Criteria for finite injective dimension of modules over a local ring

2025/04/10 by Kosaka, Shinnosuke, Mifune, Yuki, Shimizu, Kenta
#13D05 #13D07 #13H10 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.07536

Abstract

Let R be a commutative Noetherian local ring. We prove that the finiteness of the injective dimension of a finitely generated R-module C is determined by the existence of a Cohen--Macaulay module M that satisfies an inequality concerning multiplicity and type, together with the vanishing of finitely many Ext modules. As applications, we recover a result of Rahmani and Taherizadeh and provide sufficient conditions for a finitely generated R-module to have finite injective dimension.

Related

• The existence of finitely generated modules of finite Gorenstein injective dimension
• A characterization of modules locally of finite injective dimension
• Finiteness of homological dimensions of Ext modules
• On the existence of certain modules of finite Gorenstein homological dimensions
• On Ext-finite modules, quasi-injective dimension and width of modules