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 whether such can be extended $n$-Tor-rigid modules, and obtain an affirmative answer $2$-Tor-rigid that are generically free. Furthermore, in appendix, use Dao's eta function determine new classes hypersurfaces quotient un...

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally, variants of the homological theorems are shown to hold in equal characteristic. This theory is then applied to Noetherian local rings in order to get: (i) over...