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n-representation infinite algebras

Authors
  • Herschend, Martin
  • Iyama, Osamu
  • Oppermann, Steffen1, 2, 3, 4
  • 1 Graduate School of Mathematics
  • 2 Nagoya University
  • 3 Institutt for matematiske fag
  • 4 NTNU
Type
Published Article
Journal
Advances in Mathematics
Publication Date
Jan 01, 2013
Accepted Date
Sep 30, 2013
Volume
252
Pages
292–342
Identifiers
DOI: 10.1016/j.aim.2013.09.023
Source
Elsevier
Keywords
License
Unknown

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 Ext1-orthogonal families of modules. Moreover we give general constructions of n-representation infinite algebras.Applying Minamotoʼs 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.

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