Homological conjectures over idealization rings and test modules for finiteness of homological dimensions

• Ano de defesa: 2025
• Autor(a) principal: Nascimento, Igor Jose do
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Anéis de idealização; Betti numbers; Conjecturas homológicas; Dimensões homológicas; Homological conjectures; Homological dimensions; Idealization rings; Módulos de teste; Números de Betti; Test modules
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-30042025-113735/

In this thesis, our main focus is to provide positive answers to some long-standing homological conjectures over the idealization ring R ⋉ M, including the Auslander-Reiten, Buchsbaum- Eisenbud-Horrocks, Total-Rank, Jorgensen-Leuschke, and Zariski-Lipman conjectures. Addi- tionally, we obtain a characterization of the Betti numbers of an R-module over the idealization ring R ⋉ M. We also study the class of test modules for projectivity and injectivity, investigating when S is a test R-module, provided that R → S is a finite local homomorphism of rings, and explore test modules with finite homological dimensions.