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

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.