arXiv subjects

Sonja Mapes

Publications and source records attributed to Sonja Mapes.

7 recordsLinked to original sources

arXiv2021

Projective dimension of Hypergraphs

Given a square-free monomial ideal $I$, satisfying certain hypotheses, in a polynomial ring $R$ over a field $\mathbb{K}$, we compute the projective dimension of $I$. Specifically, we focus on the cases where the 1-skeleton of an associated hypergraph is either a string or a cycle. We investigate the impact on the projective dimension when higher dimensional edges are removed. We prove that the higher dimensional edge either has no effect on the projective dimension or the projective dimension only goes up by one with the extra higher dimensional edge.

math.AC↗

arXiv2019

Lattices and Hypergraphs associated to square-free monomial ideals

Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field $\mathbb{K}$, one can associate it with its LCM-lattice and its hypergraph. In this short note, we establish the connection between the LCM-lattice and the hypergraph, and in doing so we provide a sufficient condition for removing higher dimension edges of the hypergraph without impacting the projective dimension of the square-free monomial ideal. We also offer algorithms to compute the projective dimension of a class of square-free monomial ideals built using the new result and previous results of Lin-Mantero.

math.AC↗

arXiv2015

Constructing monomial ideals with a given minimal resolution

This paper gives a description of various recent results which construct monomial ideals with a given minimal free resolution. We show that these are all instances of coordinatizing a finite atomic lattice as defined by Mapes. Subsequently, we explain how in some of these cases (in one case work by Faridi, and in another work by Fl\oystad), where questions still remain, this point of view can be applied. We also prove an equivalence in the case of trees between the notion of maximal defined by Fl\oystad and a notion of being maximal in a Betti stratum.

math.AC↗

arXiv2014

The Betti poset in monomial resolutions

Let $P$ be a finite partially ordered set with unique minimal element $\hat{0}$. We study the Betti poset of $P$, created by deleting elements $q\in P$ for which the open interval $(\hat{0}, q)$ is acyclic. Using basic simplicial topology, we demonstrate an isomorphism in homology between open intervals of the form $(\hat{0},p)\subset P$ and corresponding open intervals in the Betti poset. Our motivating application is that the Betti poset of a monomial ideal's lcm-lattice encodes both its $\mathbb{Z}^{d}$-graded Betti numbers and the structure of its minimal free resolution. In the case of rigid monomial ideals, we use the data of the Betti poset to explicitly construct the minimal free resolution. Subsequently, we introduce the notion of rigid deformation, a generalization of Bayer, Peeva, and Sturmfels' generic deformation.

math.AC↗

arXiv2013

Partially ordered sets in Macaulay2

We introduce the package Posets for Macaulay2. This package provides a data structure and the necessary methods for working with partially ordered sets, also called posets. In particular, the package implements methods to enumerate many commonly studied classes of posets, perform operations on posets, and calculate various invariants associated to posets.

math.CO↗

arXiv2011

Rigid monomial ideals

In this paper we investigate the class of rigid monomial ideals. We give a characterization of the minimal free resolutions of certain classes of these ideals. Specifically, we show that the ideals in a particular subclass of rigid monomial ideals are lattice-linear and thus their minimal resolution can be constructed as a poset resolution. We then use this result to give a description of the minimal free resolution of a larger class of rigid monomial ideals by using $\mathcal{L}(n)$, the lattice of all lcm-lattices of monomial ideals with $n$ generators. By fixing a stratum in $\mathcal{L}(n)$ where all ideals have the same total Betti numbers we show that rigidity is a property which is upward closed in $\mathcal{L}(n)$. Furthermore, the minimal resolution of all rigid ideals contained in a fixed stratum is shown to be isomorphic to the constructed minimal resolution.

math.AC↗

arXiv2010

Finite atomic lattices and resolutions of monomial ideals

In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated LCM lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial ideal whose LCM lattice is P, and we give a complete characterization of all such coordinatizations. We prove that all relations in the lattice L(n) of all finite atomic lattices with n ordered atoms can be realized as deformations of exponents of monomial ideals. We also give structural results for L(n). Moreover, we prove that the cellular structure of a minimal free resolution of a monomial ideal M can be extended to minimal resolutions of certain monomial ideals whose LCM lattices are greater than that of M in L(n).

math.AC↗