K‐flat complexes and derived categories

Abstract

Let R $R$ be a ring with identity. Inspired by recent work in Emmanouil, Preprint, 2021, we show that the derived category of R $R$ is equivalent to the chain homotopy category of all K-flat complexes with pure-injective components. This is implicitly related to a recollement we exhibit. It expresses D p u r ( R ) $\mathcal {D}_{pur}(R)$ , the pure derived category of R $R$ , as an attachment of the usual derived category D ( R ) $\mathcal {D}(R)$ with Emmanouil's quotient category D K − f l a t ( R ) : = K ( R ) / K − F l a t $\mathcal {D}_{\textnormal {K-flat}}(R):=K(R)/K\textnormal {-Flat}$ , which we call here the K-flat derived category. It follows that this Verdier quotient is a compactly generated triangulated category. We obtain our results by using methods of cotorsion pairs to construct (cofibrantly generated) monoidal abelian model structures on the exact category of chain complexes along with the degreewise pure exact structure. In fact, most of our model structures are obtained as corollaries of a general method which associates an abelian model structure to any class of so-called C $\mathcal {C}$ -acyclic complexes, where C $\mathcal {C}$ is any given class of chain complexes. Finally, we also give a new characterization of K-flat complexes in terms of the pure derived category of R $R$ .