In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question of whether such a depth inequality can be extended for $n$-Tor-rigid modules, and obtain an affirmative answer for $2$-Tor-rigid modules that are generically free. Furthermore, in the appendix, we use Dao's eta function and determine new classes of Tor-rigid modules over hypersurfaces that are quotient of unramified regular local rings.
Celikbas was partly supported by WVU Mathematics Excellence and Research Funds (MERF). Matsui was partly supported by JSPS Grant-in-Aid for JSPS Fellows 19J00158.
The authors are grateful to Shunsuke Takagi for his help and explaining the arguments of 3.1 and 3.2 to them. The authors thank to Ehsan Tavanfar for comments and discussions related to a previous version of the manuscript. The authors are grateful to W. Frank Moore for his help and for showing them Macaulay2 codes to find the presentations of the modules $M$ and $N$ in Example B.1.
The authors are also grateful to the referee for their valuable help and suggestions that significantly improved the paper; Examples 1.2 and B.1 are suggested to the authors by the referee.
"An Extension of a Depth Inequality of Auslander." Taiwanese J. Math. 26 (5) 903 - 926, October, 2022. https://doi.org/10.11650/tjm/220501