Remarks on Auslander's depth 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 $$\mathrm{depth}M+\mathrm{depth}N=\mathrm{depth}R+\mathrm{depth}({\mathrm{Tor}}_q^R(M,N))-q,$$ where $q:=\sup \{i\ge 0\mid {\mathrm{Tor}}_i^R(M,N)\ne 0\}$. 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 $\mathrm{depth}({\mathrm{Tor}}_q^R(M,N))\le 1$. 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.

Some minor corrections from V3. Accepted for publication in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas