The following article is an introduction to higher Auslander-Reiten theory:

• [1] Osamu Iyama: Auslander-Reiten theory revisited
  (Proceeding of the ICRA XII conference, Torun, August 2007, 47 pages, arXiv version)
  

The goal of the seminar is to cover the main results and topics mentioned of Iyama's survey article.

The proofs of the results mentioned in [1] are given in the following two papers:

• [2] Osamu Iyama: Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories
  (Adv. Math. 210 (2007), no. 1, 22--50, arXiv version)


• [3] Osamu Iyama: Auslander correspondence
  (Adv. Math. 210 (2007), no. 1, 51--82, arXiv version)

The topics of the first research paper [2] will be taken upon in talks 3,4,7 and 9. Iyama's second research paper [3] is mainly relevant for talk 8. Further examples of algebras admitting cluster tilting subcategories will be presented in talk 10, which concerns the following paper:

• [4] Osamu Iyama: Cluster tilting for higher Auslander algebras (Adv. Math. 226 (2011), no. 1, 1--61, arXiv version)

Seminar schedule

#1	20.10.2015	Wassilij Gnedin	Classical Auslander-Reiten theory
			recall of classical Auslander-Reiten theory, classical Auslander-Reiten formula (Theorems 1.6 and 1.7 in [1]), examples for knitting Auslander-Reiten quivers
#2	20.10.2015	Julian Külshammer	Classical Auslander correspondence
			statement and examples for classical Auslander correspondence (Theorem 1.2 in [1]), Yoneda's lemma in the context of Auslander algebras (first equivalence in the proof of Theorem 2.6 (a) in [2]), main steps in the proof of classical Auslander correspondence
#3	3.11.2015	Ruari Walker	Cluster-tilting categories and their properties
			Definition of cluster-tilting category C (Subsection 2.1 in [1], or Subsections 2.1 and 2.2 in [2]), equivalent characterizations of cluster-tilting subcategories (Proposition 2.2 in [1] or Proposition 2.2.2 in [2]), basic examples (Example 2.4(a) in [1]).
#4	10.11.2015	Armin Shalile	Higher Auslander-Reiten formula
			Definition of higher Auslander-Reiten translation, statement of higher Auslander-Reiten formula for cluster-tilting categories (Theorems 2.8 and 2.9 in [1], or Theorem 1.4.1, 1.5 in [2]), basic examples, main steps in the proof of higher Auslander-Reiten formula.
#5	24.11.2015	Gustavo Jasso	A short proof of the defect formula for d-exact sequences
#6	1.12.2015	Pin Liu	Tau-tilting and cluster-tilting in higher cluster categories
			Relationship between higher cluster-tilting objects in higher cluster categories and tau-tilting modules over a cluster-tilted algebra.
#7	8.12.2015	Steffen König	Higher Auslander-Reiten sequences
			Existence and properties of higher Auslander-Reiten sequences (Theorem 2.11 in [1] or Theorem 3.3.1 in [2])
#8	15.12.2015	Christian Lomp	Higher Auslander correspondence
			main steps in the proof of higher Auslander correspondence (Theorem 2.6 in [1] or Theorem 4.2.2 in [3])
#9	12.1.2016	René Marczinzik	Cluster-tilting for self-injective algebras
			Cluster tilting objects for representation-finite self-injective algebras (Section 4 in [2]), general results on the existence of cluster-tilting objects for self-injective algebras (Theorem 2.22 in [1] by Erdmann and Holm)
#10	19.1.2016	Yiping Chen	n-complete algebras
			Inductive construction of algebras admitting cluster subcategories (Section 2.4 in [1], or [4])
#11	26.1.2016	Bernhard Böhmler	Introduction to representation theory of orders
			Classical Auslander-Reiten theory for Cohen-Macaulay modules over orders (Section 3.1 in [1])
#12	26.1.2016	Wassilij Gnedin	Cluster-tilting modules over quotient singularities
			higher Auslander-Reiten theory for Cohen-Macaulay modules over orders, main steps in the proof of Theorem 3.10 a) in [1]
#13	2.2.2016	Wassilij Gnedin	Higher algebraic McKay correspondence
			main steps in the proof that McKay quiver is the higher AR quiver (Theorem 3.10 b), c) in [1] )
#14	9.2.2016	Wassilij Gnedin	Higher Auslander orders
			the depth of cluster tilted algebras, higher Auslander correspondence for orders, derived equivalence of 2-cluster tilted algebras (Theorem 3.15 in [1]), ;
#15	9.2.2016	Steffen König	Non-commutative crepant resolutions
			geometric and algebraic notions of resolutions, Bondal-Orlov and van den Bergh's conjectures, main steps of the proof of Theorem 3.18 in [1]

Talks related to higher Auslander-Reiten theory at Stuttgart:

Darstellungstheorie-Tage:

13.11.2015, 16:00, (V57.04)

Julian Külshammer

Higher Nakayama algebras

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