SoSe18 S4A2
Graduate Seminar on Representation Theory

τ-tilitng theory and related topics

Meeting time Fridays 12:00–13:45, room 1.008
Organiser

Dr. Gustavo Jasso

URLs

BASIS

Description

The class of τ-tilting modules was introduced by Adachi, Iyama and Reiten in [1] as a completion of the class of tilting modules with respect to mutations. In this seminar we will introduce this class of modules and study their relationship with other important classes of objects in representation theory of finite dimensional algebras such as torsion classes, bricks, wide subcategories, stability conditions, etc..

Prerequisites

Solid background in representation theory of finite dimensional algebras, Auslander–Reiten theory, basic homological algebra, triangulated categories (in particular the bounded derived category and the bounded homotopy category of projectives for a finite dimensional algebra).


  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
  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
Organisational meeting Friday 13.04 @ 12:00, room 1.008
Tentative schedule
Date Speaker Topic References
13.04 Gustavo Overview and distribution of the talks  
20.04   No lecture  
27.04 TBD Introduction to τ-tilting theory [1], [2]
04.05 TBD τ-titling finite algebras and g-vectors [3]
11.05 TBD τ-tilting modules over preprojective algebras of Dynkin type [4]
18.05 TBD τ-tilting modules over gentle algebras [5]
25.05   No lecture  
01.06 TBD τ-tilting reduction [6]
08.06 TBD τ-tilting modules and semibricks [7], [3]
15.06 TBD Lattice theory of torsion classes [8], [9]
22.06 TBD τ-tilting theory and stability conditions [10], [11]
29.06 TBD Torsion classes, wide subcategories and localisations [12], [13]
06.07 TBD Silted algebras [14], [15]
13.07 TBD Silting modules [16], [17]
20.07 TBD Silting modules and universal localisations [18], [19]
Bibliography
  1. Adachi, Takahide and Iyama, Osamu and Reiten, Idun. τ-tilting theory. Compos. Math. 150 (3) (2014): 415–452
  2. Aihara, Takuma and Iyama, Osamu. Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85 (3) (2012): 633–668
  3. Demonet, Laurent and Iyama, Osamu and Jasso, Gustavo. τ-Tilting Finite Algebras, Bricks, and g-Vectors. Int. Math. Res. Not. IMRN (00) (2017): 1–41
  4. Mizuno, Yuya. Classifying τ-tilting modules over preprojective algebras of Dynkin type. Math. Z. 277 (3-4) (2014): 665–690
  5. Palu, Yann and Pilaud, Vincent and Plamondon, Pierre-Guy. Non-kissing complexes and tau-tilting for gentle algebras. (2017)
  6. Jasso, Gustavo. Reduction of τ-tilting modules and torsion pairs. Int. Math. Res. Not. IMRN (16) (2015): 7190–7237
  7. Asai, Sota. Semibricks. (2016)
  8. Iyama, Osamu and Reiten, Idun and Thomas, Hugh and Todorov, Gordana. Lattice structure of torsion classes for path algebras. Bull. Lond. Math. Soc. 47 (4) (2015): 639–650
  9. Demonet, Laurent and Iyama, Osamu and Reading, Nathan and Reiten, Idun and Thomas, Hugh. Lattice theory of torsion classes. (2017)
  10. Brüstle, Thomas and Smith, David and Treffinger, Hipolito. Stability conditions, τ-tilting Theory and Maximal Green Sequences. (2017)
  11. Thomas, Hugh. Stability, shards, and preprojective algebras. (2017)
  12. Marks, Frederik and Šťovíček, Jan. Torsion classes, wide subcategories and localisations. Bull. Lond. Math. Soc. 49 (3) (2017): 405–416
  13. Marks, Frederik. Homological embeddings for preprojective algebras. Math. Z. 285 (3-4) (2017): 1091–1106
  14. Buan, Aslak Bakke and Zhou, Yu. A silting theorem. J. Pure Appl. Algebra 220 (7) (2016): 2748–2770
  15. Buan, Aslak Bakke and Zhou, Yu. Silted algebras. Adv. Math. 303 (2016): 859–887
  16. Angeleri Hügel, Lidia and Marks, Frederik and Vitória, Jorge. Silting modules. Int. Math. Res. Not. IMRN (4) (2016): 1251–1284
  17. Angeleri Hügel, Lidia and Marks, Frederik and Vitória, Jorge. A characterisation of τ-tilting finite algebras. (2018)
  18. Marks, Frederik. Universal localisations and tilting modules for finite dimensional algebras. J. Pure Appl. Algebra 219 (7) (2015): 3053–3088
  19. Marks, Frederik and Stovicek, Jan. Universal localisations via silting. (2016)