Resolutions of mesh algebras: periodicity and Calabi-Yau dimensions

TL;DR

This paper investigates the Calabi-Yau properties of stable categories of self-injective algebras of finite type, determining their dimensions through projective resolutions and covering theory, with connections to derived categories.

Contribution

It provides a classification of Calabi-Yau stable categories for self-injective algebras and computes their dimensions using novel resolution and covering techniques.

Findings

Identifies which stable categories are Calabi-Yau and their dimensions

Uses projective resolutions of the stable Auslander algebra

Reduces problems to preprojective algebras of Dynkin graphs

Abstract

A triangulated category is said to be Calabi-Yau of dimension d if the dth power of its suspension is a Serre functor. We determine which stable categories of self-injective algebras A of finite representation type are Calabi-Yau and compute their Calabi-Yau dimensions. We achieve this by studying the minimal projective resolution of the stable Auslander algebra of A over its enveloping algebra, and use covering theory to reduce to (generalized) preprojective algebras of Dynkin graphs. We also describe how this problem can be approached by realizing the stable categories in question as orbit categories of the bounded derived categories of hereditary algebras.