simplicial complex. For any integer $i\geq -1$ the suspension $S\Delta$ has homology $\Hred_i(S\Delta)=\Hred_{i-1}(\Delta)$.

The suspension $S\Delta$ is the join of $\Delta$ with the complex consisting of two disjoint vertices $u$ and $v$. We can decompose $\Delta$ into the subcomplexes $A\cup B$ where $\Delta \ast \{u\}$ and $B=\Delta \ast\{v\}$. Both of these subcomplexes are cones and are hence acyclic by Corollary <ref>. They intersect at $\Delta$ itself, and hence for each $i\geq -1$, the Mayer-Vietoris sequence gives us the isomorphism
$$0\rightarrow \Hred_i(S\Delta)\rightarrow \Hred_{i-1}(\Delta)\rightarrow 0.$$

We end this section by presenting a particularly useful lemma for finding deformation retracts. It provides sufficient conditions for ensuring that a complex $\Delta$ deformation retracts on to $\Delta - g$, for a given face $g\in \Delta$ (recall that $\Delta-g$ is the deletion of $g$ from $\Delta$, as defined in Definition <ref>). We suspect this lemma exists in the literature, but have been unable to find it.

Let $\Delta$ be a simplicial complex, and let $g\subsetneqq f$ be faces of $\Delta$ such that every facet of $\Delta$ that contains $g$ also contains $f$. There is a deformation retraction $\Delta \rightsquigarrow \Delta - g$, obtained by identifying $g$ with a vertex in $f-g$.

Suppose $\Delta$ has $n$ vertices $v_1,\dots,v_n$ with $g=\{v_1,\dots,v_k\}$ and $v_{k+1}$ in $f-g$.

Let $X=X_{\Delta}$ be the geometric realization of $\Delta$ in $\RR^n$ and $X_{\Delta-g}$ the geometric realization of $\Delta -g$. For each nonempty face $\sigma =\{v_{i_1},\dots,v_{i_r}\}$ of $\Delta$, we define $X_\sigma$ to be the set $\left\{\lambda_1 e_{i_1}+\dots \lambda_r e_{i_r} : \lambda_1, \dots, \lambda_r > 0, \sum_{j=1}^r \lambda_j = 1\right\} \subset X$, where $\{e_1,\dots,e_n\}$ is the canonical basis of $\RR^n$. Note that $X=\bigcup_{\sigma \in \Delta} X_\sigma$, while $X_{\Delta-g}=\bigcup_{\sigma \in \Delta, \sigma \nsupseteq g} X_\sigma$.

Let $\bp$ be a point in $X$. We may write it as $\bp = \sum_{j=1}^n \lambda_j e_j$ where the coefficients $\lambda_j$ are all nonnegative and sum to $1$. In particular, $\bp$ lies inside $X_\sigma$ for some $\sigma\in \Delta$ containing $g$ if and only if all of $\lambda_1,\dots,\lambda_k$ are positive.

We define $\lambda = \min \{\lambda_1, \dots, \lambda_k\}$. This allows us to rewrite the point $\bp$ as

\begin{equation}\label{Equation: bp}
\bp = \lambda(e_1 + \dots + e_k) + \sum_{i=1}^{k} (\lambda_i - \lambda) e_i + \lambda_{k+1} e_{k+1} + y
\end{equation}

for some $y$ in the span of $\{e_{k+2},\dots,e_n\}$. Note that the coefficients $(\lambda_i-\lambda)$ are all nonnegative, and (by the definition of $\lambda$) at least one of them is zero. Note also that $\lambda$ itself is nonzero if and only if all of $\lambda_1,\dots,\lambda_k$ are positive, which occurs if and only if $\bp$ is in $X_\sigma$ for some $\sigma$ containing $g$. In other words, we have $\lambda = 0$ if and only if $\bp$ lies inside $X_{\Delta-g}$.

Using this notation for the points in $X$, we can define a function $\varphi: X \times [0,1] \rightarrow \RR^n$ as follows. For $\bp$ as in Equation (<ref>) and $0\leq t \leq 1$ we define

\begin{equation}\label{Equation: varphi}
\varphi(\bp, t) = \lambda (1-t)(e_1 + \dots + e_k) + \sum_{i=1}^{k} (\lambda_i - \lambda) e_i + (\lambda_{k+1} + k \lambda t) e_{k+1} + y.
\end{equation}

We claim that $\varphi$ is a deformation retraction from $X$ to $X_{\Delta-g}$.

First, note that $\varphi$ is continuous. Indeed, the function $\lambda=\min \{\lambda_1,\dots,\lambda_k\}$ is continuous in the variables $\lambda_i$, and hence so is each summand in Equation (<ref>). Each summand is also continuous in $t$.

Next, we note that $\varphi(*,0)$ is the identity on $X$. Moreover, for $\bp$ in $X_{\Delta-g}$ we have $\lambda=0$, and hence $\varphi(\bp,t)=\bp$ for every $0\leq t \leq 1$.

It remains to show that the image of $\varphi(*,t)$ lies inside $X$ for every value of $t$, and in particular that the image of $\varphi(*,1)$ is $X_{\Delta-g}$.

For the former claim, we note that the sum of the coefficients of $e_1,\dots,e_n$ in the decomposition of $\varphi(\bp,t)$ is the same as the sum of these coefficients in the decomposition of $\bp$ (which is $1$), and all of these coefficients are nonnegative. We may assume that $\bp$ lies inside $X_{\sigma}$ for some face $\sigma$ in $\Delta$ containing $g$ (otherwise $\bp$ lies inside $X_{\Delta-g}$ and we are already done). By our assumption on $g$ and $f$ we have that $\sigma\cup \{v_{k+1}\}$ is also a face of $\Delta$. If $\sigma = \{v_1,\dots,v_k,v_{i_1},\dots,v_{i_r}\}$ for some $k<i_1<\dots<i_r\leq n$, then the vectors in the decomposition of $\varphi(\bp,t)$ with strictly positive coefficients are all contained in $\{e_1,\dots,e_k,e_{i_1},\dots,e_{i_r}\}\cup\{e_{k+1}\}$. We conclude that $\varphi(\bp,t)$ lies inside $X_{\tau}$ for some $\tau \subseteq \sigma \cup \{v_{k+1}\}$, and hence inside $X$.

For the latter claim, we note that the coefficients of the vectors $e_1,\dots, e_k$ in $\varphi(\bp,1)$ are $\lambda_1-\lambda,\dots,\lambda_k-\lambda$, and at least one of these must be zero by the definition of $\lambda$. Thus $\varphi(\bp,1)$ lies inside $X_{\Delta-g}$.

In particular we have the following corollary, which will be sufficient for our needs in most (but not all) cases.

Let $\Delta$ be a simplicial complex, and let $a$ and $b$ be distinct vertices of $\Delta$ satisfying the following conditions.

* There is at least one facet of $\Delta$ containing $a$ (i.e. it is not a missing vertex).

* Every facet of $\Delta$ containing $a$ also contains $b$.

There is a deformation retraction $\Delta \leadsto \Delta - \{a\}$ given by the map $a\mapsto b$.

This comes from Lemma <ref>, setting $g=\{a\}$ and $f=\{a,b\}$. Note that $g$ is a face of $\Delta$ by assumption (1), and $f$ is a face of $\Delta$ by assumption (2).

§.§ Graphs and Edge Ideals

A $1$- or $0$-dimensional simplicial complex with no missing vertices contains the same data as a graph. The following graph-theoretic definitions are standard, and fundamental to our work going forwards.

A graph $G=(V,E)$ consists of a set of vertices $V$ and a set of edges $E$ consisting of subsets of $V$ of size 2.

We often denote the vertex set and edge set of a graph $G$ as $V(G)$ and $E(G)$ respectively.

Let $G=(V,E)$ be a graph, and $m$ any positive integer.

* We define the complement $G^c$ of $G$ to be the graph on vertex set $V$ with edge set $\{\{x,y\}: x, y \in V \{x,y\}\notin E\}$.

* For a subset $U\subset V$ we define the induced subgraph of $G$ to be the graph on vertex set $U$ with edge set $\{\{x,y\}:x,y\in U,\{x,y\}\in E\}$.

* A vertex cover for $G$ is a subset $U\subset V$ such that for every edge $e\in E$ we have $e\cap U \neq \emptyset$.

* A matching in $G$ is a collection $M$ of pairwise disjoint edges in $G$. We say $M$ is a maximal matching if it is maximal with respect to inclusion.

* A cycle of length $m$ in $G$ (also called an $m$-cycle) is a sequence of edges in $G$ of the form $\{v_1,v_2\},\dots,\{v_{m-1},v_m\},\{v_m,v_1\}$ for some vertices $v_1,\dots,v_m\in V$.

* We say a vertex $v\in V$ is isolated in $G$ if it is contained in no edges of $G$. We say $v$ is universal in $G$ if it is contained in every edge of $G$.

Let $G$ be the graph on vertex set $[5]$ with edges $\{1,2\}$, $\{1,4\}$, $\{2,3\}$, $\{2,4\}$ and $\{3,4\}$. We can draw $G$ as follows.

[scale = 0.8]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point, label=left: $1$] at (0,0) ;
(b)[point, label=left: $2$] at (0,2) ;
(c)[point, label=right: $3$] at (2,2) ;
(d)[point, label=right: $4$] at (2,0) ;
(e)[point, label=right: $5$] at (3.5,1) ;

(a.center) – (b.center) – (c.center) – (d.center) – cycle;
(b.center) – (d.center);

* Every edge in $G$ contains either $2$ or $4$ so the set $\{2,4\}$ is a vertex cover for $G$.

* The edges $\{1,2\}$ and $\{3,4\}$ form a maximal matching in $G$, as do the edges $\{1,4\}$ and $\{2,3\}$.

* $G$ contains two 3-cycles (the cycles $\{1,2\},\{2,4\},\{4,1\}$ and $\{2,3\},\{3,4\},\{4,2\}$) and one 4-cycle (the cycle $\{1,2\},\{2,3\},\{3,4\},\{4,1\}$).

* The complement $G^c$ of $G$ is the graph with edges $\{1,5\},\{2,5\},\{3,5\},\{4,5\}$ and $\{1,3\}$, which we can draw as follows.

[scale = 0.8]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(1)[point, label=left: $1$] at (0,0) ;
(2)[point, label=right: $2$] at (3.4,2) ;
(3)[point, label=left: $3$] at (0,2) ;
(4)[point, label=right: $4$] at (3.4,0) ;
(5)[point, label=above: $5$] at (1.7,1) ;

(1.center) – (3.center);
(1.center) – (5.center);
(2.center) – (5.center);
(3.center) – (5.center);
(4.center) – (5.center);

* Note that the vertex $5$ is isolated in $G$, and therefore universal in $G^c$.

As mentioned above, a graph may be viewed as a $1$-dimensional simplicial complex with no missing vertices. However, there are (at least) two other important ways of associating complexes to graphs, as laid out below.

Let $G=(V,E)$ be a graph.

* We define the complex of cliques of $G$, $\Cl(G)$, to be the complex on $V$ whose faces are the cliques in $V$ - i.e. those subsets $\{v_1,\dots,v_r\}\subset V$ such that for each $1\leq i < j \leq r$ we have $\{v_i,v_j\}\in E$.

* We define the independence complex of $G$, $\Ind(G)$, to be the complex on $V$ whose faces are all the independent subsets of $V$ - i.e. those subsets $\{v_1,\dots,v_r\}\subset V$ such that for each $1\leq i < j \leq r$ we have $\{v_i,v_j\}\notin E$.

The independence complex of $G$ is equal to the complex of cliques of $G^c$ (i.e. $\Ind(G)=\Cl(G^c)$).

We now review some examples of important graphs, which will be crucial to our work going forward.

Let $m$ be any positive integer.

* The complete graph on $m$ vertices, denoted $K_m$, is the graph on vertex set $[m]$ with all possible edges (i.e. its edge set is ${[m] \choose 2} = \{U\subseteq [m] : |U| = 2\}$).
We sometimes denote the graph $K_2$, consisting of a single edge, as $L$.

* The empty graph on $m$ vertices, denoted $E_m$, is the graph on vertex set $[m]$ with no edges (i.e. its edge set is $\emptyset$).

* The cyclic graph of order $m$, denoted $C_m$, is the graph on vertex set $[m]$ with edge set $\{\{1,2\},\{2,3\},\dots,\{m-1,m\},\{m,1\}\}$ (i.e. the edges of $C_m$ form a single cycle of length $m$).

Just as with our convention for the vertex sets of $\Delta^j$ and $\partial \Delta^j$ in Remark <ref>, our convention for the vertex sets of the graphs $K_m$, $E_m$ and $C_m$ is arbitrary. Again, these graphs are better understood as isomorphism classes, and we could have chosen a graph on any vertex set of size $m$ as our representative for these classes. Similarly, we could have chosen any $m$-cycle for the edges of $C_m$.

For the case $m=5$ we have

[scale = 1.25]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point] at (0,1) ;
(b)[point] at (0.951,0.309) ;
(c)[point] at (0.588,-0.809) ;
(d)[point] at (-0.588,-0.809) ;
(e)[point] at (-0.951,0.309) ;

(a.center) – (b.center) – (c.center) – (d.center) – (e.center) – cycle;
(a.center) – (c.center) – (e.center) – (b.center) – (d.center) – cycle;
[scale = 1.25]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point] at (0,1) ;
(b)[point] at (0.951,0.309) ;
(c)[point] at (0.588,-0.809) ;
(d)[point] at (-0.588,-0.809) ;
(e)[point] at (-0.951,0.309) ;

[scale = 1.25]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point] at (0,1) ;
(b)[point] at (0.951,0.309) ;
(c)[point] at (0.588,-0.809) ;
(d)[point] at (-0.588,-0.809) ;
(e)[point] at (-0.951,0.309) ;

(a.center) – (b.center) – (c.center) – (d.center) – (e.center) – cycle;

$K_5$        $E_5$         $C_5$

Just as the Stanley-Reisner correspondence allows us to assign a unique squarefree monomial ideal to every simplicial complex, there is a similar assignment of ideals to graphs.

Let $G$ be a graph on vertex set $[n]$. We define the edge ideal of $G$ to be the ideal
$$I(G)=\langle x_ix_j : \{i,j\}\in E(G) \rangle$$

As with Stanley-Reisner ideals we extend this definition to graphs $G$ on arbitrary vertex sets $V$ of size $n$ by first choosing an ordering on $V$. Again, all possible orderings give us the same ideal up to isomorphism, which we refer to as `the' edge ideal.

While Stanley-Reisner ideals account for all squarefree monomial ideals, edge ideals account for all squarefree monomial ideals generated entirely in degree $2$. Because the edge ideal of a graph is a squarefree monomial ideal, Proposition <ref> tells us that it must be the Stanley-Reisner ideal of some complex. In fact, it is the Stanley-Reisner ideal of the independence complex of the graph.

Let $G$ be a graph on vertex set $[n]$. We have
$$I(G) = I_{\Ind(G)}$$

The nonfaces of $\Ind(G)$ are all the sets $U\subseteq V$ which contain an edge of $G$. Thus the minimal nonfaces of $\Ind(G)$ are the edges of $G$. The result follows.

CHAPTER: DIMENSIONS OF BETTI CONES

In this chapter we study the dimensions of the Betti cones on Stanley-Reisner ideals and Edge ideals, which we denote by $\Dn$ and $\Cn$ respectively. We also fix a positive integer $h < n$, and introduce the subcones $\Dnh$ and $\Cnh$, generated respectively by Stanley-Reisner ideals and edge ideals of height $h$. Recall that these cones live inside the infinite-dimensional vector space $\Vn=\bigoplus_{d\in \ZZ} \QQ^{n+1}$.

We make use of some notational short-hands throughout: for a complex $\Delta$ we use $\beta(\Delta)$ to denote the Betti diagram $\beta(I_\Delta)$; and similarly for a graph $G$ we use $\beta(G)$ to denote the Betti diagram $\beta(I(G))$.

As observed at the start of Section <ref>, every graph is a $1$- or $0$-dimensional simplicial complex. Nevertheless, using the notation $\beta(G)$ to represent the diagram of the edge ideal of $G$ (as opposed to the Stanley-Reisner ideal obtained from viewing $G$ as a simplicial complex) should not result in ambiguity. It will usually be clear both from context and from our choice of notation whether we are viewing the structure in question as a graph or a complex, and hence whether its corresponding ideal is obtained using the edge ideal or Stanley-Reisner construction. In the few cases where there is a genuine risk of misinterpretation we will use the notation $\beta(I_\Delta)$ and $\beta(I(G))$ for clarity.

Specifically we prove the following results about our four cones.

Let $\Dn$ be the Betti cone generated by all Betti diagrams of Stanley-Reisner ideals in $R$. We have

$$\dim \Dn = \frac{n(n+1)}{2}.$$

Let $\Cn$ be the Betti cone generated by all Betti diagrams of edge ideals in $R$. We have

$$\dim \Cn = \begin{cases}
r^2  &\text{ if } n=2r \\
r^2 + r &\text{ if } n=2r+1.\\
\end{cases}$$

Let $\Dnh$ be the Betti cone generated by all Betti diagrams of Stanley-Reisner ideals in $R$ of height $h$. We have

$$\dim \Dnh = \frac{n(n-1)}{2}-\frac{h(h-1)}{2}+1.$$

Let $\Cnh$ be the Betti cone generated by all Betti diagrams of edge ideals in $R$ of height $h$. We have
$$\dim \Cnh = h(n-h-1)+1.$$

Some definitions of simplicial complexes require that the complexes have no missing vertices. This is equivalent to requiring that the generators of the corresponding Stanley-Reisner ideals have degree at least two. We do not use this convention in this thesis, and as such, our cones $\Dn$ and $\Dnh$ contain diagrams corresponding to Stanley-Reisner ideals with generators of degree one.

However, for the benefit of readers who prefer this convention, we denote the cone generated by diagrams of Stanley-Reisner ideals whose generators have degree at least two by $\Dntilde$. The dimension of this slightly smaller cone can be computed as
$$\dim \Dntilde = \dim \Dn - n = \frac{n(n-1)}{2}.$$
Similarly, we let $\Dnhtilde$ denote the cone generated by diagrams of Stanley-Reisner ideals of height $h$ whose generators have degree at least two. The dimension of this cone is given by
$$\dim \Dnhtilde = \dim \Dnh - h = \frac{n(n-1)}{2}-\frac{h(h+1)}{2}+1.$$

Note that the generators of edge ideals all have degree exactly two, and thus we have the following inclusion of cones.

[r, phantom, "⊂"]
[r, phantom, "⊂"]

[u, phantom, sloped, "⊂"] [r, phantom, "⊂"]
[u, phantom, sloped, "⊂"] [r, phantom, "⊂"]
[u, phantom, sloped, "⊂"]

We will show how our proof for the dimension of $\Dn$ can be modified to account for the subcone $\Dntilde$ in remarks along the way; and we will present our proofs for the cones $\Dnh$ and $\Dnhtilde$ together.

§ MOTIVATION

There are a number of motivations for finding the dimensions of these cones. Most immediately, computing dimension is a crucial first step in understanding any cone: not only because the cone's dimension is a fundamental property in itself, but also because it may prove useful in further classifications. Specifically, any cone $\calC$ in $\Vn$ can be classified as the intersection of a number of defining halfspaces (see [8], Theorem 4.5). These are halfspaces $\calH$ of $\Vn$ of the form $\{\beta \in \Vn : f(\beta) \geq 0 \}$ for some linear form $f(\beta)$, such that

* $\calC$ is completely contained in $\calH$.

* The intersection of $\calC$ with the boundary of $\calH$ (that is, the hyperplane $\{\beta \in \Vn : f(\beta) = 0 \}$)  has dimension $\dim \calC - 1$.

To determine whether a given halfspace $\calH$ of $\Vn$ is a defining halfspace for $\calC$, we need to check whether it satifies these two properties; and for the latter property, this is far easier to do if we know the dimension of $\calC$.

Moreover, our proofs for these results will, in each case, require us to find the cone's minimal ambient vector space (i.e. the minimal subspace of $\Vn$ containing the cone). In this sense our four theorems above can be seen as analogues to Corollary <ref> of the first Boij-Söderberg Conjecture, which told us the minimal subspace of $\Vn$ containing the cone $\calC(\ba,\bfb)$.

Corollary <ref> is, in essence, a result about the linear dependency relations satisfied by the diagrams in the cone $\calC(\ba,\bfb)$. Indeed, it tells us that (up to linear combination) the Herzog-Kühl Equations are the only relations satisfied by all diagrams of Cohen-Macaulay modules of codimension $h$. Similarly, in finding the minimal subspaces for our cones, we will present (again, up to linear combination) every linear dependency relation which is satisfied by all diagrams in the cone.

§ KEY TOOLS

Our proofs for all four results will proceed in roughly the same way: first, we bound the dimension from above by finding a finite-dimensional subspace of $\Vn=\bigoplus_{d\in \ZZ} \QQ^{n+1}$ containing the cone; then we bound it from below by exhibiting an appropriately sized linearly independent set of Betti diagrams lying in the cone (thus, in the process, showing that the subspace we found is in fact minimal). We begin by presenting some key tools which will help us in constructing our linearly independent sets of Betti diagrams.

§.§ Diagrams of Specific Families of Complexes and Graphs

In this section we present the Betti diagrams of some specific families of complexes and graphs, along with an important proposition and corollary (Lemma <ref> and Corollary <ref>) which help us construct Betti diagrams of slightly more complicated complexes and graphs.

In what follows, we use the notation $\Delta^j$, $\partial \Delta^j$ and $\Skel_r(V)$ from Definitions <ref> and <ref>, for integers $j\geq 0$ and $r\geq -1$ and vertex set $V$; and the notation $K_m$, $E_m$, $C_m$ and $L$ from Definition <ref>, for positive integers $m$. We also write the disjoint union of graphs and complexes in additive notation, so for example we would use $C_5+2L$ to denote the graph

[scale = 1.25]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point] at (0,1) ;
(b)[point] at (0.951,0.309) ;
(c)[point] at (0.588,-0.809) ;
(d)[point] at (-0.588,-0.809) ;
(e)[point] at (-0.951,0.309) ;

(f)[point] at (1.5,1) ;
(g)[point] at (1.5,-0.809) ;

(h)[point] at (2.5,1) ;
(i)[point] at (2.5,-0.809) ;

(a.center) – (b.center) – (c.center) – (d.center) – (e.center) – cycle;
(f.center) – (g.center);
(h.center) – (i.center);

We begin by finding the homology (and hence the Betti diagrams) of skeleton complexes, which will allow us to derive the diagrams of a number of other families of complexes and graphs.

Let $m$ and $r$ be integers and set $\Delta = \Skel_r([m])$ with $m\geq 1$ and $r\geq -1$. For any integer $i\geq -1$ we have
$$\dim_{\KK} \Hred_i(\Delta)=\begin{cases}
{m-1 \choose r+1}	& \text{ if } i=r\\
0 					& \text{ otherwise.}
\end{cases}$$

We proceed by double induction on $r\geq -1$ and $m\geq 1$. Note that for any $m\geq 1$, the complex $\Skel_{-1}([m])$ is equal to the irrelevant complex $\{\emptyset\}$, which has only $\nth[st]{(-1)}$ homology of dimension ${m - 1 \choose 0}=1$. And for any $r\geq 0$, the complex $\Skel_{r}([1])$ is a single point. This is acyclic, which means it has $\nth{r}$ homology of dimension ${0 \choose r+1}=0$. This proves the result in the base cases $r=-1$ and $m=1$.

Now suppose that $m>1$,  and define, for $\Delta=\Skel_r([m])$,

\begin{align*}
A &= \langle \sigma \in \Delta : m \in \sigma \rangle \\
B &= \langle \sigma \in \Delta : m \notin \sigma \rangle\
\end{align*}

so that $\Delta$ decomposes into the union $A\cup B$. Note that $A$ is a cone over the vertex $m$, and is therefore acyclic. Also $B$ contains every subset of $[m-1]$ of dimension less than or equal to $r$, so it is equal to $\Skel_r([m-1])$. By induction this has only $\nth{r}$ homology of dimension ${m-2 \choose r+1}$. Meanwhile the intersection $A\cap B$ contains every subset of $[m-1]$ of dimension less than or equal to $r-1$, so it is equal to $\Skel_{r-1}([m-1])$, which has only $\nth[st]{(r-1)}$ homology of dimension ${m-2 \choose r}$ by induction.

The Mayer-Vietoris Sequence (Proposition <ref>) gives us the following short exact sequence

\begin{equation*}
0 \rightarrow \Hred_r(B) \rightarrow \Hred_r(\Delta) \rightarrow \Hred_{r-1}(A\cap B)\rightarrow 0\, .
\end{equation*}

Thus $\Delta$ has only $\nth{r}$ homology, and the dimension of this homology is ${m-2 \choose r+1}+{m-2 \choose r}={m-1 \choose r+1}$.

Let $m$ and $r$ be integers and set $\Delta = \Skel_r([m])$. For any integers $i$ and $d$, we have

\begin{cases}
{m \choose i+r+2}	{i+r+1 \choose r+1}	& \text{if } 0 \leq i \leq m-r-2 \text{ and } d=i+r+2\\
0 		& \text{otherwise.}
\end{cases}$$

In particular, the diagram $\beta(\Skel_r([m]))$ has the following shape.

\begin{equation*}
\begin{bmatrix}
\beta_{0,r+2} & \dots & \beta_{m-r-2,m}\\
\end{bmatrix}
\end{equation*}

For any subset $U\subseteq [m]$ of size $d$, the induced subcomplex $\Delta_U$ is isomorphic to $\Skel_r([d])$. Thus Hochster's Formula tells us that
$$\beta_{i,d}(I_\Delta) =\sum_{U\in {[m] \choose d}} \dim_\KK \Hred_{d-i-2}(\Skel_r([d])).$$
By Lemma <ref>, this must be zero unless $d=r+i+2$, in which case we have ${m \choose r+i+2}$ subsets $U$ of $[m]$ of size $d$, each of which gives us an induced subcomplex with homology of dimension ${i+r+1 \choose r+1}$. The result follows.

Let $j$ be a nonnegative integer, and let $\Delta^j$ and $\partial \Delta^j$ denote, respectively, the $j$-simplex and its boundary, on vertex set $[j+1]$. We have

* $\beta(\Delta^j)=0$.

* $\beta_{i,d}(\partial \Delta^j)=\begin{cases}
1 & \text{ if } (i,d) = (0,j+1)\\
0 & \text{ otherwise.}
\end{cases}$

In particular, $\beta(\partial \Delta^j)$ has the shape $\begin{bmatrix} \beta_{0,j+1} \end{bmatrix}$.

These results follow immediately from Lemma <ref>, noting that $\Delta^j=\Skel_j([j+1])$ and $\partial \Delta^j = \Skel_{j-1}([j+1])$.

These results can also be seen from the fact that $\Delta^j$ has no missing faces and hence its Stanley-Reisner ideal is the zero ideal; and $\partial \Delta^j$ has a single missing face of degree $j+1$ and hence its Stanley-Reisner ideal admits the resolution $R(-(j+1))\rightarrow I_{\partial \Delta^j}$.

We now move on to the Betti diagrams of our three important families of graphs.

Let $m$ be a positive integer, and let $E_m$ denote the empty graph on $m$ vertices. We have $\beta(E_m)=0$.

The independence complex of $E_m$ is $\Delta^{m-1}$, so the result follows from Corollary <ref>.

Again, this result can also be seen directly from the fact that the edge ideal of $E_m$ is the zero ideal.

Let $m$ be a positive integer, and let $K_m$ denote the complete graph on $m$ vertices. For any integers $i$ and $d$, we have

\begin{cases}
(i+1){m\choose i+2} & \text{if } 0\leq i\leq m-2 \text{ and } d=i+2\\
0 & \text{otherwise.}
\end{cases}$$

In particular, the diagram $\beta(K_m)$ has the following shape.

\begin{equation*}
\begin{bmatrix}
\beta_{0,2} & \dots & \beta_{m-2,m}\\
\end{bmatrix}
\end{equation*}

The independence complex of $K_m$ is equal to $\Skel_0([m])$, so the result follows from Lemma <ref>.

Let $m$ be a positive integer, and suppose $G$ is the complement of the cyclic graph on $m$ vertices, $C_m$. For any integers $i$ and $d$, we have

\begin{equation*}
\beta_{i,d}(G)=
\begin{cases}
\frac{m(i+1)}{m-i-2}{m-2\choose i+2}& \text{if } 0\leq i \leq m-4 \text{ and } d=i+2\\
1 & \text{if } (i,d) = (m-3,m) \\
0 & \text{otherwise.}
\end{cases}
\end{equation*}

In particular, the diagram $\beta((C_m)^c)$ has the following shape.

\begin{equation*}
\begin{bmatrix}
\beta_{0,2} & \dots & \beta_{m-4,m-2} & \\
& & &\beta_{m-3,m}\\
\end{bmatrix}
\end{equation*}

See Theorem 2.3.3 in [34].

We end with some results about how the operations of coning and suspension affect (or rather, in the former case do not affect) Betti diagrams. In particular, Lemma <ref> and Corollary <ref> will be crucial to our construction of Betti diagrams in this section, as they allow us to take a diagram we understand already and increase its regularity by exactly 1.

Let $\Delta$ be a simplicial complex and $C\Delta$ the cone over $\Delta$. We have $\beta(\Delta)=\beta(C\Delta)$.

The minimal nonfaces of $C\Delta$ are the same as the minimal nonfaces of $\Delta$. This means that their Stanley-Reisner ideals have the same generators, and thus have the same Betti diagrams by Remark <ref>.

Let $G$ be a graph and let $G+v$ denote the graph obtained by adding a single isolated vertex $v$ to $G$. We have $\beta(G+v) = \beta(G)$.

The independence complex $\Ind(G+v)$ is equal to $C\Ind(G)$, so this follows from Lemma <ref>.

Let $\Delta$ be a simplicial complex and $S\Delta$ the suspension of $\Delta$. For any integers $0\leq i\leq n$ and $d$, we have
\beta_{0,2}(\Delta) + 1 &\text{ if } (i,d)=(0,2)\\
\beta_{i,d}(\Delta)+\beta_{i-1,d-2}(\Delta) & \text{ otherwise.}
\end{cases}$$

Let $i$ and $d$ be integers with $0\leq i \leq n$, and let $V$ denote the vertex set of $\Delta$. We can denote the vertex set of $S\Delta$ as $\tV=V\cup \{x,y\}$ for some additional vertices $x$ and $y$.

Suppose $U$ is a subset of $\tV$ of size $d$. If exactly one of the vertices $x$ or $y$ is in $U$, then $(S\Delta)|_U$ is a cone over this vertex, and is hence acyclic. Thus in order for $(S\Delta)|_U$ to have homology, it must either contain neither of the vertices $x$ and $y$, or both of them. In the former case, the induced subcomplex $(S\Delta)|_U$ is equal to $\Delta_U$. In the latter, it is equal to $S(\Delta_{U-\{x,y\}})$.

Thus, by Hochster's Formula and Corollary <ref>, we have

\begin{align*}
\beta_{i,d}(S\Delta)&=\sum_{U\in {V \choose d}} \dim_\KK \Hred_{d-i-2}(\Delta_U)+ \sum_{U\in {V \choose d-2}} \dim_\KK \Hred_{d-i-2}(S(\Delta_U))\\
&= \beta_{i,d}(\Delta) +\sum_{U\in {V \choose d-2}}\dim_\KK \Hred_{d-i-3}(\Delta_U)\\
&= \beta_{i,d}(\Delta) +\sum_{U\in {V \choose d-2}}\dim_\KK \Hred_{(d-2)-(i-1)-2}(\Delta_U)\\
\beta_{0,2}(\Delta) + 1 &\text{ if } (i,d)=(0,2)\\
\beta_{i,d}(\Delta)+\beta_{i-1,d-2}(\Delta) & \text{ otherwise.}
\end{cases}
\end{align*}

Let $G$ be a graph, and $L$ the graph consisting of a single edge between two vertices. For any integers $i$ and $d$, we have

\beta_{0,2}(G) + 1 &\text{ if } (i,d)=(0,2)\\
\beta_{i,d}(G)+\beta_{i-1,d-2}(G) & \text{ otherwise.}
\end{cases}$$

The independence complex $\Ind(G+L)$ is equal to $S \Ind(G)$, so this follows from Lemma <ref>.

As an aside, Lemma <ref> and Corollary <ref> allow us to compute the Betti diagram of the cross polytope $O^{m-1}$, or equivalently of the graph consisting of $m$ disjoint edges.

Let $m$ be a nonnegative integer. The Betti numbers of the $(m-1)$-dimensional cross-polytope $O^{m-1}$, or equivalently of the graph consisting of $m$ disjoint edges $mL$, are given by

{m \choose i+1} & \text{ if } d = 2i+2\\
0 & \text{ otherwise.}
\end{cases}$$

In particular, the diagram $\beta(O^{m-1})=\beta(mL)$ has the following shape.

\begin{equation*}
\begin{bmatrix}
\beta_{0,2} & & \\
& \ddots & \\
& &\beta_{m-1,2m}\\
\end{bmatrix}
\end{equation*}

First note that the minimal nonfaces of the cross polytope $O^{m-1}$ are $m$ disjoint edges, so we have $\Ind(mL)=O^{m-1}$ and hence $\beta(mL)=\beta(O^{m-1})$. The result follows from either Lemma <ref> or Corollary <ref> by induction on $m\geq 0$.

§.§ Indexing Sets and Initiality

In the following sections, we establish formulae for the dimensions of our cones. Our proofs proceed by showing that the formulae given are both upper and lower bounds for the dimensions of the cones.

To find a lower bound $l$ for the dimension of a convex cone $\calC$, it suffices to find a linearly independent set of $l$ vectors lying in $\calC$, as this shows that the smallest vector space containing $\calC$ must have dimension at least $l$. In this section, for ease of explanation, we present terminology for a simple condition that ensures linear independence. We then describe our general method for finding the dimensions of our cones, using this condition.

Suppose $\calC$ lives inside the rational vector space $\calV=\bigoplus_{i\in \II} \QQ$ for some finite indexing set $\II$. For a vector $v$ in $\calV$ and an index $i\in \II$, let $v_i$ denote the $i^\text{th}$ coordinate of $v$. Also suppose we have a strict total ordering $\prec$ on $\II$.

Let $v\in \calV$ and $i\in \II$. We say $v$ is $i$-initial with respect to $\prec$ (which we often write as $i_\prec$-initial, or just $i$-initial when doing so does not result in ambiguity) if

* The component $v_i$ is nonzero;

* For every $j\in \II$ such that $i\prec j$, the component $v_j$ equals zero.

The rational vector space $\QQ^3$ can be thought of as $\bigoplus_{i\in [3]}\QQ e_i$, where $\{e_1,e_2,e_3\}$ is the canonical basis of $\QQ^3$. Ordering the indexing set $[3]$ in the standard way with $1\prec 2 \prec 3$, we have that the vector $(1,0,0)$ is $1_\prec$-initial, the vector $(1,1,0)$ is $2_\prec$-initial, and the vector $(1,1,1)$ is $3_\prec$-initial.

If $X=\{v^i\}_{i\in \II}$ is a set of vectors lying in $\calC$ such that for each $i\in \II$, $v^i$ is $i_\prec$-initial, then $X$ must be linearly independent. So to find a linearly independent set of vectors in $\calC$, it suffices to define an order $\prec$ on some appropriately sized subset $\JJ \subseteq \II$, and find an $i_\prec$-initial vector lying in $\calC$ for each $i$ in $\JJ$.

In our case, all of our cones live inside the infinite dimensional vector space $\Vn = \bigoplus_{d\in \ZZ} \QQ^{n+1}$. Thus we use the following method to find their dimensions.

Let $\calC$ be a cone lying inside $\Vn$. The following suffices to demonstrate that $\dim \calC = D$.

* Find a finite indexing set $\II(\calC)\subset \{0,\dots,n\}\times \ZZ$ such that $\calC \subset \bigoplus_{(i,d)\in \II(\calC)} \QQ$.

* Find a subspace $\WW(\calC)\leq\bigoplus_{(i,d)\in \II(\calC)} \QQ$ containing $\calC$, with $\dim \WW(\calC)=D$ (this shows that $\dim \calC \leq D$).

* Define an ordering $\prec$ on $\II(\calC)$.

* Find a subset $\JJ\subseteq \II(\calC)$ of size $D$, and a set of diagrams $\{B^{i,d}:(i,d)\in \JJ\}$ inside $\calC$ such that for each $(i,d)\in \JJ$, the diagram $B^{i,d}$ is $(i,d)_\prec$-initial (this shows that $\dim \calC \geq D$).

§ DIMENSION OF $\DN$

In this section, we prove Theorem <ref> on the dimension of the cone $\Dn$, generated by all diagrams of Stanley-Reisner ideals.

§.§ Upper Bound

We start by bounding the dimension from above, by following the first two steps of Method <ref>. In particular we work towards finding a finite indexing set $\II(\Dn)\subset \{0,...,n\}\times\ZZ$ of size $\frac{n(n+1)}{2}$ such that for every diagram $\beta\in \Dn$, and any pair of integers $0\leq i\leq n$ and $d$ with $(i,d)\notin \II(\Dn)$, we have $\beta_{i,d}=0$. This will demonstrate that $\Dn$ actually lies inside the finite-dimensional vector space $\WW(\Dn) = \bigoplus_{(i,d)\in \II(\Dn)}\QQ$, and hence we have $\dim \Dn\leq \dim \WW(\Dn) = \frac{n(n+1)}{2}$.

We begin with the following well-known restrictions on the Betti diagrams of Stanley-Reisner ideals.

Let $\Delta$ be a simplicial complex on vertex set $[n]$ and fix $\beta=\beta(\Delta)$. Suppose $i$ and $d$ are integers with $0\leq i\leq n$. We have $\beta_{i,d}=0$ if either of the following conditions hold:

* $d > n$

* $d \leq i$

Both of these conditions are immediate results of Hochster's Formula.

* There are no subsets of $[n]$ of size greater than $n$.

* If $d\leq i$ then $d-i-2\leq -2$, so no induced subcomplex of $\Delta$ has homology at degree $d-i-2$.

Part (2) of this result can also be seen from more elementary considerations. Specifically, the degrees of the generators of a squarefree monomial ideal are at least one, so the result must hold for $i=0$; and the degrees of the maps in any minimal free resolution are at least one, so the general result follows by induction on $i$.

These inequalities give us a much clearer picture of the shape of the Betti diagrams in $\Dn$. In particular they look like the following.

\begin{equation}\label{Equation: Dn Matrix}
\begin{bmatrix}
\beta_{0,1} & \beta_{1,2} & \dots & \dots & \dots & \beta_{n-1,n}  \\
\beta_{0,2} & \beta_{1,3} & \dots & \dots & \beta_{n-2,n} & \\
\beta_{0,3} & \beta_{1,4} & \dots & \beta_{n-3,n} &\\
\vdots &\vdots & \vdots & & &\\
\beta_{0,n-1} &\beta_{1,n} & & & & \\
\beta_{0,n}
\end{bmatrix}
\end{equation}

Thus, we may define our indexing set $\II(\Dn)$ and subspace $\WW(\Dn)$ as follows.

We define

* $\II(\Dn) := \left\{(i,d)\in \{0,...,n\}\times \ZZ: 0\leq i < d\leq n \right\}$.

* $\WW(\Dn) := \bigoplus_{(i,d)\in \II(\Dn)}\QQ$.

By Proposition <ref>, the cone $\Dn$ must lie in $\WW(\Dn)$ as desired.

For ease of explanation, it will sometimes be useful for us to refer to individual rows of $\II(\Dn)$.

Let $(i,d)\in \II(\Dn)$. We say $(i,d)$ is in row $\rho$ if we have $d-i-1=\rho$.

We can arrange the elements of $\II(\Dn)$ in rows as in Equation (<ref>).

\begin{equation*}\label{Equation: I(Dn) Matrix}
\begin{matrix}
(0,1) & (1,2) & \dots & \dots & \dots & (n-1,n)\\
(0,2) & (1,3) & \dots & \dots & (n-2,n)&\\
(0,3)& (1,4) & \dots & (n-3,n) & &\\
\vdots& \vdots & \vdots & & &\\
(0,n-1)& (1,n) & & & & \\
(0,n)& & & & &
\end{matrix}
\end{equation*}

We can see that row 0 of $\II(\Dn)$ has $n$ elements, row 1 has $n-1$ elements, and so on. In general, for each $0\leq i \leq n-1$, row $i$ of $\II(\Dn)$ has $n-i$ elements. Hence we have

\begin{align*}
|\II(\Dn)|&= \sum_{i=0}^{n-1} (n-i)\\
&= \frac{n(n+1)}{2}
\end{align*}

which means that the expression in Theorem <ref> is an upper bound for $\dim \Dn$.

We can modify the above argument to find an indexing set for the cone $\Dntilde$, generated by diagrams of complexes with no missing vertices. Specifically, if $\Delta$ is a complex with no missing vertices, then for any nonempty subset $\emptyset \neq U \subset V(\Delta)$, the induced subcomplex $\Delta_U$ cannot be equal to $\{\emptyset\}$, and hence $\Hred_{-1}(\Delta_U)=0$. By Hochster's formula this means that $\beta_{0,1}(\Delta)=\beta_{1,2}(\Delta)=...=\beta_{n-1,n}(\Delta)=0$. Thus for diagrams in the cone $\Dntilde$, the top row of Betti numbers in Equation (<ref>) are all zero. In particular the indexing set $\II(\Dntilde)$ is equal to $\II(\Dn)-\{(0,1),\dots,(n-1,n)\}$, which has size $\frac{n(n-1)}{2}$.

§.§ Lower Bound

To complete our proof of Theorem <ref>, we need to show that the space $\WW(\Dn)$ is in fact the minimal subspace of $\Vn$ containing $\Dn$, by following the last two steps of Method <ref>.

In particular we present an ordering $\prec$ on $\II(\Dn)$ and a set of $(i,d_\prec)$-initial diagrams in $\Dn$ for each $(i,d)$ in $\II(\Dn)$. The ordering we choose is as follows.

For any two pairs $(i,d)$ and $(i',d')$ in $\II(\Dn)$ we write $(i,d)\prec (i',d')$ if $d-i< d'-i'$, or $d-i= d'-i'$and $i<i'$.

In other words we say $(i,d)\prec(i',d')$ if $(i,d)$ lies in a lower numbered row, or if they both lie in the same row with $i<i'$.

For convenience, we extend the terminology of Definition <ref> by declaring a complex $\Delta$ to be $(i,d)_\prec$-initial if the diagram $\beta(\Delta)$ is $(i,d)_\prec$-initial.

Thus we need to find a set of $(i,d)_\prec$-initial complexes on vertex set $[n]$. The following lemma will be helpful in this, because it allows us to broaden our search from complexes with exactly $n$ vertices to complexes with at most $n$ vertices.

For any positive integer $m<n$, we have $\calD_m\subset \Dn$.

If $\Delta$ is a complex on vertex set $[m]$, then by Lemma <ref>, we can extend it to a graph on $[n]$ by taking a cone over it a total of $n-m$ times, without affecting its Betti diagram. This means the diagram $\beta(\Delta)$ lies inside $\Dn$, and the result follows.

There is now an obvious candidate for our family of $(i,d)_\prec$-initial complexes.

Let the ordering $\prec$ on $\II(\Dn)$ be as in Definition <ref>, and let $(i,d)\in \II(\Dn)$. The complex $\Skel_{d-i-2}([d])$ is $(i,d)_\prec$-initial.

By Lemma <ref> the complex $\Skel_r([m])$ is $(m-r-2,m)$-initial. The result follows from substituting $d=m$ and $i=m-r-2$.

Suppose we have $n=4$. The following diagram depicts, for each index $(i,d)\in \II(\Dn[4])$, an $(i,d)$-initial complex on up to $4$ vertices (we use a $\times$ symbol to denote missing vertices).

point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[scale=1.2] at (0,0) $\times$;

point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[scale=1.2] at (0,0) $\times$;
[scale=1.2] at (2,0) $\times$;
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[scale=1.2] at (0,0) $\times$;
[scale=1.2] at (2,0) $\times$;
[scale=1.2] at (1,1.7) $\times$;
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[scale=1.2] at (0,0) $\times$;
[scale=1.2] at (2,0) $\times$;
[scale=1.2] at (0,2) $\times$;
[scale=1.2] at (2,2) $\times$;

$\Skel_{-1}([1])$        $\Skel_{-1}([2])$         $\Skel_{-1}([3])$         $\Skel_{-1}([4])$

$(0,1)$-initial        $(1,2)$-initial         $(2,3)$-initial        $(3,4)$-initial

point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[point, scale=1.2] at (0,0) ;
[point, scale=1.2] at (2,0) ;

point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[point, scale=1.2] at (0,0) ;
[point, scale=1.2] at (2,0) ;
[point, scale=1.2] at (1,1.7) ;
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[point, scale=1.2] at (0,0) ;
[point, scale=1.2] at (2,0) ;
[point, scale=1.2] at (0,2) ;
[point, scale=1.2] at (2,2) ;

$\Skel_0([2])$        $\Skel_0([3])$         $\Skel_0([4])$

$(0,2)$-initial        $(1,3)$-initial         $(2,4)$-initial

point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[point, scale=1.2] (1) at (0,0) ;
[point, scale=1.2] (2) at (2,0) ;
[point, scale=1.2] (3) at (1,1.7) ;

(1) – (2) – (3) – (1);
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[point, scale=1.2] (1) at (0,0) ;
[point, scale=1.2] (2) at (2,0) ;
[point, scale=1.2] (3) at (0,2) ;
[point, scale=1.2] (4) at (2,2) ;

(1) – (2) – (3) – (4) – (1);
(1) – (3);
(2) – (4);

$\Skel_1([3])$        $\Skel_2([4])$

$(0,3)$-initial        $(1,4)$-initial

[scale=0.55][line join = round, line cap = round]

(4) at (0,sqrt(2),0);
(3) at (-.5*sqrt(3),0,-.5);
(2) at (0,0,1);
(1) at (.5*sqrt(3),0,-.5);

[fill=lightgray,fill opacity=.5] (2)–(1)–(4)–cycle;
[fill=gray,fill opacity=.5] (3)–(2)–(4)–cycle;

Note that while many of these complexes have fewer than $4$ vertices, their Betti diagrams all still lie inside $\Dn[4]$ by Lemma <ref>.

Putting these results together we can now prove Theorem <ref>.

We have already seen that
$$\dim \Dn \leq
|\II(\Dn)| = \frac{n(n+1)}{2}.$$

We can show that $\dim \Dn \geq |\II(\Dn)|$ by finding a linearly independent set of diagrams in $\Dn$ of size $|\II(\Dn)|$.

To that end, let $(i,d)$ be an index in $\II(\Dn)$ and define the ordering $\prec$ on $\II(\Dn)$ as in Definition <ref>. Proposition <ref> tells us that the complex $\Skel_{d-i-2}([d])$ is $(i,d)_{\prec}$-initial. It also has $d$ vertices, and because $d\leq n$, Lemma <ref> tells us that its diagram lies inside $\Dn$. Thus we have an $(i,d)$-initial diagram in $\Dn$ for each $(i,d)$ in $\II(\Dn)$. This completes the proof.

If we restrict the ordering $\prec$ to the subset $\II(\Dntilde)\subset \II(\Dn)$ (as defined in Remark <ref>), the above proof also shows that the cone $\Dntilde$ contains an $(i,d)$-initial diagram for every index $(i,d)$ in $\II(\Dntilde)$, and hence we have $\dim \Dntilde = |\II(\Dntilde)|=\frac{n(n-1)}{2}$. This is because all of the $(i,d)$-initial complexes chosen for the indices $(i,d)$ in $\II(\Dntilde)$ are $r$-skeletons for some $r\geq 0$, and therefore have no missing vertices, so their diagrams also lie inside $\Dntilde$.

Before we move on to finding the dimension of our three subcones of $\Dn$ we note the following important lemma, which we can use to construct $(i,d)_\prec$-initial diagrams in $\Dn$ recursively. While we did not require this lemma to demonstrate the dimension of $\Dn$, analogous results will be crucial in demonstrating the dimensions of the three subcones we are about to consider.

Let $\prec$ be as in Definition <ref>, suppose $n>2$, and let $(i,d)$ be an index in $\II(\Dn)$ with $d-i\geq 2$. If $\Delta$ is an $(i-1,d-2)_\prec$-initial complex then the complex $S\Delta$ is $(i,d)_\prec$-initial.

By Lemma <ref>, we have $\beta_{i,d}(S\Delta) = \beta_{i,d}(\Delta) + \beta_{i-1,d-2}(\Delta)$. By the $(i-1,d-2)$-initiality of $\beta(\Delta)$, we have $\beta_{i-1,d-2}(\Delta) \neq 0$, so $\beta_{i,d}(S\Delta)$ must be nonzero too.

Now let $(i',d')\in \II(\Dn)$ with $(i,d)\prec (i',d')$. Again, by Lemma <ref> we have $\beta_{i',d'}(S\Delta) = \beta_{i',d'}(\Delta) + \beta_{i'-1,d'-2}(\Delta)$. We must have $(i-1,d-2)\prec (i'-1,d'-2)$, and also $(i'-1,d'-2)\prec (i',d')$ because they are in different rows. Hence, by the $(i-1,d-2)$-initiality of $\beta(\Delta)$, both the terms $\beta_{i',d'}(\Delta)$ and $\beta_{i'-1,d'-2}(\Delta)$ are zero, and $\beta_{i',d'}(S\Delta)$ is zero too.

This shows that $\beta(S\Delta)$ is $(i,d)_\prec$-initial as required.

§ DIMENSION OF $\CN$

In this section, we prove Theorem <ref>, on the dimension of the subcone $\Cn\subset \Dn$ generated by diagrams of edge ideals.

§.§ Upper Bound

As before, we start by bounding the dimension from above, by finding an appropriately sized indexing set $\II(\Cn)$ at which the diagrams of $\Cn$ are all nonzero, and hence a vector space $\WW(\Cn)$ containing $\Cn$ of the appropriate dimension. Note that we must have $\II(\Cn)\subseteq \II(\Dn)$ and hence $\WW(\Cn)\leq \WW(\Dn)$ because every edge ideal is a Stanley-Reisner ideal. Moreover, every generator of an edge ideal has degree two, and hence we have that $\II(\Cn)$ is actually a subset of $\II(\Dntilde)$, so it does not contain the indices $(0,1),\dots,(n-1,n)$.

To find our indexing set $\II(\Cn)$, we need to obtain further restrictions on the positions of the nonzero values of the diagrams in $\Cn$. The following result turns out to be sufficient.

Let $G$ be a graph on vertex set $[n]$, and set $\beta=\beta(G)$. For any integers $i$ and $d$ with $d > 2i +2$, we have $\beta_{i,d} = 0$.

This is Lemma 2.2 in [28].

This additional inequality gives us a clearer picture of the shape of the Betti diagrams in $\Cn$. Specifically, if $n = 2r$ is even, then the diagrams $\beta\in \Cn$ look like this.

\begin{equation}\label{Equation: Cn Even Matrix}
\begin{bmatrix}
\beta_{0,2} & \beta_{1,3} & \beta_{2,4} & \dots & \dots & \beta_{n-3,n-1} & \beta_{n-2,n}\\
& \beta_{1,4} & \beta_{2,5} & \dots & \dots & \beta_{n-3,n} &\\
& & \ddots & & & &\\
& & & \beta_{r-1,2r} & & & \\
\end{bmatrix}
\end{equation}

If $n = 2r+1$ is odd, then they look like this.

\begin{equation}\label{Equation: Cn Odd Matrix}
\begin{bmatrix}
\beta_{0,2} & \beta_{1,3} & \beta_{2,4} & \dots & \dots & \dots & \beta_{n-3,n-1} & \beta_{n-2,n}\\
& \beta_{1,4} & \beta_{2,5} & \dots & \dots & \dots & \beta_{n-3,n} &\\
& & \ddots & & & & &\\
& & & \beta_{r-1,2r} & \beta_{r, 2r+1} & & & \\
\end{bmatrix}
\end{equation}

Thus, we may define our indexing set $\II(\Cn)$ and subspace $\WW(\Cn)$ as follows.

We define

* $\II(\Cn) := \left\{(i,d)\in \{0,...,n\}\times \ZZ: i+2\leq d\leq \min\{2i+2,n\} \right\}$.

* $\WW(\Cn) := \bigoplus_{(i,d)\in \II(\Cn)}\QQ$.

By Proposition <ref>, the cone $\Cn$ must lie in $\WW(\Cn)$ as desired.

Just as with $\II(\Dn)$ we can arrange the elements of $\II(\Cn)$ in rows to match the above Betti diagrams (we number the rows of $\II(\Cn)$ using the same convention we used for the rows of $\II(\Dn)$ in Definition <ref>).
So if $n=2r$ then the rows of $\II(\calC_{2r})$ look like the following.

\begin{equation*}
\begin{matrix}
(0,2) & (1,3) & (2,4) & \dots & \dots & (n-3,n-1) & (n-2,n)\\
& (1,4) & (2,5) & \dots & \dots & (n-3,n) &\\
& & \ddots & & & &\\
& & & (r-1,2r) & & & \\
\end{matrix}
\end{equation*}

And if $n=2r+1$ then the rows of $\II(\calC_{2r+1})$ look like the following.

\begin{equation*}
\begin{matrix}
(0,2) & (1,3) & (2,4) & \dots & \dots & \dots & (n-3,n-1) & (n-2,n)\\
& (1,4) & (2,5) & \dots & \dots & \dots & (n-3,n) &\\
& & & \ddots & & & & &\\
& & & & (r-1,2r) & (r,2r+1) &  & & \\
\end{matrix}
\end{equation*}

We can see that row 1 of $\II(\Cn)$ has $n-1$ elements, row 2 has $n-3$ elements, and so on. In general, for each $1\leq i \leq r$, row $i$ of $\II(\Cn)$ has $n-2i+1$ elements.

Hence we have

\begin{align*}
|\II(\calC_{2r})|&= \sum_{i=1}^r (2r-2i+1)\\
&= \sum_{i=1}^r (2r) - \sum_{i=1}^{r}(2i-1)\\
&= 2r^2 - r^2\\
&= r^2
\end{align*}

\begin{align*}
|\II(\calC_{2r+1})|&=\sum_{i=1}^r (2r+1-2i+1)\\
&= \sum_{i=1}^r (2r+1) - \sum_{i=1}^{r}(2i-1)\\
&= (2r^2 + r) - r^2\\
&= r^2 + r \, .
\end{align*}

Therefore, the expressions in Theorem <ref> are upper bounds for $\dim \Cn$.

§.§ Lower Bound

We are left with the task of showing that the space $\WW(\Cn)$ is in fact the minimal subspace of $\WW(\Dn)$ containing the cone $\Cn$, which we do by finding an appropriately sized linearly independent set of diagrams in $\Cn$, following Method <ref>. Many of the constructions and lemmas we need are analogues of the ones we employed for the cone $\Dn$.

For our ordering $\prec$ on $\II(\Cn)$ we simply take the restriction of the ordering on $\II(\Dn)$ given in Definition <ref>. And just as we did with complexes, we now extend the terminology of Definition <ref> to graphs by declaring a graph $G$ to be $(i,d)_\prec$-initial if the diagram $\beta(G)$ is $(i,d)_\prec$-initial.

The cone $\Cn$ also admits a direct analogue of Lemma <ref>, which allows us to broaden our search from graphs with exactly $n$ vertices to graphs with at most $n$ vertices.

For any positive integer $m < n$, we have $\Cn[m] \subset \Cn$.

If $G$ is a graph on vertex set $[m]$, then by Corollary <ref>, we can extend it to a graph on $[n]$ by adding some isolated vertices, without affecting its Betti diagram. This means the diagram $\beta(G)$ lies inside $\Cn$, and the result follows.

Unfortunately we cannot use the diagrams of skeleton complexes as our $(i,d)$-initial diagrams in $\Cn$, because not all skeleton complexes are independence complexes of graphs, and hence the majority of the $(i,d)$-initial diagrams we found in the last section do not lie in the cone $\Cn$.

However, the diagrams of the 0-skeletons $\Skel_0([m])$  do lie in $\Cn$, because we have $\Ind(K_m)=\Skel_0([m])$. Thus we can find $(i,d)$-initial diagrams for every $(i,d)$ in row 1 of $\II(\Cn)$. To find the rest, we use the following analogue of Lemma <ref>.

Let $\prec$ be as in Definition <ref>, suppose $n>2$, and let $(i,d)$ be an index in $\II(\Cn)$ with with $d-i\geq 3$. If $\Delta$ is an $(i-1,d-2)_\prec$-initial graph then the graph $G+L$ is $(i,d)_\prec$-initial.

This follows directly from Lemma <ref> because $\Ind(G+L)=S\Ind(G)$.

Before we explain the procedure for finding $(i,d)$-initial graphs in the general case, we present a specific example to illustrate the basic principle.

The set $\II(\calC_6)$ has size $3^2 = 9$, and it looks like the following.

\begin{equation*}
\begin{matrix}
(0,2) & (1,3) & (2,4) & (3,5) & (4,6)\\
& (1,4) & (2,5) & (3,6) &\\
& & (2,6) & &
\end{matrix}
\end{equation*}

So we want to find nine linearly independent diagrams in $\mathcal{C}_6$, one for each $(i,d)\in \II(\calC_6)$. The ordering $\prec$ on $\II(\calC_6)$ is $(0,2)\prec (1,3) \prec (2,4) \prec (3,5) \prec (4,6) \prec (1,4) \prec (2,5) \prec (3,6) \prec (2,6)$.

By Proposition <ref>, we see that the complete graph on $2$ vertices, $K_2$, is $(0,2)$-initial. Similarly, $K_3$ is $(1,3)$-initial, $K_4$ is $(2,4)$-initial, $K_5$ is $(3,5)$-initial and $K_6$ is $(4,6)$-initial.

From the above, and Lemma <ref>, we also find that $K_2+L$ is $(1,4)$-initial, $K_3+L$ is $(2,5)$-initial and $K_4+L$ is $(3,6)$-initial. Similarly, we can see that $K_2+2L$ is $(2,6)$-initial.

So placing each graph in its corresponding position in $\II(\Cn)$, we get the following.

\begin{equation*}
\begin{matrix}
K_2 & K_3 & K_4 & K_5 & K_6\\
& K_2+L & K_3+L & K_4+L &\\
& & K_2+2L & &
\end{matrix}
\end{equation*}

[scale = 0.4]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point, scale=1.2] at (1,0) ;
(b)[point, scale=1.2] at (1,1.7) ;

(a.center) – (b.center);
[scale = 0.4]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point] at (0,0) ;
(b)[point] at (2,0) ;
(c)[point] at (1,1.7) ;

(a.center) – (b.center) – (c.center) – cycle;
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[point, scale=1.2] (1) at (0,0) ;
[point, scale=1.2] (2) at (2,0) ;
[point, scale=1.2] (3) at (0,2) ;
[point, scale=1.2] (4) at (2,2) ;

(1) – (2) – (3) – (4) – (1);
(1) – (3);
(2) – (4);
[scale = 0.57]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point] at (0,1) ;
(b)[point] at (0.951,0.309) ;
(c)[point] at (0.588,-0.809) ;
(d)[point] at (-0.588,-0.809) ;
(e)[point] at (-0.951,0.309) ;

(a.center) – (b.center) – (c.center) – (d.center) – (e.center) – cycle;
(a.center) – (c.center) – (e.center) – (b.center) – (d.center) – cycle;
[scale = 0.3]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(1)[point] at (0,0) ;
(2)[point] at (2,0) ;
(3)[point] at (3,1.7) ;
(4)[point] at (2,3.4) ;
(5)[point] at (0,3.4) ;
(6)[point] at (-1,1.7) ;

(1.center) – (2.center) – (3.center) – (4.center) – (5.center) – (6.center) – cycle;
(1.center) – (3.center) – (5.center) – cycle;
(2.center) – (4.center) – (6.center) – cycle;
(1.center) – (4.center);
(2.center) – (5.center);
(3.center) – (6.center);

$K_2$       $K_3$        $K_4$        $K_5$        $K_6$

$(0,2)$-initial        $(1,3)$-initial        $(2,4)$-initial        $(3,5)$-initial        $(4,6)$-initial

point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[point, scale=1.2] (1) at (0,0) ;
[point, scale=1.2] (2) at (2,0) ;
[point, scale=1.2] (3) at (0,2) ;
[point, scale=1.2] (4) at (2,2) ;

(1) – (3);
(2) – (4);
[scale = 0.4]
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
(a)[point] at (0,0) ;
(b)[point] at (2,0) ;
(c)[point] at (1,1.7) ;

(d)[point] at (3.4,0) ;
(e)[point] at (3.4,1.7) ;

(a.center) – (b.center) – (c.center) – cycle;
(d.center) – (e.center);
point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[point, scale=1.2] (1) at (0,0) ;
[point, scale=1.2] (2) at (2,0) ;
[point, scale=1.2] (3) at (0,2) ;
[point, scale=1.2] (4) at (2,2) ;

[point, scale=1.2] (5) at (3.4,0) ;
[point, scale=1.2] (6) at (3.4,2) ;

(1) – (2) – (3) – (4) – (1);
(1) – (3);
(2) – (4);
(5) – (6);

$K_2+L$        $K_3+L$        $K_4+L$

$(1,4)$-initial        $(2,5)$-initial        $(3,6)$-initial

point=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=2pt,minimum height=2pt]
[point, scale=1.2] (1) at (0,0) ;
[point, scale=1.2] (2) at (2,0) ;
[point, scale=1.2] (3) at (0,2) ;
[point, scale=1.2] (4) at (2,2) ;
[point, scale=1.2] (5) at (4,0) ;
[point, scale=1.2] (6) at (4,2) ;

(1) – (3);
(2) – (4);
(5) – (6);

All of the graphs $K_2,K_3,K_4,K_6,K_2+L,K_3+L,K_4+L$ and $K_2+2L$ have $6$ vertices or fewer, so by Lemma <ref>, their diagrams all lie in $\mathcal{C}_6$ as required.

In the above example, the graphs associated to the top row of $\II(\Cn)$ were the complete graphs on $n$ or fewer vertices, and we found graphs for each subsequent row by adding disjoint edges to the graphs we had already found. We can generalise this process to arbitrary values of $n$, and thus prove Theorem <ref>, as below.

We have already seen that
$$\dim \Cn \leq
|\II(\Cn)| = \begin{cases}
r^2  &\text{ if } n=2r \\
r^2 + r &\text{ if } n=2r+1.\\
\end{cases}$$

We now show that $\dim\Cn \geq |\II(\Cn)|$ by exhibiting a set of graphs $\{G_{i,d}:(i,d)\in \II(\Cn)\}$ such that for each $(i,d)\in \II(\Cn)$, $G_{i,d}$ is $(i,d)$-initial (with respect to the ordering $\prec$ given in Definition <ref>) and has $d$ vertices (by Lemma <ref> this is sufficent to ensure that the diagrams $\beta(G_{i,d})$ lie in $\Cn$, because for each $(i,d)\in \II(\Cn)$ we have $d\leq n$).

We proceed by induction on $n\geq 1$. The set $\II(\calC_1)$ is empty, so for the base case $n=1$ there is nothing to prove.

For the inductive step, suppose that $n>1$ and that we have a set $\{G_{i,d}:(i,d)\in \II(\calC_{n-1})\}$ where each $G_{i,d}$ is an $(i,d)$-initial graph on $d$ vertices. The set $\II(\calC_{n-1})$ is a subset of $\II(\Cn)$, so we can extend our set of graphs to a set $\{G_{i,d}:(i,d)\in \II(\Cn)\}$ by adding graphs $G_{i,d}$ for the values of $(i,d)$ in $\II(\Cn)-\II(\calC_{n-1})$.

By Proposition <ref>, the complete graph $K_n$ is $(n-2,n)$-initial and has $n$ vertices, so we set $G_{n-2,n}=K_n$. For every other value of $(i,d)$ in $\II(\Cn)-\II(\calC_{n-1})$, the index $(i-1,d-2)$ is in $\II(\calC_{n-1})$, and hence we define $G_{i,d}=G_{i-1,d-2}+L$. This graph has $(d-2)+2=d$ vertices and by Lemma <ref>, we know it must be $(i,d)$-initial. This completes the proof.

§ DIMENSION OF $\DNH$

Now that we have found the dimension of our larger cones $\Dn$ and $\Cn$, we turn our attention to the subcones $\Dnh$ and $\Cnh$ generated by diagrams of ideals of height $h$.

We begin with the cone $\Dnh$ generated by diagrams of Stanley-Reisner ideals of height $h$ (or equivalently, all complexes of codimension $h$), and work towards proving Theorem <ref>.

§.§ Upper Bound

As before, we begin by searching for an indexing set $\II(\Dnh)$ and a minimal subspace $\WW(\Dnh)$.

Recall that the diagrams $\beta(\Delta)$ that generate this cone come from complexes of codimension $h$, by Lemma <ref>. This allows us to derive the two following results.

Let $\Delta$ be a complex on vertex set $[n]$ of codimension $h$, and set $\beta=\beta(\Delta)$. For any integers $i$ and $d$ with $d - i > n-h+1$, we have $\beta_{i,d} = 0$.

The fact that $\codim \Delta=h$ means that $\Delta$, along with all of its induced subcomplexes, has no faces of dimension higher than $n-h-1$. In particular, for any subset $U\subseteq [n]$ and any $j> n-h-1$, we must have $\Hred_j(\Delta_U) = 0$. Thus by Hochster's Formula, we have that $\beta_{i,d}=0$ whenever $d-i-2>n-h-1$. The result follows.

Let $\Delta$ be a complex on vertex set $[n]$ of codimension $h$, and set $\beta=\beta(\Delta)$. For any integers $h\leq i \leq n-1$ we have $\beta_{i,i+1}=0$.

Because $\Delta$ has codimension $h$ it must have a facet of size $n-h$. Thus it can have at most $h$ missing vertices, which means that the Betti numbers $\beta_{h,h+1},\dots,\beta_{n-1,n}$ must all be zero by Remark <ref>.

These inequalities give us a much clearer picture of what the diagrams in $\Dnh$ look like.

\begin{equation}\label{Equation: Dnh Matrix}
\begin{bmatrix}
\beta_{0,1} & \dots & \beta_{h-1,h} &  & & &\\
\beta_{0,2} & \dots & \beta_{h-1,h+1} & \dots & \dots &\dots & \beta_{n-2,n} \\
\beta_{0,3} & \dots & \beta_{h-1,h+2} & \dots & \dots & \beta_{n-3,n} & \\
\vdots &\vdots & \vdots & \vdots & \vdots & & &\\
\beta_{0,n-h} &\dots & \beta_{h-1,n-1} & \beta_{h,n} & & & & \\
\beta_{0,n-h+1} &\dots & \beta_{h-1,n} & & & & & \\
\end{bmatrix}
\end{equation}

We have now found all the restriction we need in order to define our indexing set $\II(\Dnh)$.

We define
$$\II(\Dnh) := \left\{(i,d)\in \II(\Dn): d-i \leq n-h+1 \}-\{(h,h+1),\dots, (n-1,n)\right\}.$$

Unlike in the cases of the cones $\Cn$ and $\Dn$, the minimal subspace $\WW(\Dnh)$ containing the cone $\Dnh$ is not simply the space carved out by this indexing set, $\bigoplus_{(i,d)\in \II(\Dnh)}\QQ$. This is because, as well as satisfying the conditions of Propositions <ref> and <ref>, we know from Remark <ref> that the diagrams $\beta$ in $\Dnh$ must also satisfy the Herzog-Kühl equations $\HK_1(\beta)=\dots = \HK_{h-1}(\beta)=0$. Thus the space $\WW(\Dnh)$ is actually equal to the following.

We define
$$\WW(\Dnh) := \left\{\beta \in \bigoplus_{(i,d)\in \II(\Dnh)}\QQ : \HK_1(\beta)=\dots =\HK_{h-1}(\beta)=0\right\}.$$

To prove that the formula given in Theorem <ref> is an upper bound for $\dim \Dnh$, we need to show that $\dim \WW(\Dnh)=\frac{n(n-1)}{2}-\frac{h(h-1)}{2}+1$. This is the content of the following proposition.

Let $\II(\Dnh)$ and $\WW(\Dnh)$ be as in Definitions <ref> and <ref>. We have

* $|\II(\Dnh)|=\frac{n(n-1)}{2}-\frac{h(h-1)}{2}+h$.

* $\dim \WW(\Dnh)=\frac{n(n-1)}{2}-\frac{h(h-1)}{2}+1$.

For part (1), we start by arranging the elements of $\II(\Dnh)$ in rows as in Equation (<ref>).

\begin{equation*}\label{Equation: I(Dnh) Matrix}
\begin{matrix}
(0,1) & \dots & (h-1,h) &  & & &\\
(0,2) & \dots & (h-1,h+1) & \dots & \dots &\dots & (n-2,n) \\
(0,3) & \dots & (h-1,h+2) & \dots & \dots & (n-3,n) & \\
\vdots &\vdots & \vdots & \vdots & \vdots & & &\\
(0,n-h) &\dots & (h-1,n-1) & (h,n) & & & & \\
(0,n-h+1) &\dots & (h-1,n) & & & & & \\
\end{matrix}
\end{equation*}