arXiv subjects

Paulo Martins

Publications and source records attributed to Paulo Martins.

11 recordsLinked to original sources

arXiv2026

Generalized symmetry in the vanishing of Ext

Foundational work by Avramov and Buchweitz, as well as by Huneke and Jorgensen, established symmetry results for the vanishing of Ext over local complete intersections and AB-rings, respectively. Later, several authors studied symmetry in the vanishing of Ext over local Gorenstein rings under additional homological assumptions on the modules involved. In this work, we provide a generalized symmetry in the vanishing of Ext for homologically finite complexes over a Noetherian ring with a dualizing complex, under additional homological conditions on the complexes involved. The results in this paper extend and unify several symmetry results on the vanishing of Ext previously proved over local Gorenstein rings and show that the dualizing complex has been hidden in these results.

math.AC↗

arXiv2026

Chouinard's formula for $C$-quasi-injective dimension

The $C$-quasi-injective dimension is a recently introduced homological invariant that unifies and extends the notions of quasi-injective dimension and of injective dimension with respect to a semidualizing module, previously studied by Gheibi and by Takahashi and White, respectively. In the main results of this paper, we provide extensions of the Bass' formula and a version of the Chouinard's formula for modules of finite $C$-quasi-injective dimension over an arbitatry ring.

math.AC↗

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

Finiteness of complete intersection dimensions of RHom complexes and Ext modules

In this paper, we explore the implications of the finiteness of complete intersection dimensions for RHom complexes and Ext modules. We prove various stability results and criteria for detecting finite complete intersection homological dimension of complexes and modules. In addition, we introduce and explore the concept of CI-perfect modules. We also study the vanishing of Ext when certain Hom module have finite complete intersection homological dimension. In this direction, we improve a result by Ghosh and Samanta, prove the Auslander-Reiten conjecture for finitely generated modules $M$ over a Noetherian local ring $R$ such that $\operatorname{Hom}_R(M,R)$ or $\operatorname{Hom}_R(M,M)$ has finite complete intersection injective dimension, and provide Gorenstein criteria.

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

Transformer-based Ranking Approaches for Keyword Queries over Relational Databases

Relational Keyword Search (R-KwS) systems enable naive/informal users to explore and retrieve information from relational databases without requiring schema knowledge or query-language proficiency. Although numerous R-KwS methods have been proposed, most still focus on queries referring only to attribute values or primarily address performance enhancements, providing limited support for queries referencing schema elements. We previously introduced Lathe, a system that accommodates schema-based keyword queries and employs an eager CJN evaluation strategy to filter out spurious Candidate Joining Networks (CJNs). However, Lathe still faces challenges in accurately ranking CJNs when queries are ambiguous. In this work, we propose a new transformer-based ranking approach that provides a more context-aware evaluation of Query Matches (QMs) and CJNs. Our solution introduces a linearization process to convert relational structures into textual sequences suitable for transformer models. It also includes a data augmentation strategy aimed at handling diverse and ambiguous queries more effectively. Experimental results, comparing our transformer-based ranking to Lathe's original Bayesian-based method, show significant improvements in recall and R@k, demonstrating the effectiveness of our neural approach in delivering the most relevant query results.

cs.DB↗

arXiv2024

3D Modelling to Address Pandemic Challenges: A Project-Based Learning Methodology

The use of 3D modelling in medical education is a revolutionary tool during the learning process. In fact, this type of technology enables a more interactive teaching approach, making information retention more effective and enhancing students' understanding. 3D modelling allows for the creation of precise representations of the human body, as well as interaction with three-dimensional models, giving students a better spatial understanding of the different organs and systems and enabling simulations of surgical and technical procedures. This way, medical education is enriched with a more realistic and safe educational experience. The goal is to understand whether, when students and schools are challenged, they play an important role in addressing health issues in their community. School-led projects are directed towards educational scenarios that emphasize STEM education, tackling relevant public health problems through open-school initiatives. By implementing an educational scenario focused on 3D modelling and leveraging technology, we aim to raise community awareness on public health issues.

cs.MM↗

arXiv2023

Challenges and Trends in User Trust Discourse in AI

The Internet revolution in 1990, followed by the data-driven and information revolution, has transformed the world as we know it. Nowadays, what seam to be 10 to 20 years ago, a science fiction idea (i.e., machines dominating the world) is seen as possible. This revolution also brought a need for new regulatory practices where user trust and artificial Intelligence (AI) discourse has a central role. This work aims to clarify some misconceptions about user trust in AI discourse and fight the tendency to design vulnerable interactions that lead to further breaches of trust, both real and perceived. Findings illustrate the lack of clarity in understanding user trust and its effects on computer science, especially in measuring user trust characteristics. It argues for clarifying those notions to avoid possible trust gaps and misinterpretations in AI adoption and appropriation.

cs.HC↗

arXiv2023

Human-centered trust framework: An HCI perspective

The rationale of this work is based on the current user trust discourse of Artificial Intelligence (AI). We aim to produce novel HCI approaches that use trust as a facilitator for the uptake (or appropriation) of current technologies. We propose a framework (HCTFrame) to guide non-experts to unlock the full potential of user trust in AI design. Results derived from a data triangulation of findings from three literature reviews demystify some misconceptions of user trust in computer science and AI discourse, and three case studies are conducted to assess the effectiveness of a psychometric scale in mapping potential users' trust breakdowns and concerns. This work primarily contributes to the fight against the tendency to design technical-centered vulnerable interactions, which can eventually lead to additional real and perceived breaches of trust. The proposed framework can be used to guide system designers on how to map and define user trust and the socioethical and organisational needs and characteristics of AI system design. It can also guide AI system designers on how to develop a prototype and operationalise a solution that meets user trust requirements. The article ends by providing some user research tools that can be employed to measure users' trust intentions and behaviours towards a proposed solution.

cs.HC↗

arXiv2022

Supporting Schema References in Keyword Queries over Relational Databases

Relational Keyword Search (R-KwS) systems enable naive/informal users to explore and retrieve information from relational databases without knowing schema details or query languages. These systems take the keywords from the input query, locate the elements of the target database that correspond to these keywords, and look for ways to "connect" these elements using information on referential integrity constraints, i.e., key/foreign key pairs. Although several such systems have been proposed in the literature, most of them only support queries whose keywords refer to the contents of the target database and just very few support queries in which keywords refer to elements of the database schema. This paper proposes LATHE, a novel R-KwS designed to support such queries. To this end, in our work, we first generalize the well-known concepts of Query Matches (QMs) and Candidate Joining Networks (CJNs) to handle keywords referring to schema elements and propose new algorithms to generate them. Then, we introduce an approach to automatically select the CJNs that are more likely to represent the user intent when issuing a keyword query. This approach includes two major innovations: a ranking algorithm for selecting better QMs, yielding the generation of fewer but better CJNs, and an eager evaluation strategy for pruning void useless CJNs. We present a comprehensive set of experiments performed with query sets and datasets previously used in experiments with state-of-the-art R-KwS systems and methods. Our results indicate that LATHE can handle a wider variety of keyword queries while remaining highly effective, even for large databases with intricate schemas.

cs.DB↗