arXiv subjects

Victor H. Jorge-Pérez

Publications and source records attributed to Victor H. Jorge-Pérez.

9 recordsLinked to original sources

arXiv2026

Remarks on Auslander's depth formula for quasi-projective dimension

For nonzero finitely generated $R$-modules $M$ and $N$ over a Noetherian local ring $R$, Auslander's depth formula is the equality $$ \operatorname{depth} M + \operatorname{depth} N = \operatorname{depth} R + \operatorname{depth}(\operatorname{Tor}_q^R(M,N)) - q, $$ where $ q := \sup\{ i \ge 0 \mid \operatorname{Tor}_i^R(M,N) \neq 0 \}$. Gheibi, Jorgensen, and Takahashi introduced a homological invariant called quasi-projective dimension, which generalizes projective dimension, and proved that Auslander's depth formula holds when $M$ has finite quasi-projective dimension and $q=0$. In this paper, we prove that the formula still holds when $M$ has finite quasi-projective dimension, $q<\infty$ and $\operatorname{depth}(\operatorname{Tor}_q^R(M,N)) \leq 1$. We present several applications of this result; in particular, we recover a theorem of Araya and Yoshino, extend our result to the setting of semidualizing modules, and in this framework derive an improved version of the dependency formula for quasi-projective dimension with respect to a semidualizing module recently obtained by Dey, Ferraro, and Gheibi.

math.AC↗

arXiv2026

On Extension modules of finite homological dimension

We explore the implications of the finiteness of homological dimensions for Ext modules, focusing on projective dimension, injective dimension, and their Gorenstein counterpart. In this direction, we establish several finiteness criteria for homological dimensions. Under such finiteness conditions, our main result is a new duality for certain Ext modules when tensored by canonical modules.

math.AC↗

arXiv2025

Ischebeck's formula, grade and quasi-homological dimensions

The quasi-projective dimension and quasi-injective dimension are recently introduced homological invariants that generalize the classical notions of projective dimension and injective dimension, respectively. For a local ring $R$ and finitely generated $R$-modules $M$ and $N$, we provide conditions involving quasi-homological dimensions where the equality $\sup \lbrace i\geq 0: \operatorname{Ext}_R^i(M,N)\not=0 \rbrace =\operatorname{depth} R-\operatorname{depth} M$, which we call Ischebeck's formula, holds. One of the results in this direction generalizes a well-known result of Ischebeck concerning modules of finite injective dimension, considering the quasi-injective dimension. On the other hand, we establish an inequality relating the quasi-projective dimension of a finitely generated module to its grade and introduce the concept of a quasi-perfect module as a natural generalization of a perfect module. We prove several results for this new concept similar to the classical results. Additionally, we provide a formula for the grade of finitely generated modules with finite quasi-injective dimension over a local ring, as well as grade inequalities for modules of finite quasi-projective dimension. In our study, Cohen-Macaulayness criteria are also obtained.

math.AC↗

arXiv2025

On $\operatorname{Ext}$-finite modules, quasi-injective dimension and width of modules

Let $(R,\mathfrak{m},k)$ be a commutative Noetherian local ring. It is well-known that if $M$ is a finitely generated $R$-module of finite quasi-injective dimension, then $\operatorname{qid}_RM = \operatorname{depth} R$. In this paper, we demonstrate that under the weaker condition that $M$ is $\operatorname{Ext}$-finite and has finite quasi-injective dimension, the equality $\operatorname{qid}_R M =0$ holds if and only if $\operatorname{Ext}_R^{i>0}(R/(\boldsymbol{x}),M)=0$, where $\boldsymbol{x} \in \mathfrak{m}$ is a maximal $R$-sequence and if $\operatorname{qid}_R M \neq 0$, we show then that $\operatorname{qid}_R M = \sup \lbrace i : \operatorname{Ext}_R^i(R/(\boldsymbol{x}),M) \neq 0 \rbrace$. Also, we show that if $R$ is a Cohen-Macaulay local ring and $M$ is an $\operatorname{Ext}$-finite $R$-module of finite quasi-injective dimension, then $\operatorname{depth} R = \operatorname{qid}_R M + \inf \lbrace i : \operatorname{Tor}_i^R(k,M) \neq 0 \rbrace$, provided that $\inf \lbrace i : \operatorname{Tor}_i^R(k,M) \neq 0 \rbrace< \infty$.

math.AC↗

arXiv2025

On modules whose dual is of finite Gorenstein dimension

In this paper, we aim to obtain some results under the condition that the dual of a module over a commutative Noetherian ring has finite Gorenstein dimension. In this direction, we derive results involving vanishing of Ext as well as the freeness or totally reflexivity of modules. For instance, we provide a generalization of a celebrated theorem by Auslander and Bridger, obtain criteria for the totally reflexivity of modules over Cohen-Macaulay rings as well as of locally totally reflexive modules on the punctured spectrum, and recover a result by Araya. Moreover, we prove that the Auslander-Reiten conjecture holds true for all finitely generated modules $M$ over a commutative Noetherian ring $R$ such that $\operatorname{G-dim}_R(\operatorname{Hom}_R(M,R))<\infty$ and $\operatorname{pd}_R(\operatorname{Hom}_R(M,M))<\infty$. Additionally, we derive Gorenstein criteria under the condition that the dual of certain modules is of finite Gorenstein dimension. Furthermore, we explore some applications in the theory of the modules of Kähler differentials of order $n\geq 1$, specifically concerning the $k$-torsionfreeness of these modules and the Herzog-Vasconcelos conjecture.

math.AC↗

arXiv2025

The Auslander-Reiten Conjecture, Finite $C$-Injective Dimension of $\operatorname{Hom}$, and vanishing of $\operatorname{Ext}$

Let $R$ be a Noetherian local ring, and let $C$ be a semidualizing $R$-module. In this paper, we present some results concerning the vanishing of $\operatorname{Ext}$ and finite injective dimension of $\operatorname{Hom}$. Additionally, we extend these results in terms of finite $C$-injective dimension of $\operatorname{Hom}$. We also investigate the consequences of some of these extensions in the case where $R$ is Cohen-Macaulay and $C$ is a canonical module for $R.$ Furthermore, we provide positive answers to the Auslander-Reiten conjecture for finitely generated $R$-modules $M$ such that $\mathcal{I}_C\operatorname{-id}_R(\operatorname{Hom}_R(M,R))<\infty$ or $M \in \mathcal{A}_C(R)$ with $\mathcal{I}_C \operatorname{-id}_R(\operatorname{Hom}_R(M,M))<\infty$. Moreover, we derive a number of criteria for a semidualizing $R$-module $C$ to be a canonical module for $R$ in terms of the vanishing of $\operatorname{Ext}$ and the finite $C$-injective dimension of $\operatorname{Hom}$.

math.AC↗

arXiv2025

Hilbert-Kunz Multiplicity of Fiber Product Rings and Nagata Idealizations

The main purpose of this paper is to provide formulas for the Hilbert-Kunz multiplicity of fiber product rings and Nagata idealizations. We give explicit formulas for the Hilbert-Kunz multiplicity of a fiber product $R \times_T S$, where $R$, $S$, and $T$ are Noetherian local rings sharing the same characteristic and residue field. We compute the Hilbert-Kunz multiplicity of a general Nagata idealization $R \ltimes M$, where $M$ is a finitely generated $R$-module. Additionally, we provide examples, structural results and establish new bounds for the Hilbert-Kunz multiplicity.

math.AC↗

arXiv2022

Rigid modules and homological dimensions

We make use of the concepts of Tor-rigid and rigid-test modules, among others, to investigate the interplay between cohomology vanishing and the finiteness of several homological dimensions such as projective, injective and Gorenstein injective dimensions. Besides criteria for the freeness of modules, the applications include new characterizations of Gorenstein and regular local rings, which are also of interest in the realm of algebraic geometry.

math.AC↗

arXiv2022

Generalized local duality, canonical modules, and prescribed bound on projective dimension

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module.

math.AC↗