New invariants of stable equivalences of algebras

TL;DR

This work investigates how stable and derived equivalences between Artin algebras affect key homological invariants tied to the finitistic dimension problem and the Auslander–Reiten conjecture (ARC). It proves that stable equivalences between algebras without nodes preserve the delooping level $del(A)$ and Igusa–Todorov dimensions $\phi\dim(A)$ and $\psi\dim(A)$, implying finite-finitistic-dimension transfer when these dimensions are finite. It further shows that stable equivalences between algebras with positive $\nu$-dominant dimension induce stable equivalences of their Frobenius parts, enabling ARC transfers via Frobenius components. The methods are applied to two broad algebra classes—principal centralizer matrix algebras and Frobenius-finite algebras—establishing ARC for both and demonstrating that ARC on stable equivalence classes can be reduced to Frobenius parts. Finally, the authors propose a conjecture that finiteness of the delooping level is invariant under derived equivalences, suggesting a new direction for understanding when derived-equivalent algebras share finiteness properties of their homological invariants.

Abstract

We show that stable equivalences between Artin algebras without nodes preserve homological data that provide upper bounds for finitistic dimension, and that stable equivalences between Artin algebras with positive $ν$-dominant dimensions induce stable equivalences of their Frobenius parts. As an application of our new methods developed, we verify the Auslander--Reiten conjecture on stable equivalences for two rather different classes of algebras: principal centralizer matrix algebras over arbitrary fields and Frobenius-finite algebras over algebraically closed fields.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5