On a relative dependency formula

TL;DR

This paper investigates the relative dependency formula, demonstrating that the previously known inequality can be strict and establishing a new formula with bounds on related homological measures.

Contribution

It proves that the relative dependency inequality can be strict and introduces a new relative dependency formula with bounds on s(M,N).

Findings

The inequality for the relative dependency formula can be strict.

A new relative dependency formula is established.

Bounds on the homological measure s(M,N) are obtained.

Abstract

Celikbas, Liang and Sadeghi established a one-sided inequality for the relative version of Jorgensen's dependency formula and questioned whether it would be an equality. In this paper, we show that the inequality can be indeed strict, and prove a relative dependency formula. Along the way, we obtain some bounds on s(M,N), a notion related to the vanishing of relative homology of finitely generated modules M and N over a local ring R, under specific assumptions.