Sara Faridi’s research while affiliated with Dalhousie University and other places

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.

Given {\it any} monomial ideal I minimally generated by q monomials, we define a simplicial complex \MM_q^2 that supports a resolution of $I^2$. We also define a subcomplex \MM^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 \Tq which is generated by q monomials. For any monomial ideal I with q generators, we establish that \beta(I^2) \leq \beta({{\Tq}^2}).}} We show that the simplicial complex \MM_q^2 supports the minimal resolution of {\Tq}^2. In fact, \MM_q^2 is the Scarf complex of {\Tq}^2.

Let G be a finite graph and I(G) its edge ideal. The question in which we are interested is when the square $I(G)^2$ is Cohen--Macaulay. Via the polarization technique together with Reisner's criterion, it is shown 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.

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.

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.

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 $r,q \geq 1$.

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.

Let G be a gapfree graph and let I(G) be its edge ideal. An open conjecture of Nevo and Peeva states that $I(G)^q$ has a linear resolution for $q\gg 0$. We investigate a stronger conjecture that if $I(G)^q$ has linear quotients for some integer q, then $I(G)^{q+1}$ also has linear quotients. We give a partial solution to this conjecture. It is known that if G does not contain a cricket, a diamond, or a $C_4$, then $I(G)^q$ has a linear resolution for $q \geq 2$. We construct a family of gapfree graphs G containing cricket, diamond, $C_4$ and $C_5$ as induced subgraphs of G for which $I(G)^q$ has linear quotients for $q \ge 2$.

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 $\mathcal{P}(I)$, we give conditions that allow for the lifting of an associated basic double G-link of $\mathcal{P}(I)$ 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 artinian monomial ideals and of stable Cohen-Macaulay monomial ideals are vertex decomposable, recovering and strengthening the recent result of Fl{\o}ystad and Mafi that these complexes are shellable. We then introduce and study polarization of a Gr\"obner 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.