1. Categories of quiver representations and relative cotorsion pairs.
- Author
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Argudín-Monroy, Alejandro and Mendoza-Hernández, Octavio
- Subjects
- *
CARDINAL numbers - Abstract
We study the category Rep (Q , C) of representations of a quiver Q with values in an abelian category C. For this purpose we introduce the mesh and the cone-shape cardinal numbers associated to the quiver Q and we use them to impose conditions on C that allow us to prove interesting homological properties of Rep (Q , C) that can be constructed from C. For example, we compute the global dimension of Rep (Q , C) in terms of the global one of C. We also review a result of H. Holm and P. Jørgensen which states that (under certain conditions on C) every hereditary complete cotorsion pair (A , B) in C induces the hereditary complete cotorsion pairs (Rep (Q , A) , Rep (Q , A) ⊥ 1 ) and (Ψ ⊥ 1 (B) , Ψ (B)) in Rep (Q , C) , and then we obtain a strengthened version of this and other related results. Finally, we will apply the above developed theory to study the following full abelian subcategories of Rep (Q , C) , finite-support, finite-bottom-support and finite-top-support representations. We show that the above mentioned cotorsion pairs in Rep (Q , C) can be restricted nicely on the aforementioned subcategories and under mild conditions we also get hereditary complete cotorsion pairs. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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