Gorenstein injective dimension and a generalization of Ischebeck Formula

TL;DR

This paper generalizes the Ischebeck Formula to relate the depth of modules with finite injective and Gorenstein injective dimensions over a local ring, expanding understanding of homological dimensions in commutative algebra.

Contribution

It introduces a generalized Ischebeck Formula connecting depth, Ext, and Gorenstein injective dimension for modules over local rings.

Findings

Generalized Ischebeck Formula proved

Established relation between depth and Ext for specific modules

Enhanced understanding of Gorenstein homological dimensions

Abstract

Let $(R,\frak m)$ be a commutative Noetherian local ring and let $M$ and $N$ be finitely generated $R$-modules of finite injective dimension and finite Gorenstein injective dimension, respectively. In this paper we prove a generalization of Ischebeck Formula, that is $\depth_RM+\sup\{i| {0.1cm}\Ext_R^i(M,N)\neq 0\}=\depth R.$