K-flat complexes and derived categories

TL;DR

This paper establishes an equivalence between the derived category of a ring and the homotopy category of K-flat complexes with pure-injective components, introducing the K-flat derived category and its relation to pure and usual derived categories.

Contribution

It introduces the K-flat derived category and shows its relation to the pure derived category, using cotorsion pairs and model structures, providing new insights into derived categories of rings.

Findings

The derived category is equivalent to the homotopy category of K-flat complexes with pure-injective components.

The K-flat derived category is shown to be a compactly generated triangulated category.

A new characterization of K-flat complexes in terms of the pure derived category is provided.

Abstract

Let $R$ be a ring with identity. Inspired by recent work of Emmanouil, we show that the derived category of $R$ 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 $\mathcal{D}_{pur}(R)$, the pure derived category of $R$, as an attachment of the usual derived category $\mathcal{D}(R)$ with Emmanouil's quotient category $\mathcal{D}_{K-flat}(R):=K(R)/K-Flat$, which here we call 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 $\mathcal{C}$-acyclic complexes, where $\mathcal{C}$ 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 $R$.