The vanishing of self-extensions over n-symmetric algebras of quasitilted type

TL;DR

This paper proves that all n-symmetric algebras of quasitilted type satisfy the Generalized Auslander-Reiten Condition, linking module periodicity and projective dimension in a broad class of self-injective algebras.

Contribution

It establishes that n-symmetric algebras of quasitilted type satisfy the Generalized Auslander-Reiten Condition, expanding understanding of module theory in self-injective algebras.

Findings

All modules with no high-degree self-extensions have bounded projective dimension.

n-symmetric algebras of quasitilted type satisfy the Generalized Auslander-Reiten Condition.

Modules are periodic under the Nakayama automorphism in this class.

Abstract

A ring R satisfies the Generalized Auslander-Reiten Condition if any R-module M with no self-extensions in degrees higher than m must have projective dimension at most m. We prove that this condition is satisfied by all n-symmetric algebras of quasitilted type- a broad class of self-injective algebras where every module is periodic under the Nakayama automorphism.