On a relative dependency formula

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 [Formula: see text], a notion related to the vanishing of relative homology of finitely generated modules [Formula: see text] and [Formula: see text] over a local ring [Formula: see text], under specific assumptions.