G-stable support \tau-tilting modules

Abstract

Motivated by \tau-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a nite-dimensional algebra \mathrm{\Lambda }with action by a nite group G; we introduce the notion of G-stable support \tau-tilting modules. Then we establish bijections among G-stable support \tau-tilting modules over \Lambda; G-stable two-term silting complexes in the homotopy category of bounded complexes of nitely generated projective \mathrm{\Lambda }-modules, and G-stable functorially nite torsion classes in the category of nitely generated left \mathrm{\Lambda }-modules. In the case when \mathrm{\Lambda } is the endomorphism of a G-stable cluster-tilting object T over a Hom-nite 2-Calabi-Yau triangulated category \ell with a G-action, these are also in bijection with G-stable cluster-tilting objects in \ell : Moreover, we investigate the relationship between stable support \tau-tilitng modules over \mathrm{\Lambda } and the skew group algebra \mathrm{\Lambda }G:

Key words： G-stable support\tau-tilting modules; G-stable two-term silting complexes; G-stable functorially nite torsion classes; G-stable cluster-tilting objects; bijection; skew group algebras

References

1 Adachi T. The classi_cation of _-tilting modules over Nakayama algebras. J Algebra, 2016, 452: 227–262

2 Adachi T, Iyama O, Reiten I. _-tilting theory. Compos Math, 2014, 150: 415–452

3 Aihara T, Iyama O. Silting mutation in triangulated categories. J Lond Math Soc, 2012, 85: 633–668

4 Auslander M, Reiten I. Applications of contravariantly _nite subcategories. Adv Math, 1991, 86: 111–152

5 Auslander M, Smal_ S O. Almost split sequences in subcategories. J Algebra, 1981, 69: 426–454

6 Bongartz K. Tilted algebras. In: Auslander M, Lluis E, eds. Representations of Algebras, Proceedings of the Third International Conference held in Puebla, August 4-8, 1980. Lecture Notes in Math, Vol 903. Berlin-New York: Springer, 1981, 26–38

7 Brüstle T, Dupont G, P_erotin N. On maximal green sequences. Int Math Res Not IMRN, 2014, 16: 4547–4586

8 Buan A B, Marsh R, Reineke M, Reiten I, Todorov G. Tilting theory and cluster combinatorics. Adv Math, 2006, 204: 572–618

9 Buan A B, Marsh R, Reiten I. Cluster-tilted algebras. Trans Amer Math Soc, 2007, 359: 323–332

10 Dionne J, Lanzilotta M, Smith D. Skew group algebras of piecewise hereditary algebras are piecewise hereditary. J Pure Appl Algebra, 2009, 213: 241–249

11 Happel D, Unger L. Almost complete tilting modules. Proc Amer Math Soc, 1989, 107: 603–610

12 Happel D, Unger L. On a partial order of tilting modules. Algebr Represent Theory, 2005, 8: 147–156

13 Iyama O, Jφrgensen P, Yang D. Intermediate co-t-structures, two-term silting objects,_-tilting modules, and torsion classes. Algebra Number Theory, 2014, 8: 2413–2431

14 Iyama O, Reiten I. Introduction to _-tilting theory. Proc Natl Acad Sci USA, 2014, 111: 9704–9711

15 Jasso G. Reduction of _-tilting modules and torsion pairs. Int Math Res Not IMRN, 2015, 16: 7190–7237

16 Keller B, Reiten I. Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. Adv Math, 2007, 211: 123–151

17 Mizuno Y. _-stable _-tilting modules. Comm Algebra, 2015, 43: 1654–1667

18 Reiten I, Riedtmann C. Skew group algebras in the representation theory of Artin algebras. J Algebra, 1985, 92: 224–282

19 Wei J Q. _-tilting theory and _-modules. J Algebra, 2014, 414: 1–5

20 Zhang X. _-rigid modules for algebras with radical square zero. arXiv: 1211.5622