arXiv subjects

Jason McCullough

Publications and source records attributed to Jason McCullough.

At least 19 recordsLinked to original sources

arXiv2026

Graded Betti numbers of graded M\"obius algebras of uniform matroids

Graded M\"obius algebras were a key tool in the proof of the Dowling-Wilson Top Heavy Conjecture. They are commutative algebras whose Hilbert functions recover the Whitney numbers of the second kind, i.e. the number of flats of a given rank. The graded Betti numbers of the defining ideal of a graded M\"obius algebra refine the Hilbert function and describe its minimal free resolution. In this paper we derive precise formulas for the graded Betti numbers of the defining ideals of graded M\"obius algebra for any uniform matroid. We also study when the graded M\"obius algebra of an arbitrary matroid is linearly presented.

math.AC↗

arXiv2026

Koszul Orlik--Solomon Algebras from Non-supersolvable Arrangements

The cohomology ring of the complement of a complex hyperplane arrangement is given by its Orlik--Solomon algebra. It is known that the defining ideal of the Orlik--Solomon algebra has a quadratic Gr\"obner basis in the standard presentation if and only if the intersection lattice is supersolvable; such algebras are automatically Koszul. In 1997, Shelton and Yuzvinsky posed the question as to whether all Koszul Orlik--Solomon algebras arise from supersolvable arrangements. We answer this question negatively using three related constructions that produce non-supersolvable arrangements whose Orlik--Solomon algebras are Koszul. Moreover, these arrangements may be chosen to be irreducible, realizable over $\mathbb{Q}$, and of any rank $\geq 3$. Our constructions rely on a result of Falk and Proudfoot which we strengthen and generalize. In two of the three cases, we show non-supersolvability using a corrected form of a result of Ziegler regarding supersolvability of parallel connections. We also construct Koszul Orlik--Terao algebras coming from non-supersolvable arrangements.

math.CO↗

arXiv2026

A Linear Bound on the Projective Dimension of Height 3 Quadratic Ideals

In 2016, Ananyan and Hochster gave the first proof of a positive answer to Stillman's Question, which asked for a bound on the projective dimension of a graded polynomial ideal purely in terms of the number and degrees of its generators. Explicit formulas for such a bound are limited and often not optimal. In this paper, we give a nearly optimal linear upper bound on the projective dimension of height $3$ ideals generated by any number of degree $2$ homogenous polynomials.

math.AC↗

arXiv2026

Koszul Binomial Edge Ideals

As the binomial edge ideal of a graph is always generated by homogeneous quadratic polynomials corresponding to the edges of the graph, the question of when a binomial edge ideal defines a Koszul algebra has been studied by many authors ever since the class of ideals was first defined. Several partial results are known, including a characterization of those binomial edge ideals that possess a quadratic Gr\"obner basis. However, a complete characterization of the graphs determining Koszul binomial edge ideals has remained elusive. Inspired by our recent work characterizing when the graded M\"obius algebras of graphic matroids are Koszul, we answer the question once and for all by proving that a graph defines a Koszul binomial edge ideal if and only if it is strongly chordal and claw-free.

math.AC↗

arXiv2026

F-Purity of Binomial Edge Ideals

In 2012, K. Matsuda introduced the class of weakly closed graphs and investigated when binomial edge ideals are F-pure. He proved that weakly closed binomial edge ideals are F-pure whenever the base field has positive characteristic. He conjectured that: (i) when the base field has characteristic two, every F-pure binomial edge ideal comes from a weakly closed graph; and (ii) that every binomial edge ideal is F-pure provided that the characteristic of the residue field is sufficiently large. In this paper, we resolve both of Matsuda's conjectures. We confirm Matsuda's first conjecture, showing that the binomial edge ideal of a graph defines an F-pure quotient in characteristic 2 if and only if the graph is weakly closed. We also show that Matsuda's second conjecture is false in a very strong way by showing that graphs containing asteroidal triples, such as the net, define non-F-pure binomial edge ideals in any positive characteristic. Our results yield a complete classification of F-pure binomial edge ideals of chordal graphs as well as large families of standard graded algebras that are F-injective but neither F-pure nor F-rational in all characteristics.

math.AC↗

arXiv2025

Properties of LCM Lattices of Monomial Ideals

LCM lattices were introduced by Gasharov, Peeva, and Welker as a way to study minimal free resolutions of monomial ideals. All LCM lattices are atomic and all atomic lattices arise as the LCM lattice of some monomial ideal. We systematically study other lattice properties of LCM lattices. For lattices associated to the edge ideal of a graph, we completely characterize the many standard lattice properties in terms of the associated graphs: Boolean, modular, upper semimodular, lower semimodular, supersolvable, coatomic, and complemented; edge ideals with graded LCM lattices were previously characterized by Nevo and Peeva as those associated to gap-free graphs. For arbitrary monomial ideals, we prove the Cohen-Macaulayness of minimal monomial ideals associated to modular lattices. We also prove separate necessary and sufficient lattice conditions for when the projective dimension of a monomial ideal matches the height of its LCM lattice. Finally, we show that LCM lattices of Gorenstein edge ideals are coatomic and raise questions about the lattice properties of arbitrary Gorenstein monomial ideals.

math.AC↗

arXiv2024

Koszul Graded M\"obius Algebras and Strongly Chordal Graphs

The graded M\"{o}bius algebra of a matroid is a commutative graded algebra which encodes the combinatorics of the lattice of flats of the matroid. As a special subalgebra of the augmented Chow ring of the matroid, it plays an important role in the recent proof of the Dowling-Wilson Top Heavy Conjecture. Recently, Mastroeni and McCullough proved that the Chow ring and the augmented Chow ring of a matroid are Koszul. We study when graded M\"obius algebras are Koszul. We characterize the Koszul graded M\"obius algebras of cycle matroids of graphs in terms of properties of the graphs. Our results yield a new characterization of strongly chordal graphs via edge orderings.

math.AC↗

arXiv2023

Prime ideals and three-generated ideals with large regularity

Ananyan and Hochster proved the existence of a function $\Phi(m,d)$ such that any graded ideal $I$ generated by $m$ forms of degree at most $d$ in a standard graded polynomial ring satisfies $\mathrm{reg}(I) \le \Phi(m,d)$. Relatedly, Caviglia et. al. proved the existence of a function $\Psi(e)$ such that any nondegenerate prime ideal $P$ of degree $e$ in a standard graded polynomial ring over an algebraically closed field satisfies $\mathrm{reg}(P) \le \Psi(\mathrm{deg}(P))$. We provide a construction showing that both $\Phi(3,d)$ and $\Psi(e)$ must be at least doubly exponential in $d$ and $e$, respectively. Previously known lower bounds were merely super-polynomial in both cases.

math.AC↗

arXiv2022

Depth and Singular Varieties of Exterior Edge Ideals

Edge ideals of finite simple graphs are well-studied over polynomial rings. In this paper, we initiate the study of edge ideals over exterior algebras, specifically focusing on the depth and singular varieties of such ideals. We prove an upper bound on the depth of the edge ideal associated to a general graph and a more refined bound for bipartite graphs, and we show that both are tight. We also compute the depth of several large families of graphs including cycles, complete multipartite graphs, spider graphs, and Ferrers graphs. Finally, we focus on the effect whiskering a graph has on the depth of the associated edge ideal.

math.AC↗

arXiv2021

Chow rings of matroids are Koszul

Chow rings of matroids were instrumental in the resolution of the Heron-Rota-Welsh Conjecture by Adiprasito, Huh, and Katz and in the resolution of the Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang. The Chow ring of a matroid is a commutative, graded, Artinian, Gorenstein algebra with linear and quadratic relations defined by the matroid. Dotsenko conjectured that the Chow ring of any matroid is Koszul. The purpose of this paper is to prove Dotsenko's conjecture. We also show that the augmented Chow ring of a matroid is Koszul. As a corollary, we show that the Chow rings and augmented Chow rings of matroids have rational Poincar\'{e} series.

math.AC↗

arXiv2021

G-Quadratic, LG-Quadratic, and Koszul Quotients of Exterior Algebras

This paper introduces the study of LG-quadratic quotients of exterior algebras, showing that they are Koszul, as in the commutative case. We construct an example of an LG-quadratic algebra that is not G-quadratic and another example that is Koszul but not LG-quadratic. This is only the second known Koszul algebra that is not LG-quadratic and the first that is noncommutative.

math.RA↗

arXiv2020

Quadratic Gorenstein algebras with many surprising properties

Let $k$ be a field of characteristic $0$. Using the method of idealization, we show that there is a non-Koszul, quadratic, Artinian, Gorenstein, standard graded $k$-algebra of regularity $3$ and codimension $8$, answering a question of Mastroeni, Schenck, and Stillman. We also show that this example is minimal in the sense that no other idealization that is non-Koszul, quadratic, Artinian, Gorenstein algebra, with regularity $3$ has smaller codimension. We also construct an infinite family of graded, quadratic, Artinian, Gorenstein algebras $A_m$, indexed by an integer $m \ge 2$, with the following properties: (1) there are minimal first syzygies of the defining ideal in degree $m+2$, (2) for $m \ge 3$, $A_m$ is not Koszul, (3) for $m \ge 7$, the Hilbert function of $A_m$ is not unimodal, and thus (4) for $m \ge 7$, $A_m$ does not satisfy the weak or strong Lefschetz properties. In particular, the subadditivity property fails for quadratic Gorenstein ideals. Finally, we show that the idealization of a construction of Roos yields non-Koszul quadratic Gorenstein algebras such that the residue field $k$ has a linear resolution for precisely $\alpha$ steps for any integer $\alpha \ge 2$. Thus there is no finite test for the Koszul property even for quadratic Gorenstein algebras.

math.AC↗

arXiv2019

Green-Lazarsfeld Condition for Toric Edge Ideals of Bipartite Graphs

Previously, Ohsugi and Hibi gave a combinatorial description of bipartite graphs $G$ whose toric edge ideal $I_G$ is generated by quadrics, showing that every cycle of $G$ of length at least $6$ must have a chord. This corresponds to the Green-Lazarsfeld condition $\mathbf{N}_1$. In this paper, we investigate the higher syzygies of $I_G$ and give combinatorial descriptions of the Green-Lazarsfeld conditions $\mathbf{N}_p$ of toric edge ideals of bipartite graphs for all $p \ge 1$. In particular, we show that $I_G$ is linearly presented (i.e. satisfies condition $\mathbf{N}_2$) if and only if the bipartite complement of $G$ is a tree of diameter at most $3$. We also investigate the regularity of linearly presented toric edge ideals and give criteria for polyomino ideals to satisfy the Green-Lazarsfeld conditions.

math.AC↗

arXiv2019

Singularities of Rees-like Algebras

Recently, Peeva and the second author constructed irreducible projective varieties with regularity much larger than their degree, yielding counterexamples to the Eisenbud-Goto Conjecture. Their construction involved two new ideas: Rees-like algebras and step-by-step homogenization. Yet, all of these varieties are singular and the nature of the geometry of these projective varieties was left open. The purpose of this paper is to study the singularities inherent in this process. We compute the codimension of the singular locus of an arbitrary Rees-like algebra over a polynomial ring. We then show that the relative size of the singular locus can increase under step-by-step homogenization. To address this defect, we construct a new process, we call prime standardization, which plays a similar role as step-by-step homogenization but also preserves the codimension of the singular locus. This is derived from ideas of Ananyan and Hochster and we use this to study the regularity of certain smooth hyperplane sections of Rees-like algebras, showing that they all satisfy the Eisenbud-Goto Conjecture, as expected. On a more qualitative note, while Rees-like algebras are almost never Cohen-Macaulay and never normal, we characterize when they are seminormal, weakly normal, and, in positive characteristic, F-split. Finally, we construct a finite free resolution of the canonical module of a Rees-like Algebra over the presenting polynomial ring showing that it is always Cohen-Macaulay and has a surprising self-dual structure.

math.AC↗

arXiv2018

Stillman's Question for Exterior Algebras and Herzog's Conjecture on Betti Numbers of Syzygy Modules

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo-Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that these examples are dual to modules over polynomial rings that yield counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules.

math.AC↗

arXiv2018

On the maximal graded shifts of ideals and modules

We generalize a result of Eisenbud-Huneke-Ulrich on the maximal graded shifts of a module with prescribed annihilator and prove a linear regularity bound for ideals in a polynomial ring depending only on the first $p - c$ steps in the resolution, where $p = \mathrm{pd}(S/I)$ and $c = \mathrm{codim}(I)$.

math.AC↗

arXiv2018

The projective dimension of three cubics is at most 5

Let $R$ be a polynomial ring over a field and $I$ an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension $\mathrm{pd}(R/I)$ of $R/I$ is at most 36, although the example with largest projective dimension he constructed has $\mathrm{pd}(R/I)=5$. Based on computational evidence, it had been conjectured that $\mathrm{pd}(R/I)\leq 5$. In the present paper we prove this conjectured sharp bound.

math.AC↗

arXiv2017

A Tight Bound on the Projective Dimension of Four Quadrics

Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring is at most 6; moreover, this bound is tight. We achieve this bound, in part, by giving a characterization of the low degree generators of ideals primary to height three primes of multiplicities one and two.

math.AC↗