A generalized depth formula for modules of finite quasi-projective dimension

Abstract: Recently, Gheibi, Jorgensen, and Takahashi introduced a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. They proved that the depth formula holds for two finitely gene-rated Tor-independent modules over a Noetherian local ring when one of the modules has finite quasi-projective dimension. In this paper, we extend this result by pro-ving that for finitely generated modules ( M ) and ( N ) over a Noetherian local ring ( R ) with ( \operatorname{qpd}_R M < \infty ), the equality ( \operatorname{depth} N = \operatorname{depth}(\operatorname{Tor}_q^R(M,N)) + \operatorname{qpd}_R M - q ) holds, where ( q := \sup{ i \geq 0 : \operatorname{Tor}_i^R(M,N) \neq 0 } ), provided that ( q < \infty ), and either ( q = 0 ) or ( \operatorname{depth}(\operatorname{Tor}_q^R(M,N)) \leq 1 ). This outcome generalizes a celebrated theorem by Auslander and allows us to derive new consequences and applications, for instance, we recover an important theorem of Araya and Yoshino.