The Q$Q$‐shaped derived category of a ring.

For any ring A$A$ and a small, pre‐additive, Hom‐finite, and locally bounded category Q$Q$ that has a Serre functor and satisfies the (strong) retraction property, we show that the category of additive functors Q→AMod$Q \rightarrow {}_{A} \operatorname{Mod}$ has a projective and an injective model structure. These model structures have the same trivial objects and weak equivalences, which in most cases can be naturally characterized in terms of certain (co)homology functors introduced in this paper. The associated homotopy category, which is triangulated, is called the Q$Q$‐shaped derived category of A$A$. The usual derived category of A$A$ is one example; more general examples arise by taking Q$Q$ to be the mesh category of a suitably nice stable translation quiver. This paper builds upon, and generalizes, works of Enochs, Estrada, and García‐Rozas (Math. Nachr. 281 (2008), no. 4, 525–540) and Dell'Ambrogio, Stevenson, and Šťovíček (Math. Z. 287 (2017), no. 3‐4, 1109–1155). [ABSTRACT FROM AUTHOR]