A generalization of silting modules and Tor-tilting modules

doi:10.1007/s11464-021-0926-5

Front. Math. China

2022, Vol. 17

Issue (4) : 715-730 https://doi.org/10.1007/s11464-021-0926-5

RESEARCH ARTICLE

A generalization of silting modules and Tor-tilting modules

Lixin MAO(

)

Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, China

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Abstract

We introduce the concept of weak silting modules, which is a generalization of both silting modules and Tor-tilting modules. It is shown that W is a weak silting module if and only if its character module W⁺ is cosilting. Some properties of weak silting modules are given.

Keywords Silting module cosilting module weak silting module Tor-tilting module

Corresponding Author(s): Lixin MAO

Issue Date: 19 December 2022

Cite this article:

Lixin MAO. A generalization of silting modules and Tor-tilting modules[J]. Front. Math. China, 2022, 17(4): 715-730.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0926-5
https://academic.hep.com.cn/fmc/EN/Y2022/V17/I4/715

1	T Adachi , O Iyama , I Reiten . τ-tilting theory. Compos Math, 2014, 150: 415- 452 https://doi.org/10.1112/S0010437X13007422
2	F W Anderson , K R Fuller . Rings and Categories of Modules. New York: Springer, 1974 https://doi.org/10.1007/978-1-4684-9913-1
3	L Angeleri Hügel . On the abundance of silting modules. In: Martsinkovsky A, Igusa K, Todorov G, eds. Surveys in Representation Theory of Algebras. Contemp Math, Vol 716. Providence: Amer Math Soc, 2018 1- 23 https://doi.org/10.1090/conm/716/14424
4	L Angeleri Hügel , F U Coelho . Infinitely generated tilting modules of finite projective dimension. Forum Math, 2001, 13: 239- 250
5	L Angeleri Hügel , M Hrbek . Silting modules over commutative rings. Int Math Res Not IMRN, 2017, 13: 4131- 4151
6	L Angeleri Hügel , F Marks , J Vitória . Silting modules. Int Math Res Not IMRN, 2016, 4: 1251- 1284
7	S Breaz , F Pop . Cosilting modules. Algebr Represent Theory, 2017, 20: 1305- 1321 https://doi.org/10.1007/s10468-017-9688-x
8	R R Colby , K R Fuller . Tilting, cotilting, and serially tilted rings. Comm Algebra, 1990, 18 (5): 1585- 1615 https://doi.org/10.1080/00927879008823985
9	R Colpi , C Menini . On the structure of *-modules. J Algebra, 1993, 158: 400- 419 https://doi.org/10.1006/jabr.1993.1138
10	R Colpi , A Tonolo , J Trlifaj . Partial cotilting modules and the lattices induced by them. Comm Algebra, 1997, 25 (10): 3225- 3237 https://doi.org/10.1080/00927879708826050
11	R Colpi , J Trlifaj . Tilting modules and tilting torsion theories. J Algebra, 1995, 178: 492- 510 https://doi.org/10.1006/jabr.1995.1368
12	E E Enochs , O M G Jenda . Relative Homological Algebra. De Gruyter Exp Math, Vol 30. Berlin-New York: Walter de Gruyter, 2000 https://doi.org/10.1515/9783110803662
13	R M Fossum , P Griffith , I Reiten . Trivial Extensions of Abelian Categories: Homological Algebra of Trivial Extensions of Abelian Categories with Applications to Ring Theory. Lecture Notes in Math, Vol 456. Berlin: Springer-Verlag, 1975
14	R Göbel , J Trlifaj . Approximations and Endomorphism Algebras of Modules. De Gruyter Exp Math, Vol 41. Berlin-New York: Walter de Gruyter, 2006 https://doi.org/10.1515/9783110199727
15	D Happel , C Ringel . Tilted algebras. Trans Amer Math Soc, 1976, 215: 81- 98 https://doi.org/10.1090/S0002-9947-1976-0507068-3
16	P Krylov , A Tuganbaev . Formal Matrices. Algebr Appl, Vol 23. Cham: Springer, 2017 https://doi.org/10.1007/978-3-319-53907-2
17	F Marks , J Št'ovíček . Universal localisations via silting. Proc Roy Soc Edinburgh Sect A, 2019, 149: 511- 532 https://doi.org/10.1017/prm.2018.37
18	Y Miyashita . Tilting modules of finite projective dimension. Math Z, 1986, 193: 113- 146 https://doi.org/10.1007/BF01163359
19	R S Pierce . The global dimension of Boolean rings. J Algebra, 1967, 7: 91- 99 https://doi.org/10.1016/0021-8693(67)90069-5
20	F Pop . Finitely cosilting modules. arXiv: 1712.00817
21	Y J Yang , X G Yan , X S Zhu . Weak tilting modules. J Pure Appl Algebra, 2020, 224: 86- 97 https://doi.org/10.1016/j.jpaa.2019.04.016
22	P Y Zhang , J Q Wei . Cosilting complexes and AIR-cotilting modules. J Algebra, 2017, 491: 1- 31 https://doi.org/10.1016/j.jalgebra.2017.07.022
23	X X Zhang , L L Yao . On a generalization of tilting modules. Comm Algebra, 2009, 37: 4316- 4324 https://doi.org/10.1080/00927870902828983

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