arXiv subjects
Publications and source records attributed to Trung Chau.
We show that the quotient ring by the ideal of maximal minors of a $1$-generic matrix has rational singularities. This answers a conjecture of Eisenbud (1988) that such rings are normal, and generalizes a result of Conca, Mostafazadehfard, Singh and Varbaro (2018) that generic Hankel determinantal rings have rational singularities in characteristic zero.
For any integer $n$, we classify all trees whose $n$-path ideals have linear quotients.
In this article, we study the ideal generated by $2\times 2$ permanents of a symmetric matrix. We denote this ideal by $P_2(X)$ where $X$ is a symmetric matrix. We compute a Gröbner basis, dimension, depth, minimal primes, and a primary decomposition of $P_2(X)$. It can be seen that the answer is reliant on whether the characteristic of the base field is two, and thus these ideals constitute a class of ideals whose algebraic properties depend on characteristics of the base field.
We determine all possible triples of depth, dimension, and regularity of edge ideals of weighted oriented graphs with a fixed number of vertices. Also, we compute all the possible Betti table sizes of edge ideals of weighted oriented trees and bipartite~graphs with a fixed number of vertices.
Let $I$ be any square-free monomial ideal, and $\mathcal{H}_I$ denote the hypergraph associated with $I$. Refining the concept of $k$-admissible matching of a graph defined by Erey and Hibi, we introduce the notion of generalized $k$-admissible matching for any hypergraph. Using this, we give a sharp lower bound on the (Castelnuovo-Mumford) regularity of $I^{[k]}$, where $I^{[k]}$ denotes the $k^{\text{th}}$ square-free power of $I$. In the special case when $I$ is equigenerated in degree $d$, this lower bound can be described using a combinatorial invariant $\mathrm{aim}(\mathcal{H}_I,k)$, called the $k$-admissible matching number of $\mathcal{H}_I$. Specifically, we prove that $\mathrm{reg}(I^{[k]})\ge (d-1)\mathrm{aim}(\mathcal{H}_I,k)+k$, whenever $I^{[k]}$ is non-zero. Even for the edge ideal $I(G)$ of a graph $G$, it turns out that $\mathrm{aim}(G,k)+k$ is the first general lower bound for the regularity of $I(G)^{[k]}$. In fact, when $G$ is a forest, $\mathrm{aim}(G,k)$ coincides with the $k$-admissible matching number introduced by Erey and Hibi. Next, we show that if $G$ is a block graph, then $\mathrm{reg}(I(G)^{[k]})= \mathrm{aim}(G,k)+k$, and this result can be seen as a generalization of the corresponding regularity formula for forests. Additionally, for a Cohen-Macaulay chordal graph $G$, we prove that $\mathrm{reg}(I(G)^{[2]})= \mathrm{aim}(G,2)+2$. Finally, we propose a conjecture on the regularity of square-free powers of edge ideals of chordal graphs.
In this paper, we prove that the path ideals of both paths and cycles have minimal cellular resolutions. Specifically, these minimal free resolutions coincide with the Barile-Macchia resolutions for paths, and their generalized counterparts for cycles. Furthermore, we identify edge ideals of cycles as a class of ideals that lack a minimal Barile-Macchia resolution, yet have a minimal generalized Barile-Macchia resolution.
Every monomial ideal $I$ has a Scarf complex, which is a subcomplex of its minimal free resolution. We say that $I$ is Scarf if its Scarf complex is also its minimal free resolution. In this paper, we fully characterize all pairs $(G,n)$ of a graph $G$ and an integer $n$ such that the squarefree power $I(G)^{[n]}$ or the symbolic power $I(G)^{(n)}$ of the edge ideal $I(G)$ is Scarf. We also determine all graphs $G$ such that its cover ideal $J(G)$ is Scarf, with an explicit description when $G$ is either chordal or bipartite.
We prove that monomial ideals with at most five generators and their Artinian reductions have minimal generalized Barile-Macchia resolutions. As a corollary, these ideals have minimal cellular resolutions, extending a result by Faridi, D.G, Ghorbanic, and Pour. This corollary is independently obtained by Montaner, García, and Mafi, using a different class of cellular resolutions called pruned resolutions.
Each monomial ideal over a polynomial ring admits a free resolution which has the structure of a DG-algebra, namely, the Taylor resolution. A pivot resolution of a monomial ideal, which we introduce, is a resolution that is always shorter than the Taylor resolution (unless the Taylor resolution is as short as possible) but still retains a DG-algebra structure. We study the basic properties of this family of resolutions including a characterization of when the construction is minimal. Following the work of Sobieska, we use the explicit nature of pivot resolutions to give formulae for the Eisenbud-Shamash construction of a free resolution of a given monomial ideal over complete intersections.
We identify several classes of monomial ideals that possess minimal generalized Barile-Macchia resolutions. These classes of ideals include generic monomial ideals, monomial ideals with linear quotients, and edge ideals of hypertrees. We also characterize connected unicyclic graphs whose edge ideals are bridge-friendly and, in particular, have minimal Barile-Macchia resolutions. Barile-Macchia and generalized Barile-Macchia resolutions are cellular resolutions and special types of Morse resolutions.
Let $G$ be a connected graph and let $I(G)$ denote its edge ideal. We classify when $I(G)^n$, for $n \ge 1$, admits a minimal Lyubeznik resolution. We also give a characterization for when $I(G)^n$ is bridge-friendly, which, in turn, implies that $I(G)^n$ has a minimal Barile-Macchia cellular resolution.
Using discrete Morse theory, Batzies and Welker introduced Morse resolutions of monomial ideals. In this note, we present the {\it Macaulay2} package {\tt MorseResolutions} for working with two important classes of Morse resolutions: Lyubeznik and Barile-Macchia resolutions. This package also contains procedures to search for a minimal Barile-Macchia resolution of a given monomial ideal.
We construct cellular resolutions for monomial ideals via discrete Morse theory. In particular, we develop an algorithm to create homogeneous acyclic matchings and we call the cellular resolutions induced from these matchings Barile-Macchia resolutions. These resolutions are minimal for edge ideals of weighted oriented forests and (most) cycles. As a result, we provide recursive formulas for graded Betti numbers and projective dimension. Furthermore, we compare Barile-Macchia resolutions to those created by Batzies and Welker and some well-known simplicial resolutions. Under certain assumptions, whenever the above resolutions are minimal, so are Barile-Macchia resolutions.
We study the algebraic sets of pairs of matrices defined by the vanishing of the anti-diagonal as well as the cross-diagonal of their commutator matrix. We prove that, over a field of prime characterisitic, the coordinate ring of the latter is always $F$-regular and, with exactly one exception, so is that of the former, thus proving a conjecture of Kadyrsizova and Yerlanov.