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doi:10.1007/s11464-016-0560-9

Front. Math. China

2016, Vol. 11

Issue (4) : 1057-1077 https://doi.org/10.1007/s11464-016-0560-9

RESEARCH ARTICLE

G-stable support τ-tilting modules

Yingying ZHANG,Zhaoyong HUANG(

)

Department of Mathematics, Nanjing University, Nanjing 210093, China

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Abstract

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

Keywords τ-tilting modules')" href="#">G-stable support

-tilting modules G-stable two-term silting complexes G-stable functorially nite torsion classes G-stable cluster-tilting objects bijection skew group algebras

Corresponding Author(s): Zhaoyong HUANG

Issue Date: 30 August 2016

Cite this article:

Yingying ZHANG,Zhaoyong HUANG. G-stable support τ-tilting modules[J]. Front. Math. China, 2016, 11(4): 1057-1077.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0560-9
https://academic.hep.com.cn/fmc/EN/Y2016/V11/I4/1057

1	Adachi T. The classi_cation of _-tilting modules over Nakayama algebras. J Algebra, 2016, 452: 227–262 https://doi.org/10.1016/j.jalgebra.2015.12.013
2	Adachi T, Iyama O, Reiten I. _-tilting theory. Compos Math, 2014, 150: 415–452 https://doi.org/10.1112/S0010437X13007422
3	Aihara T, Iyama O. Silting mutation in triangulated categories. J Lond Math Soc, 2012, 85: 633–668 https://doi.org/10.1112/jlms/jdr055
4	Auslander M, Reiten I. Applications of contravariantly _nite subcategories. Adv Math, 1991, 86: 111–152 https://doi.org/10.1016/0001-8708(91)90037-8
5	Auslander M, Smal_ S O. Almost split sequences in subcategories. J Algebra, 1981, 69: 426–454 https://doi.org/10.1016/0021-8693(81)90214-3
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 https://doi.org/10.1007/bfb0092982
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 https://doi.org/10.1016/j.aim.2005.06.003
9	Buan A B, Marsh R, Reiten I. Cluster-tilted algebras. Trans Amer Math Soc, 2007, 359: 323–332 https://doi.org/10.1090/S0002-9947-06-03879-7
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 https://doi.org/10.1016/j.jpaa.2008.06.010
11	Happel D, Unger L. Almost complete tilting modules. Proc Amer Math Soc, 1989, 107: 603–610 https://doi.org/10.1090/S0002-9939-1989-0984791-2
12	Happel D, Unger L. On a partial order of tilting modules. Algebr Represent Theory, 2005, 8: 147–156 https://doi.org/10.1007/s10468-005-3595-2
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 https://doi.org/10.2140/ant.2014.8.2413
14	Iyama O, Reiten I. Introduction to _-tilting theory. Proc Natl Acad Sci USA, 2014, 111: 9704–9711 https://doi.org/10.1073/pnas.1313075111
15	Jasso G. Reduction of _-tilting modules and torsion pairs. Int Math Res Not IMRN, 2015, 16: 7190–7237 https://doi.org/10.1093/imrn/rnu163
16	Keller B, Reiten I. Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. Adv Math, 2007, 211: 123–151 https://doi.org/10.1016/j.aim.2006.07.013
17	Mizuno Y. _-stable _-tilting modules. Comm Algebra, 2015, 43: 1654–1667 https://doi.org/10.1080/00927872.2013.876033
18	Reiten I, Riedtmann C. Skew group algebras in the representation theory of Artin algebras. J Algebra, 1985, 92: 224–282 https://doi.org/10.1016/0021-8693(85)90156-5
19	Wei J Q. _-tilting theory and _-modules. J Algebra, 2014, 414: 1–5 https://doi.org/10.1016/j.jalgebra.2014.04.028
20	Zhang X. _-rigid modules for algebras with radical square zero. arXiv: 1211.5622

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