Auslander–Reiten theory revisited

By Osamu Iyama. 2008

87 citations recorded by the source corpus.

We recall several results in Auslander-Reiten theory for finite-dimensional algebras over fields and orders over complete local rings. Then we introduce $n$-cluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/55e94d65f837fc51c5a6ebd4b391ad3433451c55

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In Computational Theory and Mathematics

1. Steffen Oppermann
2. Osamu Iyama

Citations in the corpus

• Cluster tilting for higher Auslander algebras (2008, background): “It also played an important role from the viewpoint of higher analogue of Auslander-Reiten theory [I3, I4, I5 ] in the study of rigid Cohen-Macaulay modules, Calabi-Yau algebras and categories, and non-commutative crepant resolutions [BIRSc, BIRSm, BIKR, GLS1, GLS2, GLS3, IR, IY, KR1, KR2, KV, KZ, ”
• Recollement of Homotopy Categories and Cohen-Macaulay Modules (Representation Theory of Finite Groups and Algebras, and Related Topics) (2009, background): “2 In the representation theory of commutative rings [Y] and orders [CR, A2], there is another notion of Cohen-Macaulay modules (see [I]).”
• n-representation-finite algebras and n-APR tilting (2009, methodology): “To study higher Auslander algebras, the notion of n -cluster tilting subcategories (=maximal ( n − 1)-orthogonal subcategories) was introduced in [Iya3], and a higher analogue of Auslander-Reiten theory was developed in a series of papers [Iya1, Iya2, IO]; see also the survey paper [Iya4].”