Cotorsion pairs in categories of quiver representations

Abstract

We study the category $\mathrm{Rep}(Q,\mathcal{M})$ of representations of a quiver $Q$ with values in an abelian category $\mathcal{M}$. Under certain assumptions, we show that every cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces two (explicitly described) cotorsion pairs $(\Phi (\mathcal{A}),\mathrm{Rep}(Q,\mathcal{B}))$ and $(\mathrm{Rep}(Q,\mathcal{A}),\Psi (\mathcal{B}))$ in $\mathrm{Rep}(Q,\mathcal{M})$. This is akin to a result by Gillespie, which asserts that a cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces cotorsion pairs $(\tilde{\mathcal{A}},\mathrm{dg}\tilde{\mathcal{B}})$ and $(\mathrm{dg}\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ in the category $\mathrm{Ch}(\mathcal{M})$ of chain complexes in $\mathcal{M}$. Special cases of our results recover descriptions of the projective and injective objects in $\mathrm{Rep}(Q,\mathcal{M})$ proved by Enochs, Estrada, and Garcia Rozas.

23 pages. This is the Final Accepted Version which was accepted 13 April 2017 for publication in the Kyoto Journal of Mathematics

References in corpus (1)

• On exact categories and applications to triangulated adjoints and model structures

Cited by in corpus (4)

• Cotorsion torsion triples and the representation theory of filtered hierarchical clustering
• Homological theory of representations having pure acyclic injective resolutions
• Flat model structures and Gorenstein objects in functor categories
• Model structures on the category of complexes of quiver representations