Czechoslovak Mathematical Journal, Vol. 73, No. 3, pp. 695-714, 2023
Finitely silting comodules in quasi-finite comodule category
Qianqian Yuan, Hailou Yao
Received April 21, 2022. Published online June 7, 2023.
Abstract: We introduce the notions of silting comodules and finitely silting comodules in quasi-finite category, and study some properties of them. We investigate the torsion pair and dualities which are related to finitely silting comodules, and give the equivalences among silting comodules, finitely silting comodules, tilting comodules and finitely tilting comodules.
References: [1] T. Adachi, O. Iyama, I. Reiten: $\tau$-tilting theory. Compos. Math. 150 (2014), 415-452. DOI 10.1112/S0010437X13007422 | MR 3187626 | Zbl 1330.16004
[2] K. Al Takhman: Equivalences of comodule categories for coalgebras over rings. J. Pure Appl. Algebra 173 (2002), 245-271. DOI 10.1016/S0022-4049(02)00013-0 | MR 1916479 | Zbl 1004.16039
[3] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules. Graduate Texts in Mathematics 13. Springer, New York (1974). DOI 10.1007/978-1-4612-4418-9 | MR 0417223 | Zbl 0301.16001
[4] L. Angeleri Hügel, M. Hrbek: Silting modules over commutative rings. Int. Math. Res. Not. 2017 (2017), 4131-4151. DOI 10.1093/imrn/rnw147 | MR 3671512 | Zbl 1405.13018
[5] L. Angeleri Hügel, F. Marks, J. Vitória: Silting modules. Int. Math. Res. Not. 2016 (2016), 1251-1284. DOI 10.1093/imrn/rnv191 | MR 3493448 | Zbl 1367.16005
[6] R. R. Colby, K. R. Fuller: Equivalence and Duality for Module Categories: With Tilting and Cotilting for Rings. Cambridge Tracts in Mathematics 161. Cambridge University Press, Cambridge (2004). DOI 10.1017/CBO9780511546518 | MR 2048277 | Zbl 1069.16001
[7] R. Colpi, J. Trlifaj: Tilting modules and tilting torsion theories. J. Algebra 178 (1995), 614-634. DOI 10.1006/jabr.1995.1368 | MR 1359905 | Zbl 0849.16033
[8] Y. Doi: Homological coalgebra. J. Math. Soc. Japan 33 (1981), 31-50. DOI 10.2969/jmsj/03310031 | MR 0597479 | Zbl 0459.16007
[9] B. Keller, D. Vossieck: Aisles in derived categories. Bull. Soc. Math. Belg., Sér. A 40 (1988), 239-253. MR 0976638 | Zbl 0671.18003
[10] H. Krause, M. Saorín: On minimal approximations of modules. Trends in the Representation Theory of Finite Dimensional Algebras. Contemporary Mathematics 229. AMS, Providence (1998), 227-236. DOI 10.1090/conm/229 | MR 1676223 | Zbl 0959.16003
[11] B. I. Lin: Semiperfect coalgebras. J. Algebra 49 (1977), 357-373. DOI 10.1016/0021-8693(77)90246-0 | MR 0498663 | Zbl 0369.16010
[12] L. Positselski: Dedualizing complexes of bicomodules and MGM duality over coalgebras. Algebr. Represent. Theory 21 (2018), 737-767. DOI 10.1007/s10468-017-9736-6 | MR 3826725 | Zbl 1394.16040
[13] D. Simsom: Coalgebras, comodules, pseudocompact algebras and tame comodule type. Colloq. Math. 90 (2001), 101-150. DOI 10.4064/cm90-1-9 | MR 1874368 | Zbl 1055.16038
[14] D. Simson: Cotilted coalgebras and tame comodule type. Arab. J. Sci. Eng., Sect. C, Theme Issues 33 (2008), 421-445. MR 2500051 | Zbl 1186.16039
[15] M. Takeuchi: Morita theorems for categories of comodules. J. Fac. Sci., Univ. Tokyo, Sect. I A 24 (1977), 629-644. MR 0472967 | Zbl 0385.18007
[16] M.-Y. Wang: Some co-hom functors and classical tilting comodules. Southeast Asian Bull. Math. 22 (1998), 455-468. MR 1811188 | Zbl 0942.16047
[17] M. Wang: Tilting comodules over semi-perfect coalgebras. Algebra Colloq. 6 (1999), 461-472. MR 1809680 | Zbl 0945.16034
[18] M. Wang: Morita Equivalence and Its Generalizations. Science Press, Beijing (2001).
[19] Q. Q. Yuan, H.-L. Yao: On silting comodules. J. Shandong. Univ. 57 (2022), 1-7. DOI 10.6040/j.issn.1671-9352.0.2021.503
Affiliations: Qianqian Yuan, Hailou Yao (corresponding author), Department of Mathematics, Faculty of Science, Beijing University of Technology, Beijing 100124, P. R. China, e-mail: qqy94824@emails.bjut.edu.cn, yaohl@bjut.edu.cn
