On Extension modules of finite homological dimension
Rafael Holanda, Victor H. Jorge-Pérez, Victor D. Mendoza-Rubio
TL;DR
The paper develops a derived-category framework to study when Ext modules and their homological dimensions are finite, focusing on projective, injective, and Gorenstein analogues. It furnishes criteria that transfer finiteness from Ext-modules to the ambient modules or complexes, and proves dualities and Gorenstein criteria that generalize Foxby-type results, including in Cohen–Macaulay contexts with canonical or semidualizing structures. Deficiency modules are analyzed to connect their finite dimensions with finiteness properties of the ambient module, yielding new Gorenstein-type conclusions. Over Cohen–Macaulay rings, the work establishes precise equivalences between finiteness of Ext-dimensions and finiteness of the corresponding dimensions for the arguments, along with natural isomorphisms involving the canonical module. The authors conclude with questions aimed at extending these results beyond the Cohen–Macaulay setting and at refining dualizing-complex formulations, highlighting the broad impact on homological dimensions and ring-theoretic characterizations.
Abstract
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. Our results include the consequences of the finiteness of the Gorenstein (injective) dimension of the deficiency modules as well as a duality for certain Ext modules, all of finite (Gorenstein) projective dimension.