arXiv subjects

Timothy B. P. Clark

Publications and source records attributed to Timothy B. P. Clark.

7 recordsLinked to original sources

arXiv2019

Minimal free resolutions of monomial ideals and of toric rings are supported on posets

We introduce the notion of a \emph{resolution supported on a poset}. When the poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of Reiner and Welker, and of Velasco, has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a \emph{homology CW-poset} that supports a minimal free resolution of the ideal. This allows one to extend to every minimal resolution, essentially verbatim, techniques initially developed to study cellular resolutions. As two demonstrations of this process, we show that minimal resolutions of toric rings are supported on what we call toric hcw-posets, and we give a new combinatorial proof of a fundamental result of Miller on the relationship between Artininizations and Alexander duality of monomial ideals.

math.AC↗

arXiv2018

Moments of the inverse participation ratio for the Laplacian on finite regular graphs

We investigate the first and second moments of the inverse participation ratio (IPR) for all eigenvectors of the Laplacian on finite random regular graphs with $n$ vertices and degree $z$. By exactly diagonalizing a large set of $z$-regular graphs, we find that as $n$ becomes large, the mean of the inverse participation ratio on each graph, when averaged over a large ensemble of graphs, approaches the numerical value $3$. This universal number is understood as the large-$n$ limit of the average of the quartic polynomial corresponding to the IPR over an appropriate $(n-2)$-dimensional hypersphere of $\mathbb{R}^n$. For a large, but not exhaustive ensemble of graphs, the mean variance of the inverse participation ratio for all graph Laplacian eigenvectors deviates from its continuous hypersphere average due to large graph-to-graph fluctuations that arise from the existence of highly localized modes.

math-ph↗

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

Regular CW-complexes and poset resolutions of monomial ideals

We use the natural homeomorphism between a regular CW-complex X and its face poset P_X to establish a canonical isomorphism between the cellular chain complex of X and the result of applying the poset construction of arXiv:0806.4532v1 to P_X. For a monomial ideal whose free resolution is supported on a regular CW-complex, this isomorphism allows the free resolution of the ideal to be realized as a CW-poset resolution. Conversely, any CW-poset resolution of a monomial ideal gives rise to a resolution supported on a regular CW-complex.

math.AC↗

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

A Minimal Poset Resolution of Stable Ideals

We use the theory of poset resolutions to construct the minimal free resolution of an arbitrary stable monomial ideal in the polynomial ring whose coefficients are from a field. This resolution is recovered by utilizing a poset of Eliahou-Kervaire admissible symbols associated to a stable ideal. The structure of the poset under consideration is quite rich and in related analysis, we exhibit a regular CW complex which supports a minimal cellular resolution of a stable monomial ideal.

math.AC↗

arXiv2008

Poset Resolutions of Monomial Ideals

We introduce the class of lattice-linear monomial ideals and use the LCM-lattice to give an explicit construction for their minimal free resolution. The class of lattice-linear ideals includes (among others) the class of monomial ideals with linear free resolution and the class of Scarf monomial ideals. Our main tool is a new construction by Tchernev that produces from a map of posets $η:P\lra\mbb{N}^n$ a sequence of multigraded modules and maps.

math.AC↗