Auslander correspondence

By Osamu Iyama. 2004

274 citations recorded by the source corpus.

We study Auslander correspondence from the viewpoint of higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories. We give homological characterizations of Auslander algebras, especially an answer to a question of M. Artin. They are also closely related to Auslander's representation dimension of artin algebras and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6dd8b4b8d25d2068f6162a9d51e82c4f507bd03d

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Citations in the corpus

Torsion Classes and t-Structures in Higher Homological Algebra (2014, methodology): “Higher homological algebra was introduced and developed by Iyama in [10], [11], [12].”
Cluster tilting for one-dimensional hypersurface singularities (2007, methodology): “s in CM(R) for these R, in particular, we give a bijection between cluster tilting objects in CM(R) and elements in a symmetric group. Our method is based on a higher theory of almost split sequences [I1, I2], and a crucial role is played by the endomorphism algebras EndR(T) (called ‘three-dimension”
Tilting and cluster tilting for quotient singularities (2010, methodology): “-modules for a quotient singularity R. We will show that the Koszul complex of S is a direct sum of higher almost split sequences for graded Cohen-Macaulay modules. This is an analogue of a result in [I2] which assumes that S is a formal power series ring. Results in this section will be used in the”
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].”
An introduction to higher Auslander–Reiten theory (2016, background): “Higher Auslander–Reiten theory was introduced by Iyama in 2004 in [18, 19] (see also the survey article [20]).”
n-Representation infinite algebras (2012, background): “· → C1 → C0 → X → 0 such that Y ∼= τ n(X) and X ∼= τ− n (Y ). It is worthnoting that n-representationinﬁnite algebrasform an answerto the following question in homological algebra, which was asked in [I2] and at ICRA XI: Question 1.2. Is there a ﬁnite dimensional algebra Λ with an inﬁnite set {Xi}i∈”
Auslander–Reiten theory revisited (2008, methodology): “Then we have the following result [85, 86].”
Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday (2011, background): “e isomorphisms End CM(C)(Be) ≃ B and End CM(Cop)(eB) ≃ B op. (d) Be is (d− 1)-cluster tilting in CM(C). The above statements (c) and (d) show that B is a higher Auslander algebra of C in the sense of [Iya07b]. If moreover B is a graded k-algebra, we have the following additional information. Proposi”