Categorical properties and homological conjectures for bounded extensions of algebras

TL;DR

This paper studies bounded extensions of finite-dimensional algebras, showing their homological properties and equivalences, and investigates several homological conjectures with applications to specific algebra classes.

Contribution

It introduces new criteria for functors inducing equivalences between Gorenstein categories and explores homological conjectures for bounded algebra extensions.

Findings

A bounded extension implies singular equivalence of Morita type with level.

Under mild conditions, stable categories of Gorenstein modules are equivalent.

Several homological conjectures hold for bounded extensions.

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. Additionally, under mild 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 proof, we give some handy criteria for a functor between module categories to induce 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.