Narrow Results

Let ${ℳ}$ be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in ${ℳ}$. This is based on recent work of Holm and Jørgensen on cotorsion pairs in categories of quiver representations. An application on Ding projective and Ding injective representations of quivers over Ding–Chen rings is given.

Let R be an associative ring with identity. Denote by ((R-mod)^op, Ab) the category consisting of contravariant functors from the category of finitely presented left R-modules R-mod to the category of abelian groups Ab. An object in ((R-mod)^op, Ab) is said to be a stable functor if it vanishes on the regular module R. Let $\mathcal{T}$ be the subcategory of stable functors. There are two torsion pairs $t_1=\left(\text{Gen}\left(-,R\right),\mathcal{T}\right)$ and $t_2=\left(\mathcal{T},{ℱ}_1\right)$, where ℱ₁ is the subcategory of ((R-mod)^op, Ab) consisting of functors with flat dimension at most 1. In this article, let R be a ring of weakly global dimension at most 1, and assume R satisfies that for any exact sequence 0 → M → N → K → 0, if M and N are pure injective, then K is also pure injective. We calculate the cotorsion pair $\left({}^{\perp }\mathcal{T},{\left({}^{\perp }\mathcal{T}\right)}^{\perp }\right)$ cogenerated by $\mathcal{T}$ clearly. It is shown that $G\in {}^{\perp }\mathcal{T}$ if and only if G/t₁(G) is a projective object in $\mathcal{T}$, i.e., G/t₁(G) = (−,M) for some R-module M; and $G\in {\left({}^{\perp }\mathcal{T}\right)}^{\perp }$ if and only if G/t₂(G) is of the form (−, E), where E is an injective R-module.

It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian. In this article, we consider a special class of hearts of twin cotorsion pairs induced by $d$-cluster tilting subcategories in extriangulated categories. We give a necessary and sufficient condition for such hearts to be abelian. In particular, we can also see that such hearts are hereditary. As an application, this generalizes the work by Liu in the exact case, thereby providing new insights into the triangulated case.