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Susan Morey

Publications and source records attributed to Susan Morey.

At least 19 recordsLinked to original sources
arXiv2026

Regular sequences of linear forms on monomial ideals

In this paper we establish a means of using the combinatorics associated to a general monomial ideal $I$ in a polynomial ring $R$ to find a regular sequence of linear forms on $R/I$. The sequence of linear forms provides an effective lower bound on ${\rm{depth}}(R/I)$. When $I$ is the edge ideal of a graph, we provide conditions under which this bound is an equality, allowing the realization of the depth via a regular sequence of linear polynomials. In addition, we explicitly describe the minimal primes of $(I, f_1, \ldots, f_q)$, when $f_1, \ldots, f_q$ are homogeneous polynomials of degree one with pairwise disjoint support and $I$ is any monomial ideal. Finally, we propose a conjecture on the form of all associated primes of the ideal $(I, f_1, \ldots, f_q)$, when $I$ is the edge ideal of a graph and $f_i$ are disjoint stars on $I$.

math.AC↗
arXiv2026

Cellular resolutions of second powers of square-free monomial ideals with divisibility relations

Using divisibility relations between the generators of a square-free monomial ideal $I$, we describe divisibility relations between the generators of the second power $I^2$. We then employ discrete Morse theory to produce a cellular free resolution of $I^2$ which is minimal for specific ideals that are extremal with respect to a given divisibility relation. In particular, we provide sharp bounds on the projective dimension of $I^2$ when the generators of $I$ satisfy at least one divisibility relation.

math.AC↗
arXiv2026

Symbolic powers and integral closures via extremal ideals

This paper demonstrates that extremal ideals can be used to great effect to compute integral closures of powers and symbolic powers of square-free monomial ideals. We show that the generators of these powers are images of the generators of the corresponding powers of extremal ideals under a specific ring homomorphism. Extremal ideals provide sharp bounds for a variety of invariants widely studied in the literature, including resurgence, asymptotic resurgence, and symbolic defect, as well as Betti numbers of symbolic powers and of integral closures of powers of square-free monomial ideals. When restricted to the class of extremal ideals, algebraic computations are reduced to problems of discrete geometry and linear programming, allowing the use of a wide variety of techniques. As a result, in situations where computations are feasible for extremal ideals, we provide concrete sharp bounds for many of these invariants. Our methods reduce finding homological invariants and algebraic constructions for infinitely many ideals to computations for a single highly symmetric ideal, based solely on the number of generators.

math.AC↗
arXiv2025

Divisibility Relations and $\mathcal{D}$-Extremal ideals

A divisibility relation between the generators of a square-free monomial ideal formally encodes the situation when one generator divides the least common multiple of some other generators. The divisibility relations contribute to the deletion of some parts of the Taylor resolution of the ideal, and therefore lead to finding a resolution closer to the minimal one. Motivated by this observation, for a given set $\mathcal{D}$ of divisibility relations, we study all square-free monomials satisfying the relations in $\mathcal{D}$. We define a class of square-free monomial ideals called $\mathcal{D}$-extremal ideals $\mathcal{E}_\mathcal{D}$ , and show it is optimal in the sense that it is an ideal satisfying exactly those divisibility relations coming from $\mathcal{D}$, and no others. We then show that $\mathcal{E}_\mathcal{D}$ is extremal in the sense that the resolution and betti numbers of the powers of any square-free monomial ideal satisfying the relations in $\mathcal{D}$ are bounded by those of the same powers of $\mathcal{E}_\mathcal{D}$.

math.AC↗
arXiv2025

Realizing resolutions of powers of extremal ideals

Extremal ideals are a class of square-free monomial ideals which dominate and determine many algebraic invariants of powers of all square-free monomial ideals. For example, the $r^{th}$ power ${\mathcal{E}_q}^r$ of the extremal ideal on $q$ generators has the maximum Betti numbers among the $r^{th}$ power of any square-free monomial ideal with $q$ generators. In this paper we study the combinatorial and geometric structure of the (minimal) free resolutions of powers of square-free monomial ideals via the resolutions of powers of extremal ideals. Although the end results are algebraic, this problem has a natural interpretation in terms of polytopes and discrete geometry. Our guiding conjecture is that all powers ${\mathcal{E}_q}^r$ of extremal ideals have resolutions supported on their Scarf simplicial complexes, and thus their resolutions are as small as possible. This conjecture is known to hold for $r \leq 2$ or $q \leq 4$. In this paper we prove the conjecture holds for $r=3$ and any $q\geq 1$ by giving a complete description of the Scarf complex of ${\mathcal{E}_q}^3$. This effectively gives us a sharp bound on the betti numbers and projective dimension of the third power of any square-free momomial ideal. For large $i$ and $q$, our bounds on the $i^{th}$ betti numbers are an exponential improvement over previously known bounds. We also describe a large number of faces of the Scarf complex of ${\mathcal{E}_q}^r$ for any $r,q \geq 1$.

math.AC↗
arXiv2024

Regular Edges, Matchings and Hilbert Series

When $I$ is the edge ideal of a graph $G$, we use combinatorial properities, particularly Property $P$ on connectivity of neighbors of an edge, to classify when a binomial sum of vertices is a regular element on $R/I(G)$. Under a mild separability assumption, we identify when such elements can be combined to form a regular sequence. Using these regular sequences, we show that the Hilbert series and corresponding $h$-vector can be calculated from a related graph using a simplified calculation on the $f$-vector, or independence vector, of the related graph. In the case when the graph is Cohen-Macaulay with a perfect matching of regular edges satisfying the separability criterion, the $h$-vector of $R/I(G)$ will be precisely the $f$-vector of the Stanley-Reisner complex of a graph with half as many vertices as $G$.

math.AC↗
arXiv2024

Gapfree graphs and powers of edge ideals with linear quotients

Let $I(G)$ be the edge ideal of a gapfree graph $G$. An open conjecture of Nevo and Peeva states that $I(G)^q$ has linear resolution for $q\gg 0$. We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if $I(G)^q$ has linear quotients for some integer $q\geq 1$, then $I(G)^{s}$ has linear quotients for all $s\geq q$. We give a partial solution to this conjecture, and identify conditions under which only finitely many powers need to be checked. It is known that if $G$ does not contain a cricket, a diamond, or a $C_4$, then $I(G)^q$ has linear resolution for $q \geq 2$. We construct a family of gapfree graphs $G$ containing cricket, diamond, $C_4$ together with $C_5$ as induced subgraphs of $G$ for which $I(G)^q$ has linear quotients for $q \ge 2$.

math.AC↗
arXiv2024

Scarf complexes of graphs and their powers

Every multigraded free resolution of a monomial ideal I contains the Scarf multidegrees of I. We say I has a Scarf resolution if the Scarf multidegrees are sufficient to describe a minimal free resolution of I. The main question of this paper is which graphs G have edge ideal I(G) with a Scarf resolution? We show that I(G) has a Scarf resolution if and only if G is a gap-free forest. We also classify connected graphs for which all powers of I(G) have Scarf resolutions. Along the way, we give a concrete description of the Scarf complex of any forest. For a general graph, we give a recursive construction for its Scarf complex based on Scarf complexes of induced subgraphs.

math.AC↗
arXiv2023

Hilbert Coefficients and Sally Modules: A Survey of Vasconcelos' Contributions

This paper surveys and summarizes Wolmer Vasconcelos' results surrounding multiplicities, Hilbert coefficients, and their extensions. We particularly focus on Vasconcelos' results regarding multiplicities and Chern coefficients, and other invariants which they bound. The Sally module is an important instrument introduced by Vasconcelos for this study, which naturally relates Hilbert coefficients to reduction numbers.

math.AC↗
arXiv2022

Simplicial Resolutions of Powers of Square-free Monomial Ideals

The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of generators of the ideal. More precisely, for every pair of fixed integers $r$ and $q$, we construct a simplicial complex that supports a free resolution of the $r$-th power of any square-free monomial ideal with $q$ generators. The resulting resolution is significantly smaller than the Taylor resolution, and is minimal for special cases. Considering the relations on the generators of a fixed ideal allows us to further shrink these resolutions. We also introduce a class of ideals called "extremal ideals", and show that the Betti numbers of powers of all square-free monomial ideals are bounded by Betti numbers of powers of extremal ideals. Our results lead to upper bounds on Betti numbers of powers of any square-free monomial ideal that greatly improve the binomial bounds offered by the Taylor resolution.

math.AC↗
arXiv2021

Regular Sequences On Squares Of Monomial Ideals

We use initially regular sequences that consist of linear sums to explore the depth of $R/I^2$, when $I$ is a monomial ideal in a polynomial ring $R$. We give conditions under which these linear sums form regular or initially regular sequences on $R/I^2$. We then obtain a criterion for when $\depth R/I^2>1$ and a lower bound on $\depth R/I^2$.

math.AC↗
arXiv2021

Powers of graphs & applications to resolutions of powers of monomial ideals

This paper is concerned with the question of whether geometric structures such as cell complexes can be used to simultaneously describe the minimal free resolutions of all powers of a monomial ideal. We provide a full answer in the case of square-free monomial ideals of projective dimension one, by introducing a combinatorial construction of a family of (cubical) cell complexes whose 1-skeletons are powers of a graph that supports the resolution of the ideal.

math.AC↗
arXiv2019

Algebraic algorithms for even circuits in graphs

We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs.

math.CO↗
arXiv2019

Depth of powers of squarefree monomial ideals

We derive two general bounds for the depths of powers of squarefree monomial ideals corresponding to hyperforests. These bounds generalize known bounds for the depths of squarefree monomial ideals, which were given in terms of the edgewise domination number of the corresponding hypergraphs and the lengths of initially regular sequences with respect to the ideals.

math.AC↗
arXiv2018

Initially regular sequences and depths of ideals

For an arbitrary ideal $I$ in a polynomial ring $R$ we define the notion of initially regular sequences on $R/I$. These sequences share properties with regular sequences. In particular, the length of an initially regular sequence provides a lower bound for the depth of $R/I$. Using combinatorial information from the initial ideal of $I$ we construct sequences of linear polynomials that form initially regular sequences on $R/I$. We identify situations where initially regular sequences are also regular sequences, and we show that our results can be combined with polarization to improve known depth bounds for general monomial ideals.

math.AC↗
arXiv2018

Edge ideals of oriented graphs

Let $\mathcal{D}$ be a weighted oriented graph and let $I(\mathcal{D})$ be its edge ideal. Under a natural condition that the underlying (undirected) graph of $\mathcal{D}$ contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen-Macaulayness of $I(\mathcal{D})$. We also completely characterize the Cohen-Macaulayness of $I(\mathcal{D})$ when the underlying graph of $\mathcal{D}$ is a bipartite graph. When $I(\mathcal{D})$ fails to be Cohen-Macaulay, we give an instance where $I(\mathcal{D})$ is shown to be sequentially Cohen-Macaulay.

math.AC↗