Constructions of symmetric separable equivalences and their applications

TL;DR

This paper explores the construction and properties of symmetric separable equivalences between Artin algebras, demonstrating their invariance of key algebraic properties and providing methods for their construction.

Contribution

It introduces new methods to construct symmetric separable equivalences and studies their invariance properties for various algebraic conjectures and dimensions.

Findings

Existence of symmetric separable equivalences between certain endomorphism algebras.

Invariance of rigidity dimension, Frobenius-finite type, and several conjectures under these equivalences.

Methods to construct symmetric separable equivalences from existing ones.

Abstract

Let $\Lambda$ and $\Gamma$ be symmetrically separably equivalent Artin algebras. We prove that there exist symmetrical separable equivalences between certain endomorphism algebras of modules. As applications, we provide several methods to construct symmetrical separable equivalences from given ones and discuss when the rigidity dimension is an invariant under symmetrical separable equivalences. Moreover, we show that a symmetrical separable equivalence preserves the Frobenius-finite type, Auslander-type condition, the (strong) Nakayama conjecture, the Auslander-Gorenstein conjecture and so on.