Criteria for finite injective dimension of modules over a local ring
Shinnosuke Kosaka, Yuki Mifune, Kenta Shimizu
TL;DR
The paper addresses when a finitely generated module C over a Noetherian local ring has finite injective dimension by linking this finiteness to the existence of a Cohen–Macaulay module M with dimension s that satisfies a numerical inequality involving multiplicity and type, together with the vanishing of finitely many Ext modules. The authors establish a main theorem showing that such data force the ring to be Cohen–Macaulay and C to be maximal Cohen–Macaulay with finite injective dimension, with the added feature that every CM module of dimension s satisfies the same conditions. The result recovers Rahmani–Taherizadeh and yields multiple corollaries, including criteria for a ring to be Gorenstein and for CM modules with rank or endomorphism conditions to have finite injective dimension. The proof hinges on multiplicity-length relations, regular sequences that align with Ext-vanishing, and a chain of Ext computations that culminate in the vanishing of $\Ext_R^{r+1}(k,C)$. Overall, the work provides concrete, computable criteria linking numerical invariants to homological finiteness properties in local algebra.
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.