Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories

By Osamu Iyama. 2004

438 citations recorded by the source corpus.

Abstract We introduce the concept of maximal ( n − 1 ) -orthogonal subcategories over Artin algebras and orders, and develop ( n + 1 ) -dimensional Auslander–Reiten theory on them. We give the n -Auslander–Reiten translation and the n -Auslander–Reiten duality, then show the existence of n -almost split sequences and n -fundamental sequences. We give some examples.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f91c44443c1fc4d2611b9dbe2a846224d5211c52

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In Algebra and Number Theory

1. Osamu Iyama
2. B. Keller
3. I. Reiten
4. Gustavo Jasso
5. Bernard Leclerc
6. C. Geiss
7. J. Schröer
8. K. Yamagata

In Geometry and Topology

1. Osamu Iyama
2. B. Keller
3. I. Reiten
4. Bernard Leclerc
5. Gustavo Jasso
6. C. Geiss
7. J. Schröer
8. Aslak Bakke Buan

In Statistical and Nonlinear Physics

1. Osamu Iyama
2. B. Keller
3. Bernard Leclerc
4. I. Reiten
5. Gustavo Jasso
6. C. Geiss
7. J. Schröer
8. Andrzej Skowronski

Citations in the corpus

• n-angulated categories (2010, background): “to construct n-angulated categories inside triangulated categories. Namely, if Tis a triangulated category with suspension functor Σ and Sis an (n−2)-cluster tilting subcategory in the sense of Iyama [14] which is stable under Σn−2, then Snaturally becomes an n-angulated category with higher suspens”
• Higher Nakayama algebras I: Construction (2016, background): “t split sequences are the main object of study in classical Auslander{Reiten theory. The analogous concept in higher Auslander{Reiten theory is that of a d-almost split sequence, see Denition 3.1 in [Iya07b]. Denition 1.21. Let M be a Krull{Schmidt additive category. A sequence in M of the form : 0 ”
• Selfinjective quivers with potential and 2-representation-finite algebras (2010, background): “icts to the exactness of (1) since b : Pi →Pe(b) is not surjective. 3.2. Structure theorem of 2-representation-finite algebras. For a finite dimensonal algebra A, we say that M ∈modA is cluster tilting [I1, I3] if addM = {X ∈modA |Ext1 A(M,X) = 0} = {X ∈modA |Ext1 A(X,M) = 0}. It is known that the cat”
• n-Representation infinite algebras (2012, methodology): “ A (−,X) → 0. A right minimal and right almost split morphism is called a sink morphism. (b) Dually we define a left almost split morphism and a source morphism. The following notion was introduced in [I1] in higher dimensional Auslander-Reiten theory. 16 HERSCHEND, IYAMA, AND OPPERMANN Definition 4.2”
• Cluster algebra structures and semicanonical bases for unipotent groups (2007, background): “0 0 CC 0 0 1 0 1 0 0 0 8˙˙ 888 8 CC 0 0 0 0 1 1 1 0 BB 1 0 1 0 1 0 0 0 Figure 2. A category C M⊂mod(Λ) with C triangle equivalent to CA1×A1 The notion of a maximal 1-orthogonal module is due to Iyama [Iy1]. These modules are also called cluster tilting objects. Theorem 2.2. Let Mbe a terminal KQ-mod”
• Representations of Algebras and Related Topics (2011, background): “modules over a preprojective algebra of type Q associated with any element in the Coxeter group of Q (see [22], [23], [50], [51]); stable Cohen-Macaulay modules over isolated singularities (see [14], [60], [29]); and generalized cluster categories associated with finite dimensional algebras of global”
• $\tau $-tilting theory (2013, neutral): “例外列 (exceptional sequence)の変異 [7] と団傾対象 (cluster tilting object)の変異 [6, 10]の2種類が広く知られているが, いずれもある種の三 角圏の構造を解析するものであり, 特に後者は高次元Auslander-Reiten理論 [9]や団代数 (cluster algebra)の圏論化 [12]にも応用される重要なものである.”
• Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, methodology): “The autofunctor Sn of C is a triangulated analog of the n-Auslander-Reiten translation functor in [ 40 ].”
• The classification of \tau-tilting modules over Nakayama algebras (2013, background): “ion pairs in tube categories [BBM], cluster-tilting objects in cluster categories of type A [CCS] and D [S], cluster-tilting modules over self-injective algebras and Gorenstein orders of type A and D [I]. Next, we study a relationship between support τ-tilting Λ-modules over an algebra Λ (not necess”
• Categorification of Skew-symmetrizable Cluster Algebras (2009, methodology): “In this framework, the notion of cluster-tilting object introduced by [ 42 ] is very useful.”
• Tilting and cluster tilting for quotient singularities (2010, methodology): “e dimK(L ∗G) = (dimK L)2 = dimK EndK(L), we have that φP is bijective. 4. Clustertilting subcategories forquotient singularities The notion of maximal (n−1)-orthogonal subcategories was introduced in [I1] as a domain of higher dimensional analogue of Auslander-Reiten theory, and later renamed cluste”
• 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-finite algebras and n-APR tilting (2009, positive): “logical properties of Auslander algebras, and vice versa. 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 AuslanderReiten theory was developed in a series of papers [Iya1, I”
• Derived equivalences and stable equivalences of Morita type, I (2008, background): “For instance, if Δ = An, then ωAn(p, q) = (p+ q − n, n+ 1− q) (see [23], Section 4).”