Auslander correspondence.
type: Item
instance of: scholarly article
title: Auslander correspondence.
zbMATH Open document ID: 1115.16006
author: Osamu Iyama
published in: Advances in Mathematics
publication date: 2007-03-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0411631
review text: The study of maximal \(l\)-orthogonal subcategories [Adv.~Math. 210, No. 1, 22-50 (2007; see the preceding review Zbl 1115.16005)] is continued. In the present article, the concept of Auslander algebra is developed in an utmost general context. The investigation starts with an adaption of higher Auslander-Reiten theory to dualizing \(R\)-varieties. For an Abelian category \(\mathcal A\) with enough projectives over a commutative local ring \(R\), and with length-finite \(\text{Ext}^1\)-groups, let \(\mathcal B\) be a resolving subcategory with enough injectives. Then maximal \(l\)-orthogonal subcategories \(\mathcal C\) of \(\mathcal B\) are defined, and the higher Auslander-Reiten formula is proved. In case \(\mathcal C\) is a Krull-Schmidt category, existence and properties of \(n\)-almost split sequences are derived. Now let \(R\) be a complete regular local ring of dimension \(d\), and let \(\Lambda\) be a Cohen-Macaulay \(R\)-order which represents an isolated singularity. The category \(\Lambda\text{CM}\) of maximal Cohen-Macaulay modules over \(\Lambda\) is given by the standard \(d\)-cotilting module \(\Lambda^*=\Hom_R(\Lambda,R)\). More generally, the author considers an arbitrary \(m\)-cotilting module \(T\) in \(\Lambda\text{CM}\). Thus \(T\) gives rise to a resolving subcategory \(\mathcal B={^\perp T}\) of \(\mathcal A=\Lambda\text{-}\mathbf{mod}\) with an injective cogenerator \(T\). Within \(\mathcal B\), the author considers maximal \((n-1)\)-orthogonal subcategories \(\mathcal C\) with an additive generator \(M\). In this general context, depending on \(d,m\), and \(n\), the Auslander algebra \(\Gamma=\text{End}_\Lambda(M)\) is characterized. For example, in case \(d=0\), this leads to the condition that \(\text{gld\,}\Gamma\leq n+1\) and \(\text{dom.dim\,}\Gamma\geq n+1\). The general conditions are, of course, more complicated. In particular, Auslander algebras of type \(d=m=n+1\) are regular of global dimension \(d\). The author provides several …
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16E65
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E10
zbMATH DE Number: 5132550
zbMATH Keywords: Auslander-Reiten theory
zbMATH Keywords: Auslander algebras
zbMATH Keywords: maximal orthogonal subcategories
zbMATH Keywords: Auslander condition
zbMATH Keywords: non-commutative crepant resolutions
zbMATH Keywords: representation dimension
zbMATH Keywords: almost split sequences
zbMATH Keywords: Cohen-Macaulay orders
zbMATH Keywords: cotilting modules
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
OpenAlex ID: W4210799175
arXiv ID: math/0411631
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Auslander correspondence. (DOI)
DOI: 10.1016/J.AIM.2006.06.003