Notes on modules of finite injective dimension

Abstract

Motivated by Bass' conjecture, we study finitely generated modules of finite injective dimension and the additional constraints they impose on the ambient ring. Beyond ensuring the Cohen--Macaulay property, the presence of such modules enforces further conditions on the ring, including reducedness, normality, being an integral domain, and various singularity conditions such as complete intersection, Gorenstein, and beyond. This continues to detect non-singularity as well. We also address the reflexivity (and also torsionlessness) of modules with finite injective dimension and show that this forces the ring to be quasi-normal. In the same vein, we investigate the injective dimension of tensor products and endomorphism rings. Finally, we study the behavior of R when high syzygies of ksurject onto a non-zero R-module of finite injective dimension.