n-Representation infinite algebras

TL;DR

This paper introduces n-representation infinite algebras, a new class of finite-dimensional algebras with global dimension n, exploring their modules, homological properties, and connections to non-commutative geometry.

Contribution

It defines n-representation infinite algebras, studies their module categories, and generalizes classical results on regular modules to higher dimensions.

Findings

First examples of algebras with infinite Ext^1-orthogonal families

Description of n-regular modules via preprojective algebra

Lower bound of n+2 for the representation dimension

Abstract

From the viewpoint of higher dimensional Auslander-Reiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call n-representation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: n-preprojective, n-preinjective and n-regular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext^1-orthogonal families of modules. Moreover we give general constructions of n-representation infinite algebras. Applying Minamoto's theory on Fano algebras in non-commutative algebraic geometry, we describe the category of n-regular modules in terms of the corresponding preprojective algebra. Then we introduce n-representation tame algebras, and show that the category of n-regular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra. This generalizes the classical description of regular modules over tame hereditary algebras. As an application, we show that the representation dimension of an n-representation tame algebra is at least n+2.