
SoSe19 S4A2
Graduate Seminar on Representation Theory
Triangulated categories in the representation theory of finitedimensional algebras

Organisational meeting 
Monday 28.01.2019 @ 11:00, room 3.008 (Endenicher Alle 60) 
Meeting time 
Fridays 10:15–12:45, room 1.008 (Endenicher Allee 60) 
Organiser 
Dr. Gustavo Jasso

URLs 
BASIS

Description 
Triangulated categories are a fundamental tool in homological algebra. In this seminar we will focus both on the abstract theory of triangulated categories and on those triangulated categories which arise naturally in representation theory of finite dimensional algebras.

Prerequisites 
 Elementary category theory, abelian categories
 Rings and modules, basic homological algebra (Ext and Tor)
 Basic representation theory of finite dimensional algebras
 Assem, Ibrahim and Simson, Daniel and Skowroński, Andrzej. Elements of the representation theory of associative algebras. Vol. 1. London Mathematical Society Student Texts, Cambridge University Press, Cambridge 65 (2006): x+458
 Leinster, Tom. Basic category theory. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge 143 (2014): viii+183

Schedule 
Date 
Speaker(s) 
Topic 
05.04 
Gustavo 
Overview and motivation 
12.04 
Gregory 
Frobenius exact categories and Happel’s theorem 
19.04 

No talk 
26.04 
Sil 
Stable module categories and homotopy categories of chain complexes 
03.04 
Thilo/Yizhou 
Verdier quotients, derived categories, dg categories 
10.04 
Jona 
Total derived functors, the derived category of a dg category 
17.05 
Zbigniew 
Auslander–Reiten theory in triangulated categories, derived categories of Dynkin and Euclidean quivers 
24.05 
Aleksander/Lukas 
Algebraic triangulated categories: Keller’s theorem 
31.05 
Vincent 
Tilting, Silting and (co)tstructures 
07.06 

No talk 
14.06 
Andrea 
The theorems of Buchweitz and Orlov after Iyama and Yang 
21.06 
Alfio 
Calabi–Yau triangulated categories, cluster categories 
28.06 
Martin 
Amiot–Guo–Keller cluster categories 
05.07 
Manuel/Robin 
Brown representability after Krause 
12.07 
Jendrik 
Differential graded enhancements of triangulated categories 
Each talk should last between 75 and 90 minutes, except for shared talks which should be divided in two parts of 60 minutes each.

Outline 
 Frobenius exact categories and Happel’s theorem
 Exact categories (Sections 2 and 3 in [1])
 Frobenius exact categories and Happel’s Theorem (Section I.2 in [2])
 Stable module categories and homotopy categories of chain complexes
 Stable module categories of QuasiFrobenius rings (see for example Chapter 24 in [3] and Section 1.10 in [4])
 The homotopy category of chain complexes in an additive category (Examples 5.3 and 7.1 in [5])
 Verdier quotients, derived categories, differential graded categories
 Verdier localisation (Section 2.1 in [6])
 Derived categories of abelian and exact categories (Section 11 in [5])
 Dg categories and dg modules (Sections 2 and 3.1 in [7])
 The homotopy category of dg modules (Sections 3.3 and 3.4 in [7])
 Total derived functors, the derived category of a dg category
 Functorial Kprojective and Kinjective resolutions of dg modules (Sections 3.1 and 3.2 in [8])
 The derived category of a dg category (Section 3 in [7])
 Total derived functors; the functors RHom(,) and ⊗^{L} (Section 3.7 in [4] and Section 5.3 in [9])
 Auslander–Reiten theory in triangulated categories
 Auslander–Reiten theory in triangulated categories (Section I.4 in [2])
 Bounded derived categories of finitedimensional hereditary algebras (Section I.5 in [2])
 Tilting, Silting and (co)tstructures
 Tilting and silting subcategories (Section 2 in [10])
 Silting subcategories in triangulated categories with coproducts (Section 4 in [10])
 Algebraic triangulated categories: Keller’s theorem
 Algebraic triangulated categories and Keller’s theorem (Section 7.5 in [9] and Section 4.3 in [8])
 The norder of a triangulated category (Section 1 in [11])
 Algebraic triangulated categories have infinite norder (Section 2 in [11])
 The theorems of Buchweitz and Orlov after Iyama and Yang
 Prove the theorem of Iyama and Yang (Section 3 [12])
 Prove the theorems of Buchweitz and Orlov (Sections 2.1 and 2.2 in [12])
 Calabi–Yau triangulated categories, cluster categories
 Calabi–Yau triangulated categories [13]
 BMRRT cluster categories [14]
 Amiot–Guo–Keller cluster categories
 Explain the construction of the Amiot–Guo–Keller cluster categories [15], [16], [17] using the theorem of Iyama and Yang [12]
 Brown representability after Krause
 Theorems A and B in [18] (see also Section 4 in [9])
 Functorial resolutions and total derived functors via Brown representability (Sections 5 and 6 in [9])
 Differential graded enhancements of triangulated categories
 Dg categories and localisation (Section 2 in [19])
 Triangulated dg categories (Section 4.4 in [19])
 Functorial cones (Section 5.1 in [19])

Bibliography 
 Bühler, Theo. Exact categories. Expo. Math. 28 (1) (2010): 1–69
 Happel, Dieter. Triangulated categories in the representation theory of finitedimensional algebras. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge 119 (1988): x+208
 Faith, Carl. Algebra. II. , SpringerVerlag, BerlinNew York (1976): xviii+302
 Zimmermann, Alexander. Representation theory. Algebra and Applications, Springer, Cham 19 (2014): xx+707
 Keller, Bernhard. Derived categories and their uses. In Handbook of algebra, Vol. 1., NorthHolland, Amsterdam (1996): 671–701
 Neeman, Amnon. Triangulated categories. Annals of Mathematics Studies, Princeton University Press, Princeton, NJ 148 (2001): viii+449
 Keller, Bernhard. On differential graded categories. In International Congress of Mathematicians. Vol. II., Eur. Math. Soc., Zürich (2006): 151–190
 Keller, Bernhard. Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27 (1) (1994): 63–102
 Krause, Henning. Derived categories, resolutions, and Brown representability. In Interactions between homotopy theory and algebra. Contemp. Math., Amer. Math. Soc., Providence, RI 436 (2007): 101–139
 Aihara, Takuma and Iyama, Osamu. Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85 (3) (2012): 633–668
 Schwede, Stefan. The norder of algebraic triangulated categories. J. Topol. 6 (4) (2013): 857–867
 Iyama, Osamu and Yang, Dong. Quotients of triangulated categories and Equivalences of Buchweitz, Orlov and Amiot–Guo–Keller. (2017)
 Keller, Bernhard. CalabiYau triangulated categories. In Trends in representation theory of algebras and related topics. EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich (2008): 467–489
 Buan, Aslak Bakke and Marsh, Robert and Reineke, Markus and Reiten, Idun and Todorov, Gordana. Tilting theory and cluster combinatorics. Adv. Math. 204 (2) (2006): 572–618
 Amiot, Claire. Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier (Grenoble) 59 (6) (2009): 2525–2590
 Guo, Lingyan. Cluster tilting objects in generalized higher cluster categories. J. Pure Appl. Algebra 215 (9) (2011): 2055–2071
 Keller, Bernhard. On triangulated orbit categories. Doc. Math. 10 (2005): 551–581
 Krause, Henning. A Brown representability theorem via coherent functors. Topology 41 (4) (2002): 853–861
 Toën, Bertrand. Lectures on dgcategories. In Topics in algebraic and topological Ktheory. Lecture Notes in Math., Springer, Berlin 2008 (2011): 243–302
