18
views
0
recommends
+1 Recommend
0 collections
    0
    shares
      • Record: found
      • Abstract: found
      • Article: found
      Is Open Access

      K-flatness in Grothendieck categories: Application to quasi-coherent sheaves

      Preprint
      , ,

      Read this article at

      Bookmark
          There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

          Abstract

          Let \((\mathcal{G},\otimes)\) be any closed symmetric monoidal Grothendieck category. We show that K-flat covers exist universally in the category of chain complexes and that the Verdier quotient of \(K(\mathcal{G})\) by the K-flat complexes is always a well generated triangulated category. Under the further assumption that \(\mathcal{G}\) has a set of \(\otimes\)-flat generators we can show more: (i) The category is in recollement with the \(\otimes\)-pure derived category and the usual derived category, and (ii) The usual derived category is the homotopy category of a cofibrantly generated and monoidal model structure whose cofibrant objects are precisely the K-flat complexes. We also give a condition guaranteeing that the right orthogonal to K-flat is precisely the acyclic complexes of \(\otimes\)-pure injectives. We show this condition holds for quasi-coherent sheaves over a quasi-compact and semiseparated scheme.

          Related collections

          Author and article information

          Journal
          07 June 2023
          Article
          2306.04816
          4877a97b-4100-4e9c-bd3e-eac38ff78e3b

          http://creativecommons.org/licenses/by/4.0/

          History
          Custom metadata
          18N40, 18G35, 18G25
          18 pages
          math.AG math.AT math.CT math.KT

          General mathematics,Geometry & Topology
          General mathematics, Geometry & Topology

          Comments

          Comment on this article