2007/06/30 by Leovigildo Alonso, Leovigildo Alonso Tarrı́o, Ana Jeremı́as López +5 · 1 citation
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number #Algebraic structures and combinatorial models #Axiom #Coherent sheaf #Derived category #Functor #Geometry #Homotopy #Homotopy and Cohomology in Algebraic Topology #Homotopy category #Mathematical analysis #Mathematics #Noetherian #Pure mathematics #Scheme (mathematics) #Unital #math.AG #math.AT #msc:14F05 #msc:14F99 #msc:18E30
paper · pdf · doi:10.1016/j.aim.2008.03.011
published as Adv. Math. 218 (2008), no. 4, pp.1224-1252 · v2: 31 pages, some improvements in exposition; v3 updated bibliography, to appear Adv. Math
arxiv created 2008/04/22 · openalex publication_date 2008/04/23 · arxiv updated 2017/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We prove in this paper that for a quasi-compact and semi-separated (non necessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(Aqc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies Dqct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category of a usual scheme.