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

Let (G,⊗ ) ( G , ⊗ ) 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(G) K ( G ) by the K-flat complexes is always a well generated triangulated category. Under the further assumption that G G has a set of ⊗ ⊗ -flat generators we can show more: (i) The category is in recollement with the ⊗ ⊗ -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 ⊗ ⊗ -pure injectives. We show this condition holds for quasi-coherent sheaves over a quasi-compact and semiseparated scheme.