Geodesic
Cotorsion pairs in comma categories
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 715-734
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $(\mathopen {\mathcal {B}\downarrow T})$.
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 $(\mathopen {\mathcal {B}\downarrow T})$.
DOI : 10.21136/CMJ.2024.0420-23
Classification : 16E30, 18A25, 18G25
Keywords: comma category; cocompatible functor; cotorsion pair 
@article{10_21136_CMJ_2024_0420_23,
     author = {Yuan, Yuan and He, Jian and Wu, Dejun},
     title = {Cotorsion pairs in comma categories},
     journal = {Czechoslovak Mathematical Journal},
     pages = {715--734},
     year = {2024},
     volume = {74},
     number = {3},
     doi = {10.21136/CMJ.2024.0420-23},
     mrnumber = {4804956},
     zbl = {07953674},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2024.0420-23/}
}
TY  - JOUR
AU  - Yuan, Yuan
AU  - He, Jian
AU  - Wu, Dejun
TI  - Cotorsion pairs in comma categories
JO  - Czechoslovak Mathematical Journal
PY  - 2024
SP  - 715
EP  - 734
VL  - 74
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2024.0420-23/
DO  - 10.21136/CMJ.2024.0420-23
LA  - en
ID  - 10_21136_CMJ_2024_0420_23
ER  - 
%0 Journal Article
%A Yuan, Yuan
%A He, Jian
%A Wu, Dejun
%T Cotorsion pairs in comma categories
%J Czechoslovak Mathematical Journal
%D 2024
%P 715-734
%V 74
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2024.0420-23/
%R 10.21136/CMJ.2024.0420-23
%G en
%F 10_21136_CMJ_2024_0420_23
Yuan, Yuan; He, Jian; Wu, Dejun. Cotorsion pairs in comma categories. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 715-734. doi: 10.21136/CMJ.2024.0420-23

[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). | DOI | MR | JFM

[2] Chen, X.-W., Le, J.: Recollements, comma categories and morphic enhancements. Proc. R. Soc. Edinb., Sect. A, Math. 152 (2022), 567-591. | DOI | MR | JFM

[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. | DOI | MR | JFM

[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). | DOI | MR | JFM

[5] Gabriel, P., Roiter, A. V.: Representations of Finite-Dimensional Algebras. Encyclopaedia of Mathematical Sciences 73. Springer, Berlin (1992). | MR | JFM

[6] Göbel, R., Trlifaj, J.: Approximations and Endomorphism Algebras of Modules. Volume 2. Predictions. De Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin (2012). | DOI | MR | JFM

[7] Hovey, M.: Cotorsion pairs and model categories. Interactions Between Homotopy Theory and Algebra Contemporary Mathematics 436. AMS, Providence (2007), 277-296. | DOI | MR | JFM

[8] Hu, J., Zhu, H.: Special precovering classes in comma categories. Sci. China, Math. 65 (2022), 933-950. | DOI | MR | JFM

[9] Kalck, M.: Singularity categories of gentle algebras. Bull. Lond. Math. Soc. 47 (2015), 65-74. | DOI | MR | JFM

[10] Marmaridis, N.: Comma categories in representation theory. Commun. Algebra 11 (1983), 1919-1943. | DOI | MR | JFM

[11] Salce, L.: Cotorsion theories for abelian groups. Symposia Mathematica. Volume 23 Academic Press, London (1979), 11-32. | MR | JFM

Cité par Sources :