Projective Modules Research Articles

Gorenstein cohomological dimension and stable categories for groups

First we study the Gorenstein cohomological dimension Gcd R G of groups G over coefficient rings R, under changes of groups and rings; a characterization for finiteness of Gcd R G is given. Some results in literature obtained over the coefficient ring Z or rings of finite global dimension are generalized to more general cases. Moreover, we establish a model structure on the weakly idempotent complete exact category F ib consisting of fibrant RG-modules, and show that the homotopy category Ho ( F ib ) is triangle equivalent to both the stable category C of ¯ ( RG ) of Benson’s cofibrant modules, and the stable module category StMod ( RG ) . The relation between cofibrant modules and Gorenstein projective modules is discussed, and we show that under some conditions such that Gcd R G < ∞ , Ho ( F ib ) is equivalent to the stable category of Gorenstein projective RG-modules, the singularity category, and the homotopy category of totally acyclic complexes of projective RG-modules.

Pelatihan Penyusunan RPP dan Modul Proyek P5 Digital Terintegrasi Kurikulum Merdeka: Upaya Mendukung Sekolah Adiwiyata Mandiri

Sustainable education is one of the main pillars in facing global challenges such as environmental degradation and climate change. The Adiwiyata program, initiated by the Indonesian government, aims to integrate environmental values into the school curriculum, helping students not only understand environmental concepts theoretically, but also apply them in everyday life. SMPN 1 Kolaka, as a community service partner, is working to achieve the Adiwiyata Mandiri award and needs strengthening in the implementation of the Adiwiyata concept, especially in the preparation of learning tools based on environmental values. The training program includes five main stages, namely the planning, implementation, technology application, mentoring, evaluation, and program sustainability stages. In the training process, teachers are invited to be more skilled in preparing Learning Implementation Plans (RPP) and Project Modules for Strengthening the Pancasila Student Profile (P5) which are integrated with environmental values. The results of this training showed a significant improvement in teachers' abilities, as evidenced by the difference in pretest and posttest scores by 20%. In addition, feedback from participants showed that this activity received positive appreciation, with a satisfaction rate of 90%. The success of this training not only improves teachers' competence in developing environment-based learning, but also strengthens teachers' capacity in developing environment-based learning.

Development of an Artificial Intelligence-Based Mobile Application Platform: Evaluation of Prospective Science Teachers’ Project on Creating Virtual Plant Collections in terms of Plant Blindness and Knowledge

This research aims to investigate the use of an artificial intelligence-based mobile application with plants and plant identification techniques (AImPLANT) with prospective science teachers in outdoor activities. The goals of the study are: i) to develop an AI-based mobile application from scratch based customized for education ii) to enhance knowledge about common plants iii) to increase awareness of plants, animals, and the natural environment, offering solutions to reduce plant blindness iv) to support outdoor activities to observe, identify and taxonomically determine specific characteristics of plants (endemic, economic, health, aesthetic etc.) v) guide students creating an individual virtual herbarium. The research will achieve the expected goals by coding an original “AI-Based Mobile Application” using ChatGPT and PlantNet APIs through technologies like Flutter, Firebase, Firestore, involving approximately 3200 lines of code. The mobile application includes components such as an “Plant collection (herbarium) project module”, “AI-based chatbot", "AI-based plant identification module", "AI-based student assessment module", and a "Student assistance section". The application provides guidance in creating virtual plant collections (herbariums). Additionally, aids by the “help module” and “chatbot”. Furthermore, an AI-based “self-assessment module” evaluates like a teacher based on the answers. The research question and sub problems were based on prospective science teachers' knowledge about plant identification and taxonomy, plant blindness, and opinions on studying with artificial intelligence outdoors with common plant species. The research, once again showed that artificial intelligence facilitates teaching biology, increases academic success, has positive contributions to eliminating plant blindness, and reduces teacher candidates' concerns about artificial intelligence and affects their opinions positively.

Characteristic modules and Gorenstein (co-)homological dimension of groups

In this paper, we examine the Gorenstein dimension of modules over the group algebra kG of a group G with coefficients in a commutative ring k. As a Gorenstein analogue of the classical case, we bound this dimension in terms of the Gorenstein dimension of the underlying k-module and the Gorenstein dimension of G over k. Our method is based on the notion of a characteristic module for G, introduced by the second author, and uses the stability properties of the Gorenstein categories. We also examine the class of hierarchically decomposable groups defined by Kropholler and use the module of bounded Z-valued functions on such a group G to characterize the Gorenstein flat ZG-modules, in terms of flat modules, and the Gorenstein injective ZG-modules, in terms of injective modules (by complete analogy with the characterization of Gorenstein projective ZG-modules, in terms of projective modules, obtained by Dembegioti and the second author). It follows that, for a group G in Kropholler's class, (a) any Gorenstein projective ZG-module is Gorenstein flat and (b) a ZG-module is Gorenstein flat if its Pontryagin dual module is Gorenstein injective.

Implementasi Kurikulum Merdeka di Lombok NTB: Peluang, Tantangan dan Strategi

Curriculum changes are an integral part of efforts to improve the quality of education. The "Kurikulum Merdeka" (Independent Curriculum) is the latest innovation implemented by the Ministry of Education, Culture, Research, and Technology (Kemendikbudristek). This curriculum is designed to provide deeper learning experiences and focuses on strengthening students' competencies and character development. This study aims to analyze the implementation of the Kurikulum Merdeka, identify opportunities and challenges, and outline strategic steps to leverage opportunities and overcome challenges in implementing the Kurikulum Merdeka at Elementary School (SDN) 1 Sandik and Elementary School (SDN) 1 Meninting. This research employes a descriptive qualitative approach, using a case study methodology. Data collection procedures include interviews, observations, and documentation. The data analysis technique utilized was qualitative analysis. The results of this study indicate that: 1) the implementation of the Kurikulum Merdeka in both schools began with understanding government regulations, followed by preparing documents such as the Operational Curriculum of Educational Units, creating teaching modules, project modules, and assessments. 2) The opportunities available to both schools include policies from the central and local governments regarding funding for the Kurikulum Merdeka, and support from school principals, teachers, and parents to back various programs. However, challenges arise from human resources (educator competencies). 3) The strategic steps taken by both schools to leverage opportunities and address challenges include maximizing the school's learning community to discuss the Kurikulum Merdeka, organizing workshops, participating in webinars, and undergoing independent training via the "Platform Merdeka Mengajar" application.

Triviality criteria for unbounded complexes

We study properties of modules that appear as syzygies of acyclic complexes of projective, injective, flat or flat-cotorsion modules and obtain criteria for these complexes to be contractible or totally acyclic. Our results illustrate the importance of strongly fp-injective modules in the study of these properties. We examine implications of the existence of complete resolutions (in a certain weak sense) and the finiteness of the Gorenstein projective dimension of pure-projective modules. We also use the orthogonality in the homotopy category between complexes of flat modules and pure acyclic complexes of cotorsion modules, in order to study the syzygies of acyclic complexes of flat modules. Finally, we present some applications of our results to group rings, regarding complete resolutions and cohomological periodicity.

Peran Strategis Kepala Sekolah dalam Perencanaan dan Implementasi Projek Penguatan Profil Pelajar Pancasila

The values of Pancasila in the project include faith, diversity, mutual cooperation, independence, critical thinking, and creativity, with the school principal playing an essential role in guiding and managing the learning process. This study focuses on the principal's key role in planning and implementing the Pancasila Student Profile Strengthening Project at State Senior High School 1 Sawahlunto. The research employs a descriptive qualitative approach with a case study method. Data was collected through in-depth interviews, direct observation, and document analysis. Data processing involved condensation, presentation, and drawing conclusions. To ensure accuracy and reliability of the information, source triangulation was conducted. The findings indicate that the planning phase includes forming a facilitator team, identifying school readiness, organizing dimensions, themes, and time allocation for the Pancasila student profile strengthening project, creating project modules, and designing reporting strategies. In the implementation phase, each class follows the learning activities based on the assigned themes and projects, with the principal collaborating with the community, local organizations, and industries to support learning. This activity positively impacts the development of students' character, aligned with Pancasila values.

Examples of Tilting-Discrete Self-Injective Algebras Which Are Not Silting-Discrete

In this paper, we introduce the notion of \nu -stable silting-discrete algebras, which unify silting-discrete algebras and tilting-discrete self-injective algebras, where \nu is a triangle auto-equivalence of the bounded homotopy category of finitely generated projective modules. Moreover, we give an example of tilting-discrete self-injective algebras which are not silting-discrete.

Pure Rickart Modules

This paper focuses on the study of pure Rickart modules (or PR-modules for short), which are a class of modules over a commutative ring with identity. The main objective is to investigate the properties and characterizations of these modules, as well as their relationship with other classes of modules such as free, projective, and flat modules. This paper also explores the connections between PR-modules and various algebraic structures such as rings. The results obtained in this study provide a deeper understanding of the structure and behavior of PR-modules, which can have important applications in algebraic geometry, representation theory, and other areas of mathematics. Some results about PR-module have been investigated in this paper, for example we demonstrate a module is PR-module if and only if for every , is pure submodule of (or for short). Also, some kind of generalization of these rings have been constructed and demonstrated in term of PR-modules.

Prompt-Based Modality Bridging for Unified Text-to-Face Generation and Manipulation

Text-driven face image generation and manipulation are significant tasks. However, such tasks are quite challenging due to the gap between text and image modalities. It is difficult to utilize current methods to deal with both of the two problems because these methods are usually designed for one certain task, limiting their application in real scenarios. To address the two problems in one framework, we propose a U nified P rompt-based C ross- M odal Frame work (UPCM-Frame) to bridge the gap between the text modality and image modality with CLIP and StyleGAN, which are two large-scale pre-trained models. The proposed framework is combined with two main modules: a Text Embedding-to-Image Embedding projection module based on a special prompt embedding pair, and a projection module mapping Image Embeddings to semantically aligned StyleGAN Embeddings which can be used in both image generation and manipulation. The proposed framework is able to handle complicated descriptions and generate impressive results with high quality due to the utilization of large-scale pre-trained models. In order to demonstrate the effectiveness of the proposed method in the two tasks, we conduct experiments to evaluate the results of our method both quantitatively and qualitatively.

A generalized Auslander–Reiten duality and relatively projective modules

Given a field k and a subgroup H of a finite group G, we define a duality for quotients of homomorphisms of kG-modules modulo kG-homomorphisms that factors through a relatively H-projective kG-module. This generalizes the classical Auslander–Reiten duality for group algebras. As an application, we use the generalized Auslander–Reiten duality to give a new way for constructing almost split sequences.

Big pure projective modules over commutative noetherian rings: Comparison with the completion

Abstract A module over a ring R is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module M, we consider Add ⁡ ( M ) {\operatorname{Add}(M)} , which consists of direct summands of direct sums of copies of M. We are primarily interested in the case where R is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, Add ⁡ ( M ) {\operatorname{Add}(M)} can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of M and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid V * ⁢ ( M ) {V^{*}(M)} , of isomorphism classes of countably generated modules in Add ⁡ ( M ) {\operatorname{Add}(M)} with the addition induced by the direct sum. We show that V * ⁢ ( M ) {V^{*}(M)} is a submonoid of V * ⁢ ( M ⊗ R R ^ ) {V^{*}(M\otimes_{R}\widehat{R})} , this allows us to make computations with examples and to prove some realization results.

The stable category of monomorphisms between (Gorenstein) projective modules with applications

Abstract Let ( S , 𝔫 ) {(S,{\mathfrak{n}})} be a commutative noetherian local ring and let ω ∈ 𝔫 {\omega\in{\mathfrak{n}}} be non-zerodivisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective S-modules, such that their cokernels are annihilated by ω. It is shown that these categories, which will be denoted by 𝖬𝗈𝗇 ⁢ ( ω , 𝒫 ) {{\mathsf{Mon}}(\omega,\mathcal{P})} and 𝖬𝗈𝗇 ⁢ ( ω , 𝒢 ) {{\mathsf{Mon}}(\omega,\mathcal{G})} , are both Frobenius categories with the same projective objects. It is also proved that the stable category 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} is triangle equivalent to the category of D-branes of type B, 𝖣𝖡 ⁢ ( ω ) {\mathsf{DB}(\omega)} , which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} and 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒢 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{G})} are closely related to the singularity category of the factor ring R = S / ( ω ) {R=S/({\omega)}} . Precisely, there is a fully faithful triangle functor from the stable category 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒢 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{G})} to 𝖣 𝗌𝗀 ⁡ ( R ) {\operatorname{\mathsf{D_{sg}}}(R)} , which is dense if and only if R (and so S) are Gorenstein rings. Particularly, it is proved that the density of the restriction of this functor to 𝖬𝗈𝗇 ¯ ⁢ ( ω , 𝒫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} , guarantees the regularity of the ring S.

Unsupervised Deep Tensor Network for Hyperspectral-Multispectral Image Fusion.

Fusing low-resolution (LR) hyperspectral images (HSIs) with high-resolution (HR) multispectral images (MSIs) is a significant technology to enhance the resolution of HSIs. Despite the encouraging results from deep learning (DL) in HSI-MSI fusion, there are still some issues. First, the HSI is a multidimensional signal, and the representability of current DL networks for multidimensional features has not been thoroughly investigated. Second, most DL HSI-MSI fusion networks need HR HSI ground truth for training, but it is often unavailable in reality. In this study, we integrate tensor theory with DL and propose an unsupervised deep tensor network (UDTN) for HSI-MSI fusion. We first propose a tensor filtering layer prototype and further build a coupled tensor filtering module. It jointly represents the LR HSI and HR MSI as several features revealing the principal components of spectral and spatial modes and a sharing code tensor describing the interaction among different modes. Specifically, the features on different modes are represented by the learnable filters of tensor filtering layers, the sharing code tensor is learned by a projection module, in which a co-attention is proposed to encode the LR HSI and HR MSI and then project them onto the sharing code tensor. The coupled tensor filtering module and projection module are jointly trained from the LR HSI and HR MSI in an unsupervised and end-to-end way. The latent HR HSI is inferred with the sharing code tensor, the features on spatial modes of HR MSIs, and the spectral mode of LR HSIs. Experiments on simulated and real remote-sensing datasets demonstrate the effectiveness of the proposed method.

Monocular BEV Perception of Road Scenes via Front-to-Top View Projection.

HD map reconstruction is crucial for autonomous driving. LiDAR-based methods are limited due to expensive sensors and time-consuming computation. Camera-based methods usually need to perform road segmentation and view transformation separately, which often causes distortion and missing content. To push the limits of the technology, we present a novel framework that reconstructs a local map formed by road layout and vehicle occupancy in the bird's-eye view given a front-view monocular image only. We propose a front-to-top view projection (FTVP) module, which takes the constraint of cycle consistency between views into account and makes full use of their correlation to strengthen the view transformation and scene understanding. In addition, we apply multi-scale FTVP modules to propagate the rich spatial information of low-level features to mitigate spatial deviation of the predicted object location. Experiments on public benchmarks show that our method achieves various tasks on road layout estimation, vehicle occupancy estimation, and multi-class semantic estimation, at a performance level comparable to the state-of-the-arts, while maintaining superior efficiency.

Invariants and constructions of separable equivalences

In this paper, we first establish relationships between Gorenstein projective modules linked by the separable equivalence of rings, and prove that Gorenstein, CM-finite and CM-free algebras are invariant under separable equivalences. Secondly, we provide a new method to produce separable equivalences. As applications, the following results are obtained. Let Λ and Γ be Artin algebras such that Λ is separably equivalent to Γ. (1) For representation-finite algebras Λ and Γ, their Auslander algebras are separably equivalent; (2) For CM-finite algebras Λ and Γ, the endomorphism algebras of their representative generators are separably equivalent. Finally, we discuss when tilted algebras are invariant under separable equivalences, and give an example to illustrate it.

Auslander‐type conditions and weakly Gorenstein algebras

AbstractLet be an Artin algebra. Under certain Auslander‐type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander‐type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if is left quasi‐Auslander, then is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if satisfies the Auslander condition, then is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander–Reiten's conjecture, which states that is Gorenstein if satisfies the Auslander condition.

Simultaneous Dynamic Display of Meta‐Hologram and Meta‐Nanoprinting with High Frame Rate

AbstractDynamic holography can reconstruct realistic and highly interactive 3D scenes, offering a promising way for future display. Emerging metasurface with small pixel size and powerful light modulation capability demonstrates performance superior to traditional spatial light modulators, but dynamic meta‐holography with high information capacity and high frame rate remains a challenge. This paper presents a design scheme combining a preset polarization‐dependent metasurface with a high‐speed projection module to realize smooth dynamic display at multiple polarization states. With the projection module, the system can achieve dynamic meta‐holography at two orthogonal circular polarization states with 237 and 247 different holographic frames respectively, and grayscale meta‐nanoprinting at a linear polarization state with three different patterns. At the same time, the polarization multiplexed dynamic display scheme enables independent displays at three distinct polarization states with an extremely high frame rate (9523 frames per second). This advancement holds promise for applications in dynamic holographic display, optical encryption, virtual reality, and information storage.

On the Auslander–Bridger–Yoshino theory for complexes of finitely generated projective modules

Let R be a two-sided noetherian ring. Auslander and Bridger developed a theory of projective stabilization of the category of finitely generated R-modules, which is called the stable module theory. Recently, Yoshino established a stable “complex” theory, i.e., a theory of a certain stabilization of the homotopy category of complexes of finitely generated projective R-modules. We introduce higher versions of several notions introduced by Yoshino, such as ⁎torsionfreeness and ⁎reflexivity. Also, we prove the Auslander–Bridger approximation theorem for complexes of finitely generated projective R-modules.

Exploring the State of Mental Well-Being Among B40 Women

The solution projects under The All-Party Parliamentary Group Malaysia on the Sustainable Development Goals (APPGM-SDG) have played a significant role in empowering the local communities, particularly women through Income Generation (IG) and Community Learning Center (CLC) projects. The initiative consists of twelve modules on entrepreneurship skills and handholding, with a minimum of 10 to 25 beneficiaries from the B40 category who are mentored by the Solution Provider (SP) throughout the following three to four months. Each project was awarded a RM40,000 grant to provide the beneficiaries with skills and knowledge that will allow them to be independent entrepreneurs. It was discovered through the four case studies that adhered to SDG 1 (No Poverty) and SDG 3 (Good Health and Well-Being) that certain female beneficiaries were experiencing such high levels of life pressure that negatively impacted their mental health due to a lack of necessary coping strategies to manage the stress and uncomfortable emotions. The reasons vary, ranging from personal issues and societal pressures to the inherent challenges of entrepreneurship. The finding shows that the project’s method of group activities or social groups enhances the beneficiaries’ mental health by providing them with encouragement, hope, and drive to succeed in life. It is also highly recommended to provide mental health awareness as one of the compulsory modules in the solution projects to equip women with appropriate coping techniques for overall success.