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Sara Faridi

Publications and source records attributed to Sara Faridi.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2022

Cellular resolutions of monomial ideals and their Artinian reductions

The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have minimal cellular resolutions. When the ideal is generated by two monomials, we can give a precise description of the CW-complex supporting minimal free resolution of the ideal and its Artinian reduction. Also, in this case, we compute the multigraded Betti numbers, Cohen-Macaulay type and determine when the corresponding algebra is a level algebra.

math.AC↗
arXiv2022

Breaking up Simplicial Homology and Subadditivity of Syzygies

We consider the following question: if a simplicial complex $Δ$ has $d$-homology, then does the corresponding $d$-cycle always induce cycles of smaller dimension that are not boundaries in $Δ$? We provide an answer to this question in a fixed dimension. We use the breaking of homology to show the subadditivity property for the maximal degrees of syzygies of monomial ideals in a fixed homological degree.

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↗
arXiv2021

Well Ordered Covers, Simplicial Bouquets, and Subadditivity of Betti Numbers of Square-Free Monomial Ideals

Well ordered covers of square-free monomial ideals are subsets of the minimal generating set ordered in a certain way that give rise to a Lyubeznik resolution for the ideal, and have guaranteed nonvanishing Betti numbers in certain degrees. This paper is about square-free monomial ideals which have a well ordered cover. We consider the question of subadditivity of syzygies of square-free monomial ideals via complements in the lcm lattice of the ideal, and examine how lattice complementation breaks well ordered covers of the ideal into (well ordered) covers of subideals. We also introduce a family of well ordered covers called strongly disjoint sets of simplicial bouquets (generalizing work of Kimura on graphs), which are relatively easy to identify in simplicial complexes. We examine the subadditivity property via numerical characteristics of these bouquets.

math.AC↗
arXiv2019

Simplicial Complexes are Game Complexes

Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known invariants of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphic game trees, and hence equal game values. We also study a subclass of SP-games corresponding to flag complexes, showing that there is always a game whose corresponding complex is a flag complex no matter which board it is played on.

math.CO↗
arXiv2018

Chordality, $d$-collapsibility, and componentwise linear ideals

Using the concept of $d$-collapsibility from combinatorial topology, we define chordal simplicial complexes and show that their Stanley-Reisner ideals are componentwise linear. Our construction is inspired by and an extension of "chordal clutters'' which was defined by Bigdeli, Yazdan Pour and Zaare-Nahandi in 2017, and characterizes Betti tables of all ideals with linear resolution in a polynomial ring. We show $d$-collapsible and $d$-representable complexes produce componentwise linear ideals for appropriate $d$. Along the way, we prove that there are generators that when added to the ideal, do not change Betti numbers in certain degrees. We then show that large classes of componentwise linear ideals, such as Gotzmann ideals and square-free stable ideals have chordal Stanley-Reisner complexes, that Alexander duals of vertex decomposable complexes are chordal, and conclude that the Betti table of every componentwise linear ideal is identical to that of the Stanley-Reisner ideal of a chordal complex.

math.AC↗
arXiv2017

Resolutions of Monomial Ideals of Projective Dimension 1

We show that a monomial ideal $I$ has projective dimension $\leq$ 1 if and only if the minimal free resolution of $S/I$ is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the $S/I$. We also provide a new characterization of quasi-trees, which we use to give a new proof to a result by Herzog, Hibi, and Zheng which characterizes monomial ideals of projective dimension 1 in terms of quasi-trees.

math.AC↗
arXiv2017

Monomial ideals via square-free monomial ideals

We study monomial ideals using the operation polarization to first turn them into square-free monomial ideals. We focus on monomial ideals whose polarization produce simplicial trees, and show that many of the properties of simplicial trees hold for such ideals.This includes Cohen-Macaulayness of the Rees ring, and being sequentially Cohen-Macaulay. The appendix is an independent study of primary decomposition in a sequentially Cohen-Macaulay module. We demonstrate how every submodule appearing in the filtration of a sequentially Cohen-Macaulay module can be described in terms of the primary decomposition of the 0-submodule.

math.AC↗
arXiv2017

Graded Betti numbers of path ideals of cycles and lines

We use purely combinatorial arguments to give a formula to compute all graded Betti numbers of path ideals of line graphs and cycles. As a consequence we can give new and short proofs for the known formulas of regularity and projective dimensions of path ideals of line graphs.

math.AC↗
arXiv2016

Lattice Complements and the Subadditivity of Syzygies of Simplicial Forests

We prove the subadditivity property for the maximal degrees of the syzygies of facet ideals simplicial forests. For such an ideal $I$, if the $i$-th Betti number is nonzero and $i=a+b$, we show that there are monomials in the lcm lattice of $I$ that are complements in part of the lattice, each supporting a nonvanishing $a$-th and $b$-th Betti numbers. The subadditivity formula follows from this observation.

math.AC↗
arXiv2015

Betti numbers of monomial ideals via facet covers

We give a sufficient condition for a monomial ideal to have a nonzero Betti number in each multidegree. In the case of facet ideals of simplicial forests, this condition becomes a necessary one and it allows us to characterize Betti numbers, projective dimension and regularity of such ideals combinatorially. Our condition is expressed in terms of minimal facet covers of simplicial complexes.

math.AC↗
arXiv2015

Games and Complexes I: Transformation via Ideals

Placement games are a subclass of combinatorial games which are played on graphs. We will demonstrate that one can construct simplicial complexes corresponding to a placement game, and this game could be considered as a game played on these simplicial complexes. These complexes are constructed using square-free monomials.

math.CO↗
arXiv2015

Games and Complexes II: Weight Games and Kruskal-Katona Type Bounds

A strong placement game $G$ played on a board $B$ is equivalent to a simplicial complex $Δ_{G,B}$. We look at weight games, a subclass of strong placement games, and introduce upper bounds on the number of positions with $i$ pieces in $G$, or equivalently the number of faces with $i$ vertices in $Δ_{G,B}$, which are reminiscent of the Kruskal-Katona bounds.

math.CO↗
arXiv2015

Multigraded Betti numbers of simplicial forests

We prove that multigraded Betti numbers of a simplicial forest are always either 0 or 1. Moreover a nonzero multidegree appears exactly at one homological degree in the resolution. Our work generalizes work of Bouchat on edge ideals of graph forests.

math.AC↗
arXiv2014

When is a Squarefree Monomial Ideal of Linear Type ?

In 1995 Villarreal gave a combinatorial description of the equations of Rees algebras of quadratic squarefree monomial ideals. His description was based on the concept of closed even walks in a graph. In this paper we will generalize his results for all squarefree monomial ideals by using a definition of even walks in a simplicial complex.

math.AC↗