Notes on modules of finite injective dimension

TL;DR

This paper explores the properties and implications of finitely generated modules with finite injective dimension, revealing how they impose structural conditions on the ambient ring.

Contribution

It extends understanding of modules with finite injective dimension, linking their properties to ring conditions like reducedness, normality, and Gorensteinness, and examines related module behaviors.

Findings

Finite injective dimension modules imply the ring is reduced and normal.

Reflexivity and torsionlessness of such modules force the ring to be quasi-normal.

High syzygies of the residue field surjecting onto modules of finite injective dimension reveal ring properties.

Abstract

Motivated by the Bass conjecture, we study finitely generated modules of finite injective dimension and the additional constraints they impose on the ambient ring. Beyond the Cohen--Macaulay property, the existence of such modules forces further conditions on the ring, including reducedness, normality, being an integral domain, and various conditions such as being complete intersection or Gorenstein. We also address the reflexivity and torsionlessness of modules of finite injective dimension, showing that these properties force 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 a high syzygy of the residue field $k$ surjects onto a nonzero module of finite injective dimension.