Homological conjectures over idealization rings and test modules for finiteness of homological dimensions - Nascimento, Igor Jose do (2025)
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.