Remarks on Auslander's depth formula for quasi-projective dimension

TL;DR

This paper extends Auslander's depth formula to modules with finite quasi-projective dimension under broader conditions, including cases where the Tor module depth is at most one.

Contribution

It proves the depth formula holds for modules with finite quasi-projective dimension when certain Tor module depth conditions are met, generalizing prior results.

Findings

Depth formula holds for modules with finite quasi-projective dimension and Tor depth ≤ 1.

Recovers and extends known theorems in the setting of semidualizing modules.

Provides applications including an improved dependency formula for quasi-projective dimension.

Abstract

For nonzero finitely generated $R$-modules $M$ and $N$ over a Noetherian local ring $R$, Auslander's depth formula is the equality $$ \operatorname{depth} M + \operatorname{depth} N = \operatorname{depth} R + \operatorname{depth}(\operatorname{Tor}_q^R(M,N)) - q, $$ where $ q := \sup\{ i \ge 0 \mid \operatorname{Tor}_i^R(M,N) \neq 0 \}$. Gheibi, Jorgensen, and Takahashi introduced a homological invariant called quasi-projective dimension, which generalizes projective dimension, and proved that Auslander's depth formula holds when $M$ has finite quasi-projective dimension and $q=0$. In this paper, we prove that the formula still holds when $M$ has finite quasi-projective dimension, $q<\infty$ and $\operatorname{depth}(\operatorname{Tor}_q^R(M,N)) \leq 1$. We present several applications of this result; in particular, we recover a theorem of Araya and Yoshino, extend our result to the setting of semidualizing modules, and in this framework derive an improved version of the dependency formula for quasi-projective dimension with respect to a semidualizing module recently obtained by Dey, Ferraro, and Gheibi.