2021/01/15 by Henrik Holm, Holm, Henrik, Peter Jørgensen +2 · 3 citations
Mathematics · #16E35 #18E30 #18E35 #18G55 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #math.CT #math.RT #msc:16E35 #msc:18E30 #msc:18E35 #msc:18G55
paper · pdf · doi:10.48550/arxiv.2101.06176
43 pages
arxiv created 2021/01/15 · openalex publication_date 2021/01/15 · arxiv updated 2021/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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.