Searcharxiv⌕ Search

arXiv subjects

Nguyen Cong Minh

Publications and source records attributed to Nguyen Cong Minh.

16 recordsLinked to original sources
arXiv2026

Cohen-Macaulay higher conormal and Kähler differential modules of squarefree monomial ideals

Let $S=k[x_1,\ldots,x_n]$ and let $I=I_Δ\subsetneq S$ be a nonzero squarefree monomial ideal. Motivated by the classical higher-order Kähler differential modules and by the theory of higher conormal modules, we study not only the higher conormal quotients $I/I^q$, but more generally the shifted quotients $I^r/I^q$, $1\le r<q$, in the same $I$-adic conormal filtration, together with their symbolic analogues $I^{(r)}/I^{(q)}$. We prove that, for every $1\le r<q$ with $q\ge3$, the module $I^r/I^q$ is Cohen--Macaulay if and only if $I$ is a complete intersection. In sharp contrast, $I^{(r)}/I^{(q)}$ is Cohen--Macaulay if and only if $Δ$ is a matroid, where loops are allowed. Thus, the Cohen--Macaulayness of a single nonexceptional window forces the Cohen--Macaulayness of every window in the corresponding filtration. The unique exceptional pair is $(r,q)=(1,2)$: at this conormal level, we show that the Cohen--Macaulayness of $I/I^2$ forces $I^2=I^{(2)}$, and hence $I/I^{(2)}$ is Cohen--Macaulay.

math.AC↗
arXiv2026

Relative Hochster--Takayama formula and Cohen--Macaulay monomial ideal quotients

Hochster's and Takayama's formulas describes the multigraded components of local cohomology modules of monomial ideals in terms of simplicial complexes. In this paper, we develop a relative version of these formulas for quotients $I/J$ of monomial ideals, expressing the multigraded pieces of local cohomology modules of $I/J$ as reduced relative (co)homology of pairs of degree complexes. As an application, we obtain a relative Reisner criterion characterizing Cohen-Macaulay monomial ideal quotients. We further apply this relative Hochster--Takayama framework to modules arising from symbolic power filtrations, including symbolic quotients $I^{(t)}/I^{(t+1)}$ and symbolic-ordinary discrepancy module $I^{(t)}/I^t$. In particular, for a squarefree monomial ideal $I$, we give a precise classification of when $I^{(t)}/I^{(t+1)}$ is Cohen-Macaulay for all or, equivalently, for some $t \ge 2$. When $I$ is the edge ideal of a graph, we characterize the Cohen-Macaulayness of $I^{(t)}/I^t$ for all or, equivalently, for some sufficiently large $t$, and analyze the behavior of its dimension function.

math.AC↗
arXiv2025

Extremal Betti numbers of certain two-dimensional monomial ideals

In this paper, we shall provide explicit formulas for the extremal Betti numbers of $R/I$, where $I$ is the defining ideal of certain weighted hyperplanes in $\Bbb{P}^{n-1}$ and $R$ is the polynomial ring in $n$ indeterminates over a field. As a consequence, we completely classify such ideals which are pseudo-Gorenstein as in sense of V. Ene, J. Herzog, T. Hibi and S. S. Madani.

math.AC↗
arXiv2024

Stable value of depth of symbolic powers of edge ideals of graphs

Let $G$ be a simple graph on $n$ vertices. We introduce the notion of bipartite connectivity of $G$, denoted by $\operatorname{bc}(G)$ and prove that $$\lim_{s \to \infty} \operatorname{depth} (S/I(G)^{(s)}) \le \operatorname{bc}(G),$$ where $I(G)$ denotes the edge ideal of $G$ and $S = \mathrm{k}[x_1, \ldots, x_n]$ is a standard graded polynomial ring over a field $\mathrm{k}$. We further compute the depth of symbolic powers of edge ideals of several classes of graphs, including odd cycles and whisker graphs of complete graphs to illustrate the cases where the above inequality becomes equality.

math.AC↗
arXiv2023

Depth of powers of edge ideals of cycles and trees

Let $I$ be the edge ideal of a cycle of length $n \ge 5$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_n]$. We prove that for $2 \le t < \lceil (n+1)/2 \rceil$, $$\operatorname{depth} (S/I^t) = \lceil \frac{n -t + 1}{3} \rceil.$$ When $G = T_{\mathbf{a}}$ is a starlike tree which is the join of $k$ paths of length $a_1, \ldots, a_k$ at a common root $1$, we give a formula for the depth of powers of $I(T_{\mathbf{a}})$.

math.AC↗
arXiv2022

Characterization of graphs whose a small power of their edge ideals has a linear free resolution

Let $I(G)$ be the edge ideal of a simple graph $G$. We prove that $I(G)^2$ has a linear free resolution if and only if $G$ is gap-free and reg$I(G) \le 3$. Similarly, we show that $I(G)^3$ has a linear free resolution if and only if $G$ is gap-free and reg$I(G) \le 4$. We deduce these characterizations from a general formula for the regularity of powers of edge ideals of gap-free graphs $${\rm reg}(I(G)^s) = \max({\rm reg} I(G) + s-1,2s),$$ for $s =2,3$.

math.AC↗
arXiv2021

Integral closure of small powers of edge ideals and their regularity

Let $I(G)$ be the edge ideal of a simple graph $G$ over a field k. We prove that $${\rm reg}(\overline {I(G)^s}) = {\rm reg}(I(G)^s),$$ for all $s \le 4$. Furthermore, we provide an example of a graph $G$ such that $${\rm reg} I(G)^s = {\rm reg} \overline{I(G)^s} = {\rm reg} I(G)^{(s)} = \begin{cases} 5 + 2s & \text{ if char k} = 2 \\ 4 + 2s & \text{ if char k} \neq 2, \end{cases}$$ for all $s \ge 1.$

math.AC↗
arXiv2021

Comparision between regularity of small symbolic powers and ordinary powers of an edge ideal

Let $G$ be a simple graph and $I$ its edge ideal. We prove that $${\rm reg}(I^{(s)}) = {\rm reg}(I^s)$$ for $s = 2,3$, where $I^{(s)}$ is the $s$-th symbolic power of $I$. As a consequence, we prove the following bounds \begin{align*} {\rm reg} I^{s} & \le {\rm reg} I + 2s - 2, \text{ for } s = 2,3, {\rm reg} I^{(s)} & \le {\rm reg} I + 2s - 2,\text{ for } s = 2,3,4. \end{align*}

math.AC↗
arXiv2017

Regularity of symbolic powers and Arboricity of matroids

Let $Δ$ be a simplicial complex of a matroid $M$. In this paper, we explicitly compute the regularity of all the symbolic powers of a Stanley-Reisner ideal $I_Δ$ in terms of combinatorial data of the matroid $M$. In order to do that, we provide a sharp bound between the arboricity of $M$ and the circumference of its dual $M^*$.

math.AC↗
arXiv2013

Combinatorial characterizations of the Cohen-Macaulayness of the second power of edge ideals

Let $I(G)$ be the edge ideal of a simple graph $G$. In this paper, we will give sufficient and necessary combinatorial conditions of $G$ in which the second symbolic and ordinary power of its edge ideal are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay). As an application of our results, we will classify all bipartite graphs in which the second (symbolic) powers are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay).

math.AC↗
arXiv2011

Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals

We present criteria for the Cohen-Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen-Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen-Macaulayness of the second symbolic power or of all symbolic powers of a Stanley-Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen-Macaulay. In particular, all symbolic powers are Cohen-Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen-Macaulayness can pass from a symbolic power to another symbolic powers in different ways.

math.AC↗
arXiv2005

On the regularity of products and intersections of complete intersections

This paper proves that the Castelnuovo-Mumford regularities of the product and sum of two monomial complete intersection ideals are at most the sum of the regularities of the two ideals, and provides examples showing that these inequalities do not hold for general complete intersections.

math.AC↗