arXiv subjects

Rafael Holanda

Publications and source records attributed to Rafael Holanda.

12 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

Multiple Tor modules: rigidity and Mayer-Vietoris spectral sequences

We extend some properties of a pair of ideals described in terms of Tor modules to any number of ideals, including the well-known rigidity property. Those extensions require the development of a homological theory for spectral sequences arising from multiple complexes. Out of this theory, two new complexes associated with quotients by sums and quotients by products of the given ideals emerge, and their homologies are related via the Tor-independence property. In the multigraded setting, we describe the support regions of Tor modules for quotients by sums and products of ideals generated by variables in terms of each other.

math.AC↗

arXiv2025

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↗

arXiv2024

Cox-Gorenstein algebras

We study G-graded Artinian algebras having Poincar\'e duality, considering in particular their Lefschetz properties. We also prove a correspondence between the toric setup and the G-graded one, provide an application to toric geometry, and prove a Hessian criterion in the G-graded setup

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

Homology of multiple complexes and Mayer-Vietoris spectral sequences

Similarities are noted in two Mayer-Vietoris spectral sequences that generalize to any number of ideals in the Mayer-Vietoris exact sequence in local cohomology for two ideals. One has as first terms Čech cohomology with respect to sums of the given ideals and converge to cohomology with respect to the product of the ideals, the other has as first terms Čech cohomology with respect to products of the given ideals and converge to cohomology with respect to the sum of the ideals. The first one was obtained by Lyubeznik in \cite{Lyu}, while the second is constructed in \cite[Chapter 2]{Hol} and could also be deduced from results in \cite{God}. We present results on the cohomology of multiple complexes that enables us to deduce both from two related constructions on multiple complexes. A key ingredient is a fact that seems not to have been noticed before: cohomology with respect to a product of ideals is the one of a subcomplex of the Čech complex computing cohomology with respect to the sum of the given ideals; this provides a much shorter complex to compute cohomology with respect to the product of ideals.

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↗

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

Multigraded Tor and local cohomology

Notions of Castelnuovo-Mumford regularity and of $a^*$ invariant were extended from standard graded algebras to the toric setting. We here focus our attention on the standard multigraded case, which corresponds to a product of $k$ projective spaces. A natural notion for a $\mathbb Z^k$-graded module is its support: degrees in which it is not zero. A stabilized version of it is adding $-\mathbb N^k$, in order for the complement (vanishing region) to be stable by addition of $\mathbb N^k$. Cohomology of twists of a sheaf on a product of projective spaces, provided by a graded module, are given by local cohomologies with respect to the product $B$ of the ideals $B_i$ generated by the $k$ sets of variables. Our results shed some light on a central issue, the relation between shifts in graded free resolution and cohomology vanishing: it shows that stabilized support of cohomology with respect to $B$ corresponds to the union of stabilized supports for cohomologies in the $B_i$'s, while shifts in (some of the) graded free resolutions are inside the intersection of these stabilized supports. A one-to-one correspondence between stabilized supports of Tor modules and of local cohomologies with respect to the sum of the $B_i$'s is also established. We then derive a consequence on linear resolutions for truncations of a graded module.

math.AC↗

arXiv2022

Bass and Betti numbers of a module and its deficiency modules

This paper aims to provide several relations between Bass and Betti numbers of a given module and its deficiency modules. Such relations and the tools used throughout allow us to generalize some results of Foxby, characterize Cohen-Macaulay modules in equidimensionality terms and furnish a case for the Auslander-Reiten conjecture.

math.AC↗