Quasi-injective dimension

TL;DR

This paper introduces quasi-injective dimension as a generalization of injective dimension, extending known results about injective and Gorenstein-injective dimensions to this new context.

Contribution

It defines quasi-injective dimension and establishes key properties and characterizations related to Cohen-Macaulay and Gorenstein rings, generalizing classical homological dimensions.

Findings

Finite quasi-injective dimension equals the depth of the ring.

Existence of modules with certain quasi-injective and projective dimensions characterizes Gorenstein rings.

Over Gorenstein rings, quasi-injective and quasi-projective dimensions are equivalent.

Abstract

Following our previous work about quasi-projective dimension, in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module $M$ over a local ring $R$ is finite, then it is equal to the depth of $R$. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then $R$ is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then $R$ is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.