Categorical properties and homological conjectures for bounded extensions of algebras

Abstract

An extension $B\subset A$ of finite dimensional algebras is bounded if the $B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is finite and $\mathrm{Tor}_i^B(A/B, (A/B)^{\otimes_B j})=0$ for all $i, j\geq 1$. We show that for a bounded extension $B\subset A$, the algebras $A$ and $B$ are singularly equivalent of Morita type with level. Additively, under some conditions, their stable categories of Gorenstein projective modules and Gorenstein defect categories are equivalent, respectively. Some homological conjectures are also investigated for bounded extensions, including Auslander-Reiten conjecture, finististic dimension conjecture, Fg condition, Han's conjecture, and Keller's conjecture. Applications to trivial extensions and triangular matrix algebras are given. In course of proofs, we give some handy criteria for a functor between module categories induces triangle functors between stable categories of Gorenstein projective modules and Gorenstein defect categories, which generalise some known criteria and hence, might be of independent interest.