Article
| Title: | Cotorsion pairs in comma categories (English) |
| Author: | Yuan, Yuan |
| Author: | He, Jian |
| Author: | Wu, Dejun |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 3 |
| Year: | 2024 |
| Pages: | 715-734 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $\mathcal {A}$ and $\mathcal {B}$ be abelian categories with enough projective and injective objects, and $T \colon \mathcal {A}\rightarrow \mathcal {B}$ a left exact additive functor. Then one has a comma category $(\mathopen {\mathcal {B} \downarrow T})$. It is shown that if $T \colon \mathcal {A}\rightarrow \mathcal {B}$ is $\mathcal {X}$-exact, then $(^\bot \mathcal {X}, \mathcal {X})$ is a (hereditary) cotorsion pair in $\mathcal {A}$ and $(^\bot \mathcal {Y}, \mathcal {Y})$) is a (hereditary) cotorsion pair in $\mathcal {B}$ if and only if $\bigl (\binom {^\bot \mathcal {X}}{^\bot \mathcal {Y}} \bigr ), \langle {\bf h}(\mathcal {X}, \mathcal {Y})\rangle )$ is a (hereditary) cotorsion pair in $(\mathopen {\mathcal {B}\downarrow T})$ and $\mathcal {X}$ and $\mathcal {Y}$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $\mathcal {A}$ and $\mathcal {B}$ can induce special preenveloping classes in $(\mathopen {\mathcal {B}\downarrow T})$. (English) |
| Keyword: | comma category |
| Keyword: | cocompatible functor |
| Keyword: | cotorsion pair |
| MSC: | 16E30 |
| MSC: | 18A25 |
| MSC: | 18G25 |
| DOI: | 10.21136/CMJ.2024.0420-23 |
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| Date available: | 2024-10-03T12:34:29Z |
| Last updated: | 2024-10-04 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152577 |
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| Reference: | [1] Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Volume 1. Techniques of Representation Theory.London Mathematical Society Student Texts 65. Cambridge University Press, Cambridge (2006). Zbl 1092.16001, MR 2197389, 10.1017/CBO9780511614309 |
| Reference: | [2] Chen, X.-W., Le, J.: Recollements, comma categories and morphic enhancements.Proc. R. Soc. Edinb., Sect. A, Math. 152 (2022), 567-591. Zbl 1497.18021, MR 4430943, 10.1017/prm.2021.8 |
| Reference: | [3] Chen, X.-W., Shen, D., Zhou, G.: The Gorenstein-projective modules over a monomial algebra.Proc. R. Soc. Edinb., Sect. A, Math. 148 (2018), 1115-1134. Zbl 1403.16008, MR 3869172, 10.1017/S0308210518000185 |
| Reference: | [4] Fossum, R. M., Griffith, P. A., Reiten, I.: Trivial Extensions of Abelian Categories: Homological Algebra of Trivial Extensions of Abelian Categories With Applications to Ring Theory.Lecture Notes in Mathematics 456. Springer, Berlin (1975). Zbl 0303.18006, MR 389981, 10.1007/bfb0065404 |
| Reference: | [5] Gabriel, P., Roiter, A. V.: Representations of Finite-Dimensional Algebras.Encyclopaedia of Mathematical Sciences 73. Springer, Berlin (1992). Zbl 0839.16001, MR 1239447 |
| Reference: | [6] Göbel, R., Trlifaj, J.: Approximations and Endomorphism Algebras of Modules. Volume 2. Predictions.De Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin (2012). Zbl 1292.16001, MR 2985654, 10.1515/9783110218114 |
| Reference: | [7] Hovey, M.: Cotorsion pairs and model categories.Interactions Between Homotopy Theory and Algebra Contemporary Mathematics 436. AMS, Providence (2007), 277-296. Zbl 1129.18004, MR 2355778, 10.1090/conm/436 |
| Reference: | [8] Hu, J., Zhu, H.: Special precovering classes in comma categories.Sci. China, Math. 65 (2022), 933-950. Zbl 1485.18001, MR 4412784, 10.1007/s11425-020-1790-9 |
| Reference: | [9] Kalck, M.: Singularity categories of gentle algebras.Bull. Lond. Math. Soc. 47 (2015), 65-74. Zbl 1323.16012, MR 3312965, 10.1112/blms/bdu093 |
| Reference: | [10] Marmaridis, N.: Comma categories in representation theory.Commun. Algebra 11 (1983), 1919-1943. Zbl 0518.16011, MR 709023, 10.1080/00927878308822941 |
| Reference: | [11] Salce, L.: Cotorsion theories for abelian groups.Symposia Mathematica. Volume 23 Academic Press, London (1979), 11-32. Zbl 0426.20044, MR 0565595 |
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