arXiv subjects

Cleto B. Miranda-Neto

Publications and source records attributed to Cleto B. Miranda-Neto.

At least 19 recordsLinked to original sources

arXiv2026

Finite projective dimension and a question of Jorgensen

This paper studies finite projective dimension of finitely generated modules over a Noetherian local ring, by means of spectral sequence methods related to generalized local cohomology. Our main goal is to address a question raised by D. Jorgensen over fifteen years ago, concerning a prescribed bound (via Ext vanishing) for projective dimension over a complete intersection local ring. We obtain similar results involving other homological dimensions as well. Also we make use of weakly full ideals to derive further criteria for prescribed bound on projective dimension.

math.AC↗

arXiv2026

Epimorphisms of local cohomology modules, a general Peskine-Szpiro theorem, and an application to sheaf cohomology vanishing for thickenings

We study the surjectivity of certain maps involving local cohomology modules, which we can realize as a dual version of part of the investigation developed by Bhatt, Blickle, Lyubeznik, Singh and Zhang on the sheaf cohomology of thickenings (i.e., subschemes defined by powers of ideals), where injectivity played a central role. To this end, we introduce and investigate properties of cohomologically Mittag-Leffler (cML) rings, associated to a given flat local endomorphism (for instance the Frobenius map of a regular ring of prime characteristic), a class which we show to contain, in our setting, the so-called cohomologically full rings of Dao, De Stefani and Ma (in particular, Cohen-Macaulay, Stanley-Reisner, and Du Bois singularities) as well as rings with an ideal inducing a pure endomorphism of the quotient. Our two major specific goals rely upon the prime characteristic setting. First, we extend for the class of cML rings a classical result of Peskine and Szpiro that relates the cohomological dimension and the height of a given Cohen-Macaulay ideal. Second, we prove and illustrate a Kodaira type vanishing result on the sheaf cohomology of thickenings.

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arXiv2025

Bounds on Dao numbers and applications to regular local rings

The so-called Dao numbers are a sort of measure of the asymptotic behaviour of full properties of certain product ideals in a Noetherian local ring $R$ with infinite residue field and positive depth. In this paper, we answer a question of H. Dao on how to bound such numbers. The auxiliary tools range from Castelnuovo-Mumford regularity of appropriate graded structures to reduction numbers of the maximal ideal. In particular, we substantially improve previous results (and answer questions) by the authors. Finally, as an application of the theory of Dao numbers, we provide new characterizations of when $R$ is regular; for instance, we show that this holds if and only if the maximal ideal of $R$ can be generated by a $d$-sequence (in the sense of Huneke) if and only if the third Dao number of any (minimal) reduction of the maximal ideal vanishes.

math.AC↗

arXiv2024

On the factorial case of Huneke's conjecture for local cohomology modules

A conjecture raised in 1990 by C. Huneke predicts that, for a $d$-dimensional Noetherian local ring $R$, local cohomology modules of finitely generated $R$-modules have finitely many associated primes. Although counterexamples do exist, the conjecture has been confirmed in several cases, for instance if $d\leq 3$, and witnessed some progress in special cases for higher $d$. In this paper, assuming that $R$ is a factorial domain, we establish the case $d=4$, and under different additional conditions (in a couple of results) also the case $d=5$. Finally, when $R$ is regular and contains a field, we apply the Hartshorne-Lichtenbaum vanishing theorem as a tool to deal with the case $d=6$.

math.AC↗

arXiv2024

Homological dimensions, the Gorenstein property, and special cases of some conjectures

Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we investigate the impact of the finiteness of certain homological dimensions of $M$ if $R$ is local, mainly when $R$ is Cohen-Macaulay and with a partial focus on duals. Along the way, we produce various freeness criteria for modules. Finally, we give applications, including characterizations of when $R$ is Gorenstein (and other ring-theoretic properties as well, sometimes in the prime characteristic setting), particularly by means of its anticanonical module, and in addition we address special cases of some long-standing conjectures; for instance, we confirm the 1985 conjecture of Vasconcelos on normal modules in case the module of differentials is almost Cohen-Macaulay.

math.AC↗

arXiv2024

Cohen-Macaulay pairs

The main purpose of this note is to extend and establish a new approach to the concept of (relative) Cohen-Macaulayness, by investigating the cohomological dimension as well as the depth of a pair of modules over a commutative Noetherian ring. The notion also largely extends the cohomologically complete intersection property of Hellus and Schenzel. As crucial tools, we use Herzog's theory of generalized local cohomology along with spectral sequence techniques. We also provide byproducts concerning two classical problems in homological commutative algebra.

math.AC↗

arXiv2023

An Ext-Tor duality theorem, cohomological dimension, and applications

We provide a duality theorem between Ext and Tor modules over a Cohen-Macaulay local ring possessing a canonical module, and use it to prove some freeness criteria for finite modules. The applications include a characterization of codimension three complete intersection ideals and progress on a long-held multi-conjecture of Vasconcelos. By a similar technique, we furnish another theorem which in addition makes use of the notion of cohomological dimension and is mainly of interest in dimension one; as an application, we show that the celebrated Huneke-Wiegand conjecture in the case of complete intersections holds true provided that a single additional condition is satisfied.

math.AC↗

arXiv2023

Tensor products and solutions to two homological conjectures for Ulrich modules

We address the problem of when the tensor product of two finitely generated modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of Goto et al., and in particular in the original sense from the 80's. As applications, besides freeness criteria for modules, characterizations of complete intersections, and an Ulrich-based approach to the long-standing Berger's conjecture, we show that two celebrated homological conjectures, namely the Auslander-Reiten and the Huneke-Wiegand problems, are true for the class of Ulrich modules.

math.AC↗

arXiv2023

Dao's question on the asymptotic behaviour of fullness

For a local ring $(R, \M)$ of infinite residue field and positive depth, we address the question raised by H. Dao on how to control the asymptotic behaviour of the $\M$-full, full, and weakly $\M$-full properties of certain ideals (such notions were first investigated by D. Rees and J. Watanabe), by means of bounding appropriate numbers which express such behaviour. We establish upper bounds, and in certain cases even formulas for such invariants. The main tools used in our results are reduction numbers along with Ratliff-Rush closure of ideals, and also the Castelnuovo-Mumford regularity of the Rees algebra of $\M$.

math.AC↗

arXiv2023

Generalized Hartshorne's problem on finiteness properties of local cohomology modules

Our main goal in this paper is to answer new positive cases of the natural generalized version of Hartshorne's celebrated question on cofiniteness of local cohomology modules, and consequently of Huneke's conjecture on the finiteness of their sets of associated primes. Our approach, by means of which we extend several results from the literature, is essentially based on spectral sequence techniques and connections to numerical invariants such as the cohomological dimension and the Gorenstein projective dimension. We also provide, over a polynomial ring, a rather pathological example of a non weakly Laskerian module (i.e., it admits a quotient with infinitely many associated primes) whose first local cohomology module is non-zero and has finitely many associated primes.

math.AC↗

arXiv2023

On the theory of generalized Ulrich modules

In this paper we further develop the theory of generalized Ulrich modules introduced in 2014 by Goto et al. Our main goal is to address the problem of when the operations of taking the Hom functor and horizontal linkage preserve the Ulrich property. One of the applications is a new characterization of quadratic hypersurface rings. Moreover, in the Gorenstein case, we deduce that applying linkage to sufficiently high syzygy modules of Ulrich ideals yields Ulrich modules. Finally, we explore connections to the theory of modules with minimal multiplicity, and as a byproduct we determine the Chern number of an Ulrich module as well as the Castelnuovo-Mumford regularity of its Rees module.

math.AC↗

arXiv2023

Free divisors, blowup algebras of Jacobian ideals, and maximal analytic spread

Free divisors form a celebrated class of hypersurfaces which has been extensively studied in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors and investigate central aspects of the blowup algebras of their Jacobian ideals. For instance, for all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay -- a desirable feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property with tools ranging from cohomology to asymptotic depth. In addition, as an application, we give an ideal-theoretic homological criterion for homaloidal divisors, i.e., hypersurfaces whose polar maps are birational.

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

On reduction numbers and Castelnuovo-Mumford regularity of blowup rings and modules

We prove new results on the connections between reduction numbers and the Castelnuovo-Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff-Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we generalize a result of A. Mafi on ideals in two-dimensional Cohen-Macaulay local rings, by extending it to arbitrary dimension (and allowing for the setting relative to a Cohen-Macaulay module). We derive a number of applications, including a characterization of (polynomial) ideals of linear type, progress on the theory of generalized Ulrich ideals, and improvements of results by other authors.

math.AC↗

arXiv2022

Vanishing of (co)homology, freeness criteria, and the Auslander-Reiten conjecture for Cohen-Macaulay Burch rings

We establish new results on (co)homology vanishing and Ext-Tor dualities, and derive a number of freeness criteria for finite modules over Cohen-Macaulay local rings. In the main application, we settle the long-standing Auslander-Reiten conjecture for the class of Cohen-Macaulay Burch rings, among other results toward this and related problems, e.g., the Tachikawa and Huneke-Wiegand conjectures. We also derive results on further topics of interest such as Cohen-Macaulayness of tensor products and Tor-independence, and inspired by a paper of Huneke and Leuschke we obtain characterizations of when a local ring is regular, or a complete intersection, or Gorenstein; for the regular case, we describe progress on some classical differential problems, e.g., the strong version of the Zariski-Lipman conjecture. Along the way, we generalize several results from the literature and propose various questions.

math.AC↗

arXiv2022

A formula for symbolic powers

Let $S$ be a Cohen-Macaulay ring which is local or standard graded over a field, and let $I$ be an unmixed ideal that is also generically a complete intersection. Our goal in this paper is multi-fold. First, we give a multiplicity-based characterization of when an unmixed subideal $J \subseteq I^{(m)}$ equals the $m$-th symbolic power $I^{(m)}$ of $I$. Second, we provide a saturation-type formula to compute $I^{(m)}$ and employ it to deduce a theoretical criterion for when $I^{(m)}=I^m$. Third, we establish an explicit linear bound on the exponent that makes the saturation formula effective, and use it to obtain lower bounds for the initial degree of $I^{(m)}$. Along the way, we prove a conjecture (in fact, a generalized version of it) due to Eisenbud and Mazur about ${\rm ann}_S(I^{(m)}/I^m)$, and we propose a conjecture connecting the symbolic defect of an ideal to Jacobian ideals.

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↗

arXiv2022

On reduced G-perfection and horizontal linkage relative to a semidualizing module

In their investigation of horizontal linkage of modules of finite Gorenstein dimension over a commutative, Noetherian, semiperfect (e.g., local) ring, Dibaei and Sadeghi introduced the class of reduced G-perfect modules, making use of Bass' concept of reduced grade. A few years later, the same authors extended this class by considering the relative property of reduced G$_C$-perfection, where $C$ is a semidualizing module, and studied linkage even further. In the present paper, we contribute to their theory and also generalize results of Auslander and Bridger as well as of Martsinkovsky and Strooker. Our investigation includes, for example, when reduced G$_C$-perfection is preserved by relative Auslander transpose, and how to numerically characterize horizontally linked modules under suitable conditions. Along the way, we show how to produce reduced $\textrm{G}_C$-perfect modules that are also $C$-$k$-torsionless (for a given integer $k\geq 0$) but fail to be $\textrm{G}_C$-perfect, and moreover we illustrate that, in contrast to the usual grade, the relative reduced grade does depend on the choice of $C$.

math.AC↗