arXiv subjects

Sara Faridi

Publications and source records attributed to Sara Faridi.

At least 19 recordsLinked to original sources

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

Polarization and Gorenstein liaison

A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^n$ can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then, after re-embedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$, indeed it can be G-linked to a complete intersection. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely related reduced scheme. Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G-links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley--Reisner complexes. Given a monomial ideal $I$ and a vertex decomposition of the Stanley--Reisner complex of its polarization $P(I)$, we give conditions that allow for the lifting of an associated basic double G-link of $P(I)$ to a basic double G-link of $I$ itself. We use the relationship we develop in the process to show that the Stanley--Reisner complexes of polarizations of stable Cohen--Macaulay monomial ideals are vertex decomposable. We then introduce and study polarization of a Gröbner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage.

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

Simplicial Resolutions of the Quadratic Power of Monomial Ideals

Given any monomial ideal $ I $ minimally generated by $ q $ monomials, we define a simplicial complex $\mathbb{M}_q^2$ that supports a resolution of $ I^2 $. We also define a subcomplex $\mathbb{M}^2(I)$, which depends on the monomial generators of $I$ and also supports the resolution of $ I^2 $. As a byproduct, we obtain bounds on the projective dimension of the second power of any monomial ideal. We also establish bounds on the Betti numbers of $ I^2 $, which are significantly tighter than those determined by the Taylor resolution of $ I^2 $. Moreover, we introduce the permutation ideal $\mathcal{T}_q$ which is generated by $q$ monomials. For any monomial ideal $I$ with $q$ generators, we establish that $β(I^2) \leq β({\mathcal{T}_q}^2)$. We show that the simplicial complex $\mathbb{M}_q^2$ supports the minimal resolution of ${\mathcal{T}_q}^2$. In fact, $\mathbb{M}_q^2$ is the Scarf complex of ${\mathcal{T}_q}^2$.

math.AC↗

arXiv2025

When are Morse resolutions polyhedral?

It is known that the chain complex of a simplex on $q$ vertices can be used to construct a free resolution of any ideal generated by $q$ monomials, and as a direct result, the Betti numbers always have binomial upper bounds, given by the number of faces of a simplex in each dimension. It is also known that for most monomials the resolution provided by the simplex is far from minimal. Discrete Morse theory provides an algorithm called \say{Morse matchings} by which faces of the simplex can be removed so that the chain complex on the remaining faces is still a free resolution of the same ideal. An immediate positive effect is an often considerable improvement on the bounds on Betti numbers. A caveat is the loss of the combinatorial structure of the simplex we started with: the output of the Morse matching process is a cell complex with no obvious structure besides an \say{address} for each cell. The main question in this paper is: which Morse matchings lead to Morse complexes that are polyhedral cell complexes? We prove that if a monomial ideal is minimally generated by up to four generators, then there is a maximal Morse matching of the simplex such that the resulting cell complex is a polyhedral cell complex. We then give an example of a monomial ideal minimally generated by six generators whose minimal free resolution is supported on a Morse complex and the Morse complex cannot be polyhedral no matter what Morse matching is chosen, and we go further to show that this ideal cannot have any polyhedral minimal free resolution.

math.AC↗

arXiv2025

Cohen-Macaulay squares of edge ideals

Let $G$ be a finite graph and $I(G)$ its edge ideal. We give a full description of the Stanley--Reisner complex of the polarization of $I(G)^2$, naturally introducing the tools of Stanley--Reisner theory in the study of the algebraic behaviour of powers of edge ideals. As an application, we demonstrate how Reisner's criterion can be applied directly to check if $I(G)^2$ is Cohen--Macaulay. We can show that if $G$ belongs to the class of finite graphs which consists of cycles, whisker graphs, trees, connected chordal graphs and connected Cohen--Macaulay bipartite graphs, then the square $I(G)^2$ is Cohen--Macaulay if and only if either $G$ is the pentagon, the cycle of length $5$, or $G$ consists of exactly one edge.

math.AC↗

arXiv2025

Spheres and balls as independence complexes

The terms "whiskering", and more generally "grafting", refer to adding generators to any monomial ideal to make the resulting ideal Cohen-Macaulay. We investigate the independence complexes of simplicial complexes that are constructed through a whiskering or grafting process, and we show that these independence complexes are (generalized) Bier balls. More specifically, the independence complexes are either homeomorphic to a ball or a sphere. In a related direction, we classify when the independence complexes of very well-covered graphs are homeomorphic to balls or spheres.

math.CO↗

arXiv2025

Artinian Gorenstein algebras with binomial Macaulay dual generator

This paper initiates a systematic study for key properties of Artinian Gorenstein $K$-algebras having binomial Macaulay dual generators. In codimension 3, we demonstrate that all such algebras satisfy the strong Lefschetz property, can be constructed as a doubling of an appropriate 0-dimensional scheme in $\mathbb{P}^2$, and we provide an explicit characterization of when they form a complete intersection. For arbitrary codimension, we establish sufficient conditions under which the weak Lefschetz property holds and show that these conditions are optimal.

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↗

arXiv2025

Spherical complexes

In this paper we define spherical complexes as simplicial complexes with the property that every subcomplex obtained by a sequence of links and deletions either has trivial homology, or has the homology of a sphere. Examples of such complexes are independence complexes of ternary graphs and independence complexes of simplicial forests. We give criteria for when a spherical complex is acyclic, and describe the dimension of the sphere when it is not. We then apply our results to compute the Leray number of these complexes, and define combinatorial invariants for them which are counterparts to algebraic invariants of their Stanley-Reisner rings.

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

Building monomial ideals with fixed betti numbers

Motivated by the fact that as the number of generators of an ideal grows so does the complexity of calculating relations among the generators, this paper identifies collections of monomial ideals with a growing number of generators which have predictable free resolutions. We use elementary collapses from discrete homotopy theory to construct infinitely many monomial ideals, with an arbitrary number of generators, which have similar or the same betti numbers. We show that the Cohen-Macaulay property in each unmixed (pure) component of the ideal is preserved as the ideal is expanded.

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↗

arXiv2024

Counting lattice points that appear as algebraic invariants of Cameron-Walker graphs

In 2021, Hibi et. al. studied lattice points in $\mathbb{N}^2$ that appear as $(\depth R/I,\dim R/I)$ when $I$ is the edge ideal of a graph on $n$ vertices, and showed these points lie between two convex polytopes. When restricting to the class of Cameron--Walker graphs, they showed that these pairs do not form a convex lattice polytope. In this paper, for the edge ideal $I$ of a Cameron--Walker graph on $n$ vertices, we find how many points in $\mathbb{N}^2$ appear as $(\depth(R/I),\dim(R/I))$, and how many points in $\mathbb{N}^4$ appear as $(\depth(R/I),\reg(R/I),\dim(R/I),\degh(R/I)).$

math.AC↗

arXiv2024

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↗

arXiv2023

The weak Lefschetz property of whiskered graphs

We consider Artinian level algebras arising from the whiskering of a graph. Employing a result by Dao-Nair we show that multiplication by a general linear form has maximal rank in degrees 1 and $n-1$ when the characteristic is not two, where $n$ is the number of vertices in the graph. Moreover, the multiplication is injective in degrees $<n/2$ when the characteristic is zero, following a proof by Hausel. Our result in the characteristic zero case is optimal in the sense that there are whiskered graphs for which the multiplication maps in all intermediate degrees $n/2,\ldots,n-2$ of the associated Artinian algebras fail to have maximal rank, and consequently, the weak Lefschetz property.

math.AC↗

arXiv2023

The Scarf complex and betti numbers of powers of extremal ideals

This paper is concerned with finding bounds on betti numbers and describing combinatorially and topologically (minimal) free resolutions of powers of ideals generated by a fixed number $q$ of square-free monomials. Among such ideals, we focus on a specific ideal $\mathcal{E}_q$, which we call {\it extremal}, and which has the property that for each $r\ge 1$ the betti numbers of ${\mathcal{E}_q}^r$ are an upper bound for the betti numbers of $I^r$ for any ideal $I$ generated by $q$ square-free monomials (in any number of variables). We study the Scarf complex of the ideals ${\mathcal{E}_q}^r$ and use this simplicial complex to extract information on minimal free resolutions. In particular, we show that ${\mathcal{E}_q}^r$ has a minimal free resolution supported on its Scarf complex when $q\leq 4$ or when $r\leq 2$, and we describe explicitly this complex. For any $q$ and $r$, we also show that $β_1({\mathcal{E}_q}^r)$ is the smallest possible, or in other words equal to the number of edges in the Scarf complex. These results lead to effective bounds on the betti numbers of $I^r$, with $I$ as above. For example, we obtain that pd$(I^r)\leq 5$ for all ideals $I$ generated by $4$ square-free monomials and any $r\geq 1$.

math.AC↗