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Sara Faridi

Publications and source records attributed to Sara Faridi.

51 records · Page 3Linked to original sources
arXiv2013

The projective dimension of sequentially Cohen-Macaulay monomial ideals

In this short note we prove that the projective dimension of a sequentially Cohen-Macaulay square-free monomial ideal is equal to the maximal height of its minimal primes (also known as the big height), or equivalently, the maximal cardinality of a minimal vertex cover of its facet complex. This in particular gives a formula for the projective dimension of facet ideals of these classes of ideals, which are known to be sequentially Cohen-Macaulay: graph trees and simplicial trees, chordal graphs and some cycles, chordal clutters and graphs, and some path ideals to mention a few. Since polarization preserves projective dimension, our result also gives the projective dimension of any sequentially Cohen-Macaulay monomial ideal.

math.AC↗
arXiv2013

A good leaf order on simplicial trees

Using the existence of a good leaf in every simplicial tree, we order the facets of a simplicial tree in order to find combinatorial information about the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire splitting of the ideal, as well as a refinement of a recursive formula of Hà and Van Tuyl for computing the graded Betti numbers of simplicial trees.

math.AC↗
arXiv2013

Chorded complexes and a necessary condition for a monomial ideal to have a linear resolution

In this paper we extend one direction of Fröberg's theorem on a combinatorial classification of quadratic monomial ideals with linear resolutions. We do this by generalizing the notion of a chordal graph to higher dimensions with the introduction of d-chorded and orientably-d-cycle-complete simplicial complexes. We show that a certain class of simplicial complexes, the d-dimensional trees, correspond to ideals having linear resolutions over fields of characteristic 2 and also give a necessary combinatorial condition for a monomial ideal to be componentwise linear over all fields.

math.AC↗
arXiv2013

On the resolution of path ideals of cycles

We give a formula to compute all the top degree graded Betti numbers of the path ideal of a cycle. Also we will find a criterion to determine when Betti numbers of this ideal are non zero and give a formula to compute its projective dimension and regularity.

math.AC↗
arXiv2012

Monomial Resolutions Supported By Simplicial Trees

We explore resolutions of monomial ideals supported by simplicial trees. We argue that since simplicial trees are acyclic, the criterion of Bayer, Peeva and Sturmfels for checking if a simplicial complex supports a free resolution of a monomial ideal reduces to checking that certain induced subcomplexes are connected. We then use results of Peeva and Velasco to show that every simplicial tree appears as the Scarf complex of a monomial ideal, and hence supports a minimal resolution. We also provide a way to construct smaller Scarf ideals than those constructed by Peeva and Velasco.

math.AC↗
arXiv2008

The defining ideals of conjugacy classes of nilpotent matrices and a conjecture of Weyman

Tanisaki introduced generating sets for the defining ideals of the schematic intersections of the closure of conjugacy classes of nilpotent matrices with the set of diagonal matrices. These ideals are naturally labeled by integer partitions. Given such a partition $λ$, we define several methods to produce a reduced generating set for the associated ideal $I_λ$. For particular shapes we find nice generating sets. By comparing our sets with some generating sets of $I_λ$ arising from a work of Weyman, we find a counterexample to a related conjecture of Weyman.

math.AC↗
arXiv2007

Odd-Cycle-Free Facet Complexes and the König property

We use the definition of a simplicial cycle to define an odd-cycle-free facet complex (hypergraph). These are facet complexes that do not contain any cycles of odd length. We show that besides one class of such facet complexes, all of them satisfy the König property. This new family of complexes includes the family of balanced hypergraphs, which are known to satisfy the König property. These facet complexes are, however, not Mengerian; we give an example to demonstrate this fact.

math.AC↗
arXiv2006

Simplicial cycles and the computation of simplicial trees

We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplicial cycle is either a sequence of facets connected in the shape of a circle, or is a cone over such a structure. We show that a simplicial tree is a connected cycle-free simplicial complex, and use this characterization to produce an algorithm that checks in polynomial time whether a simplicial complex is a tree. We also present an efficient algorithm for checking whether a simplicial complex is grafted, and therefore Cohen-Macaulay.

math.AC↗
arXiv2005

Cohen-Macaulay Properties of Square-Free Monomial Ideals

In this paper we study simplicial complexes as higher dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen-Macaulay quotients, and a criterion for the Cohen-Macaulayness of facet ideals of simplicial trees. Along the way, we generalize several concepts from graph theory to simplicial complexes.

math.AC↗
arXiv2003

Simplicial Trees are Sequentially Cohen-Macaulay

This paper uses dualities between facet ideal theory and Stanley-Reisner theory to show that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay. The proof involves showing that the Alexander dual (or the cover dual, as we call it here) of a simplicial tree is a componentwise linear ideal. We conclude with additional combinatorial properties of simplicial trees.

math.AC↗
arXiv2002

The facet ideal of a simplicial complex

To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By generalizing the notion of a tree from graphs to simplicial complexes, we show that ideals associated to trees satisfy sliding depth condition, and therefore have normal and Cohen-Macaulay Rees rings. We also discuss connections with the theory of Stanley-Reisner rings.

math.AC↗
arXiv2002

Normal ideals of graded rings

For a graded domain $R=k[X_0,...,X_m]/J$ over an arbitrary domain $k$, it is shown that the ideal generated by elements of degree $\geq mA$, where $A$ is the least common multiple of the weights of the $X_i$, is a normal ideal.

math.AC↗