The $Q$-shaped derived category of a ring

TL;DR

This paper introduces the $Q$-shaped derived category of a ring, generalizing classical derived categories by constructing model structures on functor categories with specific properties.

Contribution

It constructs and characterizes a new class of triangulated categories called $Q$-shaped derived categories, extending previous work to broader categorical contexts.

Findings

Established projective and injective model structures on functor categories

Characterized weak equivalences via (co)homology functors

Generalized the classical derived category framework

Abstract

For any ring $A$ and a small, preadditive, Hom-finite, and locally bounded category $Q$ that has a Serre functor and satisfies the (strong) retraction property, we show that the category of additive functors from $Q$ to the category of (left) $A$-modules 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 $Q$-shaped derived category of $A$. The usual derived category of $A$ is one example; more general examples arise by taking $Q$ to be the mesh category of a suitably nice stable translation quiver. This paper builds upon, and generalizes, works of Enochs, Estrada, and Garcia-Rozas and of Dell'Ambrogio, Stevenson, and Stovicek.