arXiv subjects

Van Duc Trung

Publications and source records attributed to Van Duc Trung.

5 recordsLinked to original sources

arXiv2025

On the Stability of Bass and Betti Numbers under Ideal Perturbations in a Local Ring

Let $(R,\mathfrak{m})$ be a Noetherian local ring, and let $J$ be an arbitrary ideal of $R$. Suppose $M$ is a finitely generated $R$-module. Let $x_1,\ldots,x_r$ be a $J$-filter regular sequence on $M$. We provide an explicit number $N$ such that the Bass and Betti numbers of $M/(x_1, \ldots, x_r)M$ are preserved when we perturb the sequence $x_1, \ldots,x_r$ by $\varepsilon_1, \ldots, \varepsilon_r \in \mathfrak{m}^N$.

math.AC↗

arXiv2025

Koszul Homology Under Small Perturbations

Let $x_1,\ldots,x_s$ be a filter regular sequence in a local ring $(R,\mathfrak{m})$. Denote by $R_{x_1,\ldots,x_s}$ the Koszul complex of $x_1,\ldots,x_s$ over $R$. In this paper, we give an explicit number $N$ such that the sum of lengths $\sum_{i=1}^s (-1)^i\ell(H_i(R_{x_1,\ldots,x_s}))$ is preserved when we perturb the sequence $x_1, \ldots,x_s$ by $\varepsilon_1, \ldots, \varepsilon_s \in \mathfrak{m}^N$. Applying this result and the main Theorem of Eisenbud, we show that there exits $N >0$ such that for all $i \geq 1$ the length of $H_i(R_{x_1,\ldots,x_s})$ is preserved under small perturbation.

math.AC↗

arXiv2025

The initial ideal of generic sequences and Fröberg's Conjecture

Let $K$ be an infinite field and let $I = (f_1,\cdots,f_r)$ be an ideal in the polynomial ring $R = K[x_1,\cdots,x_n]$ generated by generic forms of degrees $d_1,\cdots,d_r$. A longstanding conjecture by Fröberg predicts the shape of the Hilbert function of $R/I.$ In 2010 Pardue stated a conjecture on the initial ideal of $n$ generic forms with respect to the deg-revlex order and he proved that it is equivalent to Fröberg's Conjecture. We study Pardue's Conjecture and we prove it under suitable conditions on the degrees of the forms. This yields a partial solution to Fröberg's Conjecture in the case $r \leq n+2$ over an infinite field of any characteristic.

math.AC↗

arXiv2025

The second Hilbert coefficient of modules with almost maximal depth

Let $\mathbb{M} = \{ M_n \}$ be a good $\mathfrak{q}$-filtration of a finitely generated $R$-module $M$ of dimension $d$, where $(R,\mathfrak{m})$ is a local ring and $\mathfrak{q}$ is an $\mathfrak{m}$-primary ideal of $R$. In case $depth(M) \geq d-1$, we give an upper bound for the second Hilbert coefficient $e_2(\mathbb{M})$ generalizing results by Huckaba-Marley and Rossi-Valla proved assuming that $M$ is Cohen-Macaulay. We also give a condition for the equality, which relates to the depth of the associated graded module $gr_{\mathbb{M}}(M)$. A lower bound on $e_2(\mathbb{M})$ is proved generalizing a result by Rees and Narita.

math.AC↗

arXiv2021

Small perturbations in generalized Cohen-Macaulay local rings

Let $(R, \frak m)$ be a generalized Cohen-Macaulay local ring of dimension $d$, and $f_1, \ldots, f_r$ a part of system of parameters of $R$. In this paper we give explicit numbers $N$ such that the lengths of all lower local cohomology modules and the Hilbert function of $R/(f_1, \ldots, f_r)$ are preserved when we perturbs the sequence $f_1, \ldots, f_r$ by $\varepsilon_1, \ldots, \varepsilon_r \in {\frak m}^N$. The second assertion extends a previous result of Srinivas and Trivedi for generalized Cohen-Macaulay rings.

math.AC↗