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Ngo Viet Trung

Publications and source records attributed to Ngo Viet Trung.

At least 19 recordsLinked to original sources
arXiv2024

A general formula for the index of depth stability of edge ideals

By a classical result of Brodmann, the function $\operatorname{depth} R/I^t$ is asymptotically a constant, i.e. there is a number $s$ such that $\operatorname{depth} R/I^t = \operatorname{depth} R/I^s$ for $t > s$. One calls the smallest number $s$ with this property the index of depth stability of $I$ and denotes it by $\operatorname{dstab}(I)$. This invariant remains mysterious til now. The main result of this paper gives an explicit formula for $\operatorname{dstab}(I)$ when $I$ is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize $\operatorname{dstab}(I)$ explicitly. The formula expresses $\operatorname{dstab}(I)$ in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.

math.AC↗
arXiv2024

Regularity functions of powers of graded ideals

This paper studies the problem of which sequences of non-negative integers arise as the functions $\operatorname{reg} I^{n-1}/I^n$, $\operatorname{reg} R/I^n$, $\operatorname{reg} I^n$ for an ideal $I$ generated by forms of degree $d$ in a standard graded algebra $R$. These functions are asymptotically linear with slope $d$. If $\dim R/I = 0$, we give a complete characterization of all numerical functions which arise as the functions $\operatorname{reg} I^{n-1}/I^n$, $\operatorname{reg} R/I^n$ and show that $\operatorname{reg} I^n$ can be any numerical function $f(n) \ge dn$ that weakly decreases until it becomes a linear function with slope $d$. The latter result gives a negative answer to a question of Eisenbud and Ulrich. If $\dim R/I \ge 1$, we show that $\operatorname{reg} I^{n-1}/I^n$ can be any numerical asymptotically linear function $f(n) \ge dn-1$ with slope $d$ and $\operatorname{reg} R/I^n$ can be any numerical asymptotically linear function $f(n) \ge dn-1$ with slope $d$ that is weakly increasing. Inspired of a recent work of Ein, Ha and Lazarsfeld on non-singular complex projective schemes, we also prove that the function of the saturation degree of $I^n$ is asymptotically linear for an arbitrary graded ideal $I$ and study the behavior of this function.

math.AC↗
arXiv2023

When does a perturbation of the equations preserve the normal cone

Let $(R,\mathfrak m)$ be a local ring and $I, J$ two arbitrary ideals of $R$. Let $\operatorname{gr}_J(R/I)$ denote the associated ring of $R/I$ with respect to $J$, which corresponds to the normal cone in geometry. The main result of this paper shows that if $I = (f_1,...,f_r)$, where $f_1,...,f_r$ is a $J$-filter regular sequence, there exists a number $N$ such that if $f_i' \equiv f_i \mod J^N$ and $I' = (f_1',...,f_r')$, then $\operatorname{gr}_J(R/I) \cong \operatorname{gr}_J(R/I')$. If $J$ is an $\mathfrak m$-primary ideal, this result implies a long standing conjecture of Srinivas and Trivedi on the invariance of the Hilbert-Samuel function under small perturbations, which has been solved recently by Ma, Quy and Smirnov. As a byproduct, the Artin-Rees number of $I$ and $I'$ with respect to $J$ are the same. Furthermore, we give explicit upper bounds for the smallest number $N$ with the above property. These results solve two problems raised by Ma, Quy and Smirnov. There are other interesting consequences on the invariance of the Achilles-Manaresi function, the relation type, the Castelnuovo-Mumford regularity, the Cohen-Macaulayness and the Gorensteiness of the Rees algebra of $R/I$ with respect to $J$ under small perturbation of $I$. We also prove a converse of the main result showing that the condition $I$ being generated by a $J$-filter regular sequence is the best possible for its validity. The main result can be also extended to perturbations with respect to filtrations of ideals. As a consequence, if $R$ is a power series ring, $f_1,...,f_r$ is a filter regular sequence, and $f_i'$ is the $n$-jet of $f_i$ for $n \gg 0$, then $I$ and $I'$ have the same initial ideal with respect to any Noetherian monomial order. A special case of this consequence was a conjecture of Adamus and Seyedinejad on approximations of analytic complete intersection singularities.

math.AC↗
arXiv2023

Decreasing behavior of the depth functions of edge ideals

Let $I$ be the edge ideal of a connected non-bipartite graph and $R$ the base polynomial ring. Then $\operatorname{depth} R/I \ge 1$ and $\operatorname{depth} R/I^t = 0$ for $t \gg 1$. We give combinatorial conditions for $\operatorname{depth} R/I^t = 1$ for some $t$ in between and show that the depth function is non-increasing thereafter. Especially, the depth function quickly decreases to 0 after reaching 1. We show that if $\operatorname{depth} R/I = 1$ then $\operatorname{depth} R/I^2 = 0$ and if $\operatorname{depth} R/I^2 = 1$ then $\operatorname{depth} R/I^5 = 0$. Other similar results suggest that if $\operatorname{depth} R/I^t = 1$ then $\operatorname{depth} R/I^{t+3} = 0$. This a surprising phenomenon because the depth of a power can determine a smaller depth of another power. Furthermore, we are able to give a simple combinatorial criterion for $\operatorname{depth} R/I^{(t)} = 1$ for $t \gg 1$ and show that the condition $\operatorname{depth} R/I^{(t)} = 1$ is persistent, where $I^{(t)}$ denotes the $t$-th symbolic powers of $I$.

math.AC↗
arXiv2021

Buchsbaumness and Castelnuovo-Mumford regularity of non-smooth monomial curves

Projective monomial curves correspond to rings generated by monomials of the same degree in two variables. Such rings always have finite Macaulayfication. We show how to characterize the Buchsbaumness and the Castelnuovo-Mumford regularity of these rings by means of their finite Macaulayfication, and we use this method to study the Buchsbaumness and to estimate the Castelnuovo-Mumford regularity of large classes of non-smooth monomial curves in terms of the given monomials.

math.AC↗
arXiv2020

Multiplicity sequence and integral dependence

We prove that two arbitrary ideals $I \subset J$ in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. We also obtain a Principle of Specialization of Integral Dependence, which gives a condition for integral dependence in terms of the constancy of the multiplicity sequence in families.

math.AC↗
arXiv2019

Depth functions of symbolic powers of homogeneous ideals

This paper addresses the problem of comparing minimal free resolutions of symbolic powers of an ideal. Our investigation is focused on the behavior of the function depth R/I^(t) = dim R - pd I^(t) - 1, where I^(t) denotes the t-th symbolic power of a homogeneous ideal I in a noetherian polynomial ring R and pd denotes the projective dimension. It has been an open question whether the function depth R/I^(t) is non-increasing if I is a squarefree monomial ideal. We show that depth R/I^(t) is almost non-increasing in the sense that depth R/I^(s) \ge depth R/I^(t) for all s \ge 1 and t \in E(s), where E(s) = \cup_{i \ge 1} {t \in N| i(s-1)+1 \le t \le is} (which contains all integers t \ge (s-1)^2+1). The range E(s) is the best possible since we can find squarefree monomial ideals I such that depth R/I^(s) < depth R/I^(t) for t \not\in E(s), which gives a negative answer to the above question. Another open question asks whether the function depth R/I^(t) is always constant for t \gg 0. We are able to construct counter-examples to this question by monomial ideals. On the other hand, we show that if I is a monomial ideal such that I^(t) is integrally closed for t \gg 0 (e.g. if I is a squarefree monomial ideal), then depth R/I^(t) is constant for t \gg 0 with lim_{t \to \infty} depth R/I^(t) = dim R - dim \oplus_{t \ge 0} I^(t)/m I^(t). Our last result (which is the main contribution of this paper) shows that for any positive numerical function ϕ(t) which is periodic for t \gg 0, there exist a polynomial ring R and a homogeneous ideal I such that depth R/I^(t) = ϕ(t) for all t \ge 1. As a consequence, for any non-negative numerical function ψ(t) which is periodic for t \gg 0, there is a homogeneous ideal I and a number c such that pd I^(t) = ψ(t) + c for all t \ge 1.

math.AC↗
arXiv2019

Symbolic powers of sums of ideals

Let $I$ and $J$ be nonzero ideals in two Noetherian algebras $A$ and $B$ over a field $k$. Let $I+J$ denote the ideal generated by $I$ and $J$ in $A\otimes_k B$. We prove the following expansion for the symbolic powers: $$(I+J)^{(n)} = \sum_{i+j = n} I^{(i)} J^{(j)}.$$ If $A$ and $B$ are polynomial rings and if chara$(k) = 0$ or if $I$ and $J$ are monomial ideals, we give exact formulas for the depth and the Castelnuovo-Mumford regularity of $(I+J)^{(n)}$, which depend on the interplay between the symbolic powers of $I$ and $J$. The proof involves a result of independent interest which states that under the above assumption, the induced map Tor$_i^A(k,I^{(n)}) \to$ Tor$_i^A(k,I^{(n-1)})$ is zero for all $i \ge 0$, $n \ge 0$. We also investigate other properties and invariants of $(I+J)^{(n)}$ such as the equality between ordinary and symbolic powers, the Waldschmidt constant and the Cohen-Macaulayness.

math.AC↗
arXiv2019

Depth functions of powers of homogeneous ideals

We settle a conjecture of Herzog and Hibi, which states that the function depth $S/Q^n$, $n \ge 1$, where $Q$ is a homogeneous ideal in a polynomial ring $S$, can be any convergent numerical function. We also give a positive answer to a long-standing open question of Ratliff on the associated primes of powers of ideals.

math.AC↗
arXiv2019

Membership criteria and containments of powers of monomial ideals

We present a close relationship between matching number, covering numbers and their fractional versions in combinatorial optimization and ordinary powers, integral closures of powers, and symbolic powers of monomial ideals. This relationship leads to several new results and problems on the containments between these powers.

math.AC↗
arXiv2018

Associated primes of powers of edge ideals and ear decompositions of graphs

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly unrelated notions of Commutative Algebra and Combinatorics. It covers all previous major results in this topic and has several interesting consequences.

math.AC↗
arXiv2018

Castelnuovo-Mumford regularity and Ratliff-Rush closure

We establish strong relationships between the Castelnuovo-Mumford regularity and the Ratliff-Rush closure of an ideal. Our results have several interesting consequences on the computation of the Ratliff-Rush closure, the stability of the Ratliff-Rush filtration, the invariance of the reduction number, and the computation of the Castelnuovo-Mumford regularity of the Rees algebra and the fiber ring. In particular, we prove that the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring are equal for large classes of monomial ideals in two variables, thereby verifying a conjecture of Eisenbud and Ulrich for these cases.

math.AC↗
arXiv2018

Depth and regularity modulo a principal ideal

We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of hypegraphs and of powers of ideals.

math.AC↗
arXiv2017

Toward a Theory of Monomial Preorders

In this paper we develop a theory of monomial preorders, which differ from the classical notion of monomial orders in that they allow ties between monomials. Since for monomial preorders, the leading ideal is less degenerate than for monomial orders, our results can be used to study problems where monomial orders fail to give a solution. Some of our results are new even in the classical case of monomial orders and in the special case in which the leading ideal defines the tangent cone.

math.AC↗
arXiv2016

Depth and regularity of powers of sums of ideals

Given arbitrary homogeneous ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $k$, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum $I+J$ in $A \otimes_k B$ in terms of those of $I$ and $J$. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.

math.AC↗
arXiv2014

Saturation and associated primes of powers of edge ideals

For the edge ideal I of an arbitrary simple graph G we describe the monomials of the saturation of a power of I in terms of (vertex) weighted graphs associated with the monomials. This description allows us to characterize the embedded associated primes of a power of I as covers of G which contain certain types of subgraphs of G. As an application, we completely classify the associated primes of the second and the third power of I in terms of G.

math.AC↗