Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories.
type: Item
instance of: scholarly article
title: Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories.
zbMATH Open document ID: 1115.16005
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/0407052
review text: While Auslander-Reiten theory is two-dimensional in the sense that almost split sequences can be regarded as projective resolutions of simple functors of length two, the present article exhibits a class of subcategories of classical representation categories with generalized almost split sequences of length \(n\). Let \(R\) be a complete regular local ring of dimension \(d\), and let \(\Lambda\) be a Cohen-Macaulay \(R\)-order, i.e., an \(R\)-algebra which is finitely generated and free over \(R\). Assume that \(\Lambda\) represents an isolated singularity in the sense of Auslander. Within the category \(\Lambda\text{CM}\) of maximal Cohen-Macaulay modules over \(\Lambda\), the author considers functorially finite full subcategories \(\mathcal C\) which are maximally \(l\)-orthogonal for some \(l>0\). This means that \(\mathcal C\) consists of all objects \(X\) in \(\Lambda\text{CM}\) which satisfy \(\text{Ext}^i(X,Y)=0\) for all \(Y\) in \(\mathcal C\) and \(0<i\leq l\), and similarly, \(\mathcal C\) consists of all \(X\) in \(\Lambda\text{CM}\) with \(\text{Ext}^i(Y,X)=0\) for all \(Y\) in \(\mathcal C\) and \(0<i\leq l\). It turns out that such categories \(\mathcal C\) admit generalized almost split sequences of length \(2+l\). Among other things, the author proves a higher Auslander-Reiten formula, he shows that long almost split sequences are found in the socle of higher Ext-groups, he generalizes Auslander's description of the Auslander-Reiten translate (in dimension \(d\)), and characterizes the higher Auslander-Reiten translate by means of derived functors. For representation-finite Gorenstein orders \(\Lambda\) of arbitrary dimension \(d\), he gives a complete classification of maximal 1-orthogonal subcategories of \(\Lambda\text{CM}\). This classification is related to configurations of bijectives in dimension 0 (Riedtmann) and 1 (Wiedemann). The author contributes to the ubiquity of Catalan numbers by showing that in case \(\mathbb{A}_n\), maximal 1-orth…
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16E65
Mathematics Subject Classification ID: 16G30
zbMATH DE Number: 5132549
zbMATH Keywords: Auslander-Reiten theory
zbMATH Keywords: maximal orthogonal subcategories
zbMATH Keywords: Auslander-Reiten translations
zbMATH Keywords: Auslander-Reiten dualities
zbMATH Keywords: almost split sequences
zbMATH Keywords: fundamental sequences
zbMATH Keywords: Cohen-Macaulay orders
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
OpenAlex ID: W2025614293
arXiv ID: math/0407052
cites work: Graded algebras of global dimension 3
cites work: Reflexive modules over rational double points
cites work: Q5586467
cites work: Q4163648
cites work: Q3770634
cites work: Rational Singularities and Almost Split Sequences
cites work: Stable module theory
cites work: The homological theory of maximal Cohen-Macaulay approximations
cites work: Stable equivalence of dualizing R-varieties
cites work: Almost split sequences in dimension two
cites work: Almost split sequences in subcategories
cites work: Q4318498
cites work: Tilting theory and cluster combinatorics.
cites work: Quivers with relations arising from clusters (𝐴_{𝑛} case)
cites work: Q3240881
cites work: Q3996909
cites work: Q5602190
cites work: Reflexive modules on quotient surface singularities.
cites work: Q3686801
cites work: Q5669455
cites work: Q3795770
cites work: Finite Modules of Finite Injective Dimension Over a Noetherian Algebra
cites work: Towards a theory of Bass numbers with application to Gorenstein algebras.
cites work: Q3994341
cites work: Q3890860
cites work: Q3125202
cites work: \(\tau\)-categories. I: Ladders.
cites work: \(\tau\)-categories. II: Nakayama pairs and rejective subcategories.
cites work: \(\tau\)-categories. III: Auslander orders and Auslander-Reiten quivers.
cites work: Auslander correspondence.
cites work: Q3897070
cites work: Q3992979
cites work: Q3903139
cites work: Algebren, Darstellungsköcher, Überlagerungen und zurück
cites work: Tame algebras and integral quadratic forms
cites work: Two-dimensional tame and maximal orders of finite representation type
cites work: Almost split sequences for integral group rings and orders
cites work: Q4271281
cites work: Q4236280
cites work: Q4669640
cites work: Q3995215
cites work: Classification of the Auslander-Reiten quivers of local Gorenstein orders and a characterization of the simple curve singularities
cites work: Q3475410
DOI: 10.1016/J.AIM.2006.06.002
description: scientific article