SoSe19 S4A2
Graduate Seminar on Representation Theory

Triangulated categories in the representation theory of finite-dimensional algebras

Organisational meeting Monday 28.01.2019 @ 11:00, room 3.008 (Endenicher Alle 60)
Meeting time Fridays 10:15–11:45, 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


  1. 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
    Notes. Techniques of representation theory
  2. Krause, Henning. Krull-Schmidt categories and projective covers. Expo. Math. 33 (4) (2015): 535–549
  3. 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
03.05 Thilo Verdier quotients
10.05 Yizhou/Jona DG categories / Total derived functors
17.05 Lukas Algebraic triangulated categories have infinite n-order
24.05 Aleksander/Zbigniew Algebraic triangulated categories: Keller’s theorem / Auslander–Reiten theory in triangulated categories I
31.05 Zbigniew/Vincent Auslander–Reiten theory in triangulated categories II / Tilting, Silting and (co-)t-structures
07.06   No talk
14.06 Andrea The theorems of Buchweitz and Orlov after Iyama and Yang
21.06 Gustavo 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
  1. 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])
  2. Stable module categories and homotopy categories of chain complexes
    • Stable module categories of Quasi-Frobenius 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])
  3. 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] and Section 10 in [1])
    • 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])
  4. Total derived functors, the derived category of a dg category
    • Functorial K-projective and K-injective 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])
  5. Auslander–Reiten theory in triangulated categories
    • Auslander–Reiten theory in triangulated categories (Section I.4 in [2])
    • Bounded derived categories of finite-dimensional hereditary algebras (Section I.5 in [2])
  6. Tilting, Silting and (co-)t-structures
    • Tilting and silting subcategories (Section 2 in [10])
    • Silting subcategories in triangulated categories with co-products (Section 4 in [10])
  7. 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 n-order of a triangulated category (Section 1 in [11])
    • Algebraic triangulated categories have infinite n-order (Section 2 in [11])
  8. 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])
  9. Calabi–Yau triangulated categories, cluster categories
    • Calabi–Yau triangulated categories [13]
    • BMRRT cluster categories [14]
  10. 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]
  11. 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])
  12. 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
  1. Bühler, Theo. Exact categories. Expo. Math. 28 (1) (2010): 1–69
  2. Happel, Dieter. Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge 119 (1988): x+208
  3. Faith, Carl. Algebra. II. , Springer-Verlag, Berlin-New York (1976): xviii+302
    Notes. Ring theory, Grundlehren der Mathematischen Wissenschaften, No. 191
  4. Zimmermann, Alexander. Representation theory. Algebra and Applications, Springer, Cham 19 (2014): xx+707
    Notes. A homological algebra point of view
  5. Keller, Bernhard. Derived categories and their uses. In Handbook of algebra, Vol. 1., North-Holland, Amsterdam (1996): 671–701
  6. Neeman, Amnon. Triangulated categories. Annals of Mathematics Studies, Princeton University Press, Princeton, NJ 148 (2001): viii+449
  7. Keller, Bernhard. On differential graded categories. In International Congress of Mathematicians. Vol. II., Eur. Math. Soc., Zürich (2006): 151–190
  8. Keller, Bernhard. Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27 (1) (1994): 63–102
  9. 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
  10. Aihara, Takuma and Iyama, Osamu. Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85 (3) (2012): 633–668
  11. Schwede, Stefan. The n-order of algebraic triangulated categories. J. Topol. 6 (4) (2013): 857–867
  12. Iyama, Osamu and Yang, Dong. Quotients of triangulated categories and Equivalences of Buchweitz, Orlov and Amiot–Guo–Keller. (2017)
  13. Keller, Bernhard. Calabi-Yau triangulated categories. In Trends in representation theory of algebras and related topics. EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich (2008): 467–489
  14. 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
  15. Amiot, Claire. Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier (Grenoble) 59 (6) (2009): 2525–2590
  16. Guo, Lingyan. Cluster tilting objects in generalized higher cluster categories. J. Pure Appl. Algebra 215 (9) (2011): 2055–2071
  17. Keller, Bernhard. On triangulated orbit categories. Doc. Math. 10 (2005): 551–581
  18. Krause, Henning. A Brown representability theorem via coherent functors. Topology 41 (4) (2002): 853–861
  19. Toën, Bertrand. Lectures on dg-categories. In Topics in algebraic and topological K-theory. Lecture Notes in Math., Springer, Berlin 2008 (2011): 243–302

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