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

• Estrada, Sergio ^[1] ; Odabaşi, Sinem ^[1] ; Gillespie, James ^[2]
  1. [1] Universidad de Murcia

     Universidad de Murcia

     Murcia, España

     
  2. [2] Ramapo College of New Jersey School of Theoretical and Applied Science, 505 Ramapo Valley Road, Mahwah, NJ, 07430, USA
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 76, Fasc. 2, 2025, págs. 435-454
• Idioma: inglés
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• Resumen

  • 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.