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Pham Hung Quy

Publications and source records attributed to Pham Hung Quy.

At least 19 recordsLinked to original sources
arXiv2026

On the perturbations of Noetherian local domains

We study how the properties of being reduced, integral domain, and normal, behave under small perturbations of the defining equations of a noetherian local ring. It is not hard to show that the property of being a local integral domain (reduced, normal ring) is not stable under small perturbations in general. We prove that perturbation stability holds in the following situations: (1) perturbation of being an integral domain for factorial excellent Henselian local rings; (2) perturbation of normality for excellent local complete intersections containing a field of characteristic zero; and (3) perturbation of reducedness for excellent local complete intersections containing a field of characteristic zero, and for factorial Nagata local rings.

math.AC↗
arXiv2026

Buchsbaum modules and parameter ideals under flat extensions

Let $(R,\frak m)$ and $(S, \frak n)$ be commutative Noetherian local rings, let $φ: R\to S$ be a flat local homomorphism. In this paper, we first characterize the ascent and descent of Buchsbaum modules and generalized Cohen-Macaulay modules via the flat extension $φ$. Then, we provide relationships between the parameter ideals of a finitely generated $R$-module $M$ and those of $M\otimes_RS$ under the assumption that $\frak{n} = \frak{m} S.$

math.AC↗
arXiv2026

A sharp bound for the Frobenius test exponents in generalized Cohen-Macaulay local rings

Let $(R,\frak m)$ be a generalized Cohen-Macaulay local ring of prime characteristic $p$. In this paper we give a sharp bound for the Frobenius test exponent of parameter ideals. Namely, we prove that $$\mathrm{Fte}(R) \le \lceil \log_p(2n_0)\rceil + \mathrm{HSL}(R),$$ where $n_0$ is the integer such that $\frak m^{n_0} \, H^i_{\frak m}(R) = 0$ for all $i < \mathrm{dim}(R)$, and $\lceil x\rceil$ is the smallest integer that is greater than or equal to $x$.

math.AC↗
arXiv2026

A Buchsbaum theory for Frobenius closure

We give a partial characterization for when the difference $e(\mathfrak{q})-\ell_R(R/\mathfrak{q}^F)$ is independent of the choice of parameter ideal $\mathfrak{q}\subseteq R$ in an excellent equidimensional local ring $(R,\mathfrak{m})$ of prime characteristic $p>0$. Here, $\mathfrak{q}^F$ is the Frobenius closure of $\mathfrak{q}$ and $e(\mathfrak{q})$ denotes the Hilbert--Samuel multiplicity of $\mathfrak{q}$. In addition to ideal-theoretic equivalences, our characterization involves the derived category and is motivated by Schenzel's criterion of the Buchsbaum property as well as similar results of Ma-Quy in the setting of tight closure.

math.AC↗
arXiv2025

The Depth Formula for modules over quotients of Gorenstein rings

A foundational result by C. Huneke and V. Trivedi provides a formula for the depth of an ideal in terms of height, computed over a finite set of prime ideals, for rings that are homomorphic images of regular rings. Building on a result by the first author for local quotients of Cohen-Macaulay rings, this paper first gives a new proof and derives a similar formula for the finiteness dimension. Our main result then establishes the depth formula for non-local rings that are homomorphic images of a finite-dimensional Gorenstein ring.

math.AC↗
arXiv2025

On the structure of finitely generated modules and the unmixed degrees

Let $(R, \frak m)$ be a homomorphic image of a Cohen-Macaulay local ring and $M$ a finitely generated $R$-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every finitely generated $R$-module $M$ is associated by a sequence of invariant modules. This modules sequence expresses the deviation of $M$ with the Cohen-Macaulay property. This result generalizes the unmixed theorem of Cohen-Macaulayness for any finitely generated $R$-module. As an application we construct a new extended degree in sense of Vasconcelos.

math.AC↗
arXiv2025

Upper bound of multiplicity in Cohen-Macaulay rings of prime characteristic

Let $(R, \frak m)$ be a local ring of prime characteristic $p$ and of dimension $d$ with the embedding dimension $v$, type $s$ and the Frobenius test exponent for parameter ideals $\mathrm{Fte}(R)$. We will give an upper bound for the multiplicity of Cohen-Macaulay rings in prime characteristic in terms of $\mathrm{Fte}(R),d,v$ and $s$. Our result extends the main results for Gorenstein rings due to Huneke and Watanabe.

math.AC↗
arXiv2024

Vanishing and non-negativity of the first normal Hilbert coefficient

Let $(R,\mathfrak{m})$ be a Noetherian local ring such that $\widehat{R}$ is reduced. We prove that, when $\widehat{R}$ is $S_2$, if there exists a parameter ideal $Q\subseteq R$ such that $\bar{e}_1(Q)=0$, then $R$ is regular and $ν(\mathfrak{m}/Q)\leq 1$. This leads to an affirmative answer to a problem raised by Goto-Hong-Mandal. We also give an alternative proof (in fact a strengthening) of their main result. In particular, we show that if $\widehat{R}$ is equidimensional, then $\bar{e}_1(Q)\geq 0$ for all parameter ideals $Q\subseteq R$, and in characteristic $p>0$, we actually have $e_1^*(Q)\geq 0$. Our proofs rely on the existence of big Cohen-Macaulay algebras.

math.AC↗
arXiv2024

Colength, multiplicity, and ideal closure operations II

Let $(R, \mathfrak{m})$ be a Noetherian local ring. This paper concerns several extremal invariants arising from the study of the relation between colength and (Hilbert--Samuel or Hilbert--Kunz) multiplicity of an $\mathfrak{m}$-primary ideal. We introduce versions of these invariants by restricting to various closures and ``cross-pollinate'' the two multiplicity theories by asking for analogues invariants already established in one of the theories. On the Hilbert--Samuel side, we prove that the analog of the Stückrad--Vogel invariant (that is, the infimum of the ratio between the multiplicity and colength) for integrally closed $\mathfrak{m}$-primary ideals is often $1$ under mild assumptions. We also compute the supremum and infimum of the relative drops of multiplicity for (integrally closed) $\mathfrak{m}$-primary ideals. On the Hilbert--Kunz side, we study several analogs of the Lech--Mumford and Stückrad--Vogel invariants.

math.AC↗
arXiv2023

Tight Hilbert Polynomial and F-rational local rings

Let $(R,\mathfrak{m})$ be a Noetherian local ring of prime characteristic $p$ and $Q$ be an $\mathfrak{m}$-primary parameter ideal. We give criteria for F-rationality of $R$ using the tight Hilbert function $H^*_Q(n)=\ell(R/(Q^n)^*$ and the coefficient $e_1^*(Q)$ of the tight Hilbert polynomial $P^*_Q(n)=\sum_{i=0}^d(-1)^ie_i^*(Q)\binom{n+d-1-i}{d-i}.$ We obtain a lower bound for the tight Hilbert function of $Q$ for equidimensional excellent local rings that generalises a result of Goto and Nakamura. We show that if $\dim R=2 $, the Hochster-Huneke graph of $R$ is connected and this lower bound is achieved then $R$ is F-rational. Craig Huneke asked if the $F$-rationality of unmixed local rings may be characterized by the vanishing of $e_1^*(Q).$ We construct examples to show that without additional conditions, this is not possible. Let $R$ be an excellent, reduced, equidimensional Noetherian local ring and $Q$ be generated by parameter test elements. We find formulas for $e_1^*(Q), e_2^*(Q), \ldots, e_d^*(Q)$ in terms of Hilbert coefficients of $Q$, lengths of local cohomology modules of $R,$ and the length of the tight closure of the zero submodule of $H^d_{\mathfrak{m}}(R).$ Using these we prove: $R$ is F-rational $\Leftrightarrow e_1^*(Q)=e_1(Q) \Leftrightarrow$ depth $R\geq 2$ and $e_1^*(Q)=0.$

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

Colength, multiplicity, and ideal closure operations

In a formally unmixed Noetherian local ring, if the colength and multiplicity of an integrally closed ideal agree, then $R$ is regular. We deduce this using the relationship between multiplicity and various ideal closure operations.

math.AC↗
arXiv2022

On the Frobenius closure of parameter ideals when the ring is F-injective on the punctured spectrum

Let $(R,\frak m)$ be an excellent generalized Cohen-Macaulay local ring of dimension $d$ that is $F$-injective on the punctured spectrum. Let $\frak q$ be a standard parameter ideal of $R$. The aim of the paper is to prove that $$\ell_R({\frak q}^F/{\frak q})\leq \sum\limits_{i=0}^{d}\binom{d}{i}\ell_R(0^F_{H^i_{\frak m}(R)}).$$ Moreover, if $\frak q$ is contained in a large enough power of $\frak m$, we have $${\frak q}^F/{\frak q} \cong \bigoplus_{i=0}^d (0^F_{H^i_{\frak m}(R)})^{\binom{d}{i}}.$$

math.AC↗
arXiv2022

A Buchsbaum theory for tight closure

A Noetherian local ring $(R,\mathfrak{m})$ is called Buchsbaum if the difference $e(\mathfrak{q}, R)-\ell(R/\mathfrak{q})$, where $\mathfrak{q}$ is an ideal generated by a system of parameters, is a constant independent of $\mathfrak{q}$. In this article, we study the tight closure analog of this condition. We prove that in an unmixed excellent local ring $(R,\mathfrak{m})$ of prime characteristic $p>0$ and dimension at least one, the difference $e(\mathfrak{q}, R)-\ell(R/\mathfrak{q}^*)$ is independent of $\mathfrak{q}$ if and only if the parameter test ideal $τ_{\text{par}}(R)$ contains $\mathfrak{m}$. We also provide a characterization of this condition via derived category which is analogous to Schenzel's criterion for Buchsbaum rings.

math.AC↗
arXiv2022

On the limit closure of a sequence of elements in local rings

We present a systematic study for the limit closure $(\underline{x})^{\lim}$ of a sequence of elements $\underline{x}$ (eg. a system of of parameters) in a local ring. Firstly, we answer the question which elements are always contained in the limit closure of a system of parameters. Then we apply this result to give a characterization of systems of parameters which is a generalization of previous results of Dutta and Roberts in \cite{DR} and of Fouli and Huneke in \cite{FH}. We also prove a topological characterization of unmixed local rings. In two dimensional case, we compute explicitly the limit closure of a system of parameters. Some interesting examples are given.

math.AC↗