2014/06/10 by James Gillespie, Gillespie, James · 1 citation
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1406.2514
openalex publication_date 2014/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a Grothendieck category G along with a choice of generator G, or equivalently a generating set \Gi\. We introduce the derived category D(G), which kills all G-acyclic complexes, by putting a suitable model structure on the category of chain complexes. It follows that the category D(G) is always a well-generated triangulated category. It is compactly generated whenever the generating set \Gi\ has each Gi finitely presented, and in this case we show that two recollement situations hold. The first is when passing from the homotopy category K(G) to D(G). The second is a G-derived analog to the recollement of Krause. We illustrate with several examples ranging from pure and clean derived categories to quasi-coherent sheaves on the projective line P1(k).