arXiv subjects

Hop Dang Nguyen

Publications and source records attributed to Hop Dang Nguyen.

4 recordsLinked to original sources

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↗

arXiv2022

A sharp bound for the resurgence of sums of ideals

We prove a sharp upper bound for the resurgence of sums of ideals involving disjoint sets of variables, strengthening work of Bisui--Hà--Jayanthan--Thomas. Complete solutions are delivered for two conjectures proposed by these authors. For given real numbers $a$ and $b$, we consider the set Res$(a,b)$ of possible values of the resurgence of $I+J$ where $I$ and $J$ are ideals in disjoint sets of variables having resurgence $a$ and $b$, respectively. Some questions and partial results about Res$(a,b)$ are discussed.

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

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↗