Artin-Schelter regular algebras and categories

TL;DR

This paper extends the concept of Artin-Schelter regular algebras to categories and infinite types, revealing they share properties like Serre duality and Frobenius structures, thus broadening their applicability in representation theory.

Contribution

It introduces a generalized notion of Artin-Schelter regularity for algebras and categories, including infinite and Auslander types, with key properties like Serre duality.

Findings

Generalized Artin-Schelter regular algebras satisfy Serre duality.

The Ext-category of simple objects forms a Frobenius category.

Includes a graded analogue for infinite representation type.

Abstract

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension $n$ to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin-Schelter regular algebra or a category of dimension $n$ is shown to have common properties with the classical Artin-Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the $\Ext$-category of nice sets of simple objects of maximal projective dimension $n$ is a finite length Frobenius category.