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Seminar on Higher Auslander-Reiten theory (winter term 2015/2016)


Seminar on higher Auslander-Reiten theory (winter term 2015/2016)

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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