arXiv subjects

Doan Trung Cuong

Publications and source records attributed to Doan Trung Cuong.

12 recordsLinked to original sources

arXiv2026

On the pertubations of E-depth, Ef-depth and sequentially Cohen-Macaulay locus

Let (R,m) be a Noetherian local ring, I an ideal of R and M a finitely generated R-module. In this paper, we define and study E-depth and Ef-depth of M in I. We prove that E-depth (resp. Ef-depth under mild conditions) of M in I is the common length of all maximal sequential sequences (resp. maximal sequential f-sequences) of M in I. We show that E-depth and Ef-depth do not decrease under small pertubations. We describe the non sequentially Cohen-Macaulay locus nSCM(M) of M in terms of support and non Cohen-Macaulay locus of deficiency modules of M. Finally, we show that the dimension of non sequentially Cohen-Macaulay locus with respect to a sequential f-sequence does not increase under small pertubations.

math.AC↗

arXiv2025

On direct summands of syzygies of the residue field of a local ring

We investigate local rings in which a syzygy of the residue field occurs as a direct summand of another syzygy of the field. This class of local rings includes Golod rings, Burch rings and non-trivial fiber products of local rings. For such rings, we prove that the Betti sequence of any finitely generated module is eventually periodically non-decreasing. As an application, we confirm the Tachikawa conjecture for all Cohen-Macaulay local rings satisfying this syzygy condition. In the second part of the paper, we show that a recursive direct sum decomposition of the syzygy of the residue field characterizes Golod rings, thereby establishing the converse to a recent theorem of Cuong-Dao-Eisenbud-Kobayashi-Polini-Ulrich [10].

math.AC↗

arXiv2025

Syzygies of the residue field over Golod rings

Let $(R,m,k)$ be a Golod ring. We show a recurrent formula for high syzygies of $k$ interms of previous ones. In the case of embedding dimension at most $2$, we provided complete descriptions of all indecomposable summands of all syzygies of $k$.

math.AC↗

arXiv2021

Componentwise linearity of projective varieties with almost maximal degree

The degree of a projective subscheme has an upper bound in term of the codimension and the reduction number. If a projective variety has an almost maximal degree, that is, the degree equals to the upper bound minus one, then its Betti table has been described explicitly. We build on this work by showing that for most of such varieties, the defining ideals are componentwise linear and in particular the componentwise linearity is suitable for classifying the Betti tables of such varieties. As an application, we compute the Betti table of all varieties with almost maximal degree and componentwise linear resolution.

math.AC↗

arXiv2019

The reduction number and degree bound of projective subschemes

In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen-Macaulay subschemes with linear resolution. Furthermore, it can be shown that there are only two types of reduced, irreducible projective varieties with almost maximal degree. We also give explicit Betti tables for almost maximal cases. Interesting examples are provided to understand our main results.

math.AG↗

arXiv2017

On the length function of saturations of ideal powers

For an ideal $I$ in a local ring $(R, \fm)$, we prove that the integer-valued function $\ell_R(H^0_\fm(R/I^{n+1}))$ is a polynomial for $n$ big enough if either $I$ is a principle ideal or $I$ is generated by part of an almost p-standard system of parameters. Furthermore, we are able to compute the coefficients of this polynomial in terms of length of certain local cohomology modules and usual multiplicity if either the ideal is principal or it is generated by part of a standard system of parameters in a generalized Cohen-Macaulay ring. We also give an example of an ideal generated by part of a (general) system of parameters such that the function $\ell_R(H^0_\fm(R/I^{n+1}))$ is not a polynomial for $n\gg 0$.

math.AC↗

arXiv2013

Local cohomology annihilators and Macaulayfication

The aim of this paper is to study a deep connection between local cohomology annihilators and Macaulayfication and arithmetic Macaulayfication over a local ring. Local cohomology annihilators appear through the notion of p-standard system of parameters. For a local ring, we prove an equivalence of the existence of Macaulayfications; the existence of a p-standard system of parameters; being a quotient of a Cohen-Macaulay local ring; and the verification of Faltings' Annihilator theorem. For a finitely generated module which is unmixed and faithful, we prove an equivalence of the existence of an arithmetic Macaulayfication and the existence of a p-standard system of parameters; and both are proved to be equivalent to the existence of an arithmetic Macaulayfication on the ground ring. A connection between Macaulayfication and universal catenaricity is also discussed.

math.AC↗

arXiv2010

On a new invariant of finitely generated modules over local rings

Let $M$ be a finitely generated module on a local ring $R$ and $\F: M_0\subset M_1\subset...\subset M_t=M$ a filtration of submodules of $M$ such that $ d_o<d_1< ... <d_t=d$, where $d_i=\dim M_i$. This paper is concerned with a non-negative integer $p_\mathcal F(M)$ which is defined as the least degree of all polynomials in $n_1, ..., n_d$ bounding above the function $$\ell(M/(x_1^{n_1}, ..., x_d^{n_d})M)-\sum_{i=0}^tn_1...n_{d_i}e(x_1,..., x_{d_i};M_i).$$ We prove that $p_\mathcal F(M)$ is independent of the choices of good systems of parameters $\underline x=x_1, ..., x_d$. When $\F$ is the dimension filtration of $M$ we also present some relations between $p_\F(M)$ and the polynomial type of each $M_i/M_{i-1}$ and the dimension of the non-sequentially Cohen-Macaulay locus of $M$.

math.AC↗

arXiv2009

Hodge cohomology of étale Nori finite vector bundles

Étale Nori finite vector bundles are those bundles defined by representations of a finite étale group scheme in the usual way. In this note we show that in many cases the dimensions of the Hodge cohomology groups of such a vector bundle and of a twist of it by an automorphism of the ground field are the same. This generalizes to the higher rank case a result of Pink-Roessler.

math.AG↗

arXiv2007

On the Structure of Sequentially Generalized Cohen-Macaulay Modules

A finitely generated module $M$ over a local ring is called a sequentially generalized Cohen-Macaulay module if there is a filtration of submodules of $M$: $M_0\subset M_1\subset ... \subset M_t=M$ such that $\dim M_0<\dim M_1< >... <\dim M_t$ and each $M_i/M_{i-1}$ is generalized Cohen-Macaulay. The aim of this paper is to study the structure of this class of modules. Many basic properties of these modules are presented and various characterizations of sequentially generalized Cohen-Macaulay property by using local cohomology modules, theory of multiplicity and in terms of systems of parameters are given. We also show that the notion of dd-sequences defined in \cite{cc} is an important tool for studying this class of modules.

math.AC↗

arXiv2005

dd-sequences and partial Euler-Poincare' characteristics of Koszul complex

The aim of this paper is to introduce a new notion of sequences called dd-sequences and show that this notion may be convenient for studying the polynomial property of partial Euler-Poincare' characteristics of the Koszul complex with respect to the powers of a system of parameters. Some results about the dd-sequences, the partial Euler-Poincare' characteristics and the lengths of local cohomology modules are presented in the paper. There are also applications of dd-sequences on the structure of sequentially Cohen-Macaulay modules.

math.AC↗

arXiv2005

On Sequentially Cohen-Macaulay Modules

In this paper we present characterizations of sequentially Cohen-Macaulay modules in terms of systems of parameters, which are generalizations of well-known results on Cohen-Macaulay and generalized Cohen-Macaulay modules. The sequentially Cohen-Macaulayness of Stanley-Reisner rings of small embedding dimension are also examined.

math.AC↗