The derived category with respect to a generator

Abstract

Let \({\mathcal {G}}\) be any Grothendieck category along with a choice of generator \(G\), or equivalently a generating set \(\{G_i\}\). We introduce the derived category \({\mathcal {D}}(G)\), which kills all \(G\)-acyclic complexes, by putting a suitable model structure on the category \(\text {Ch}({\mathcal {G}})\) of chain complexes. It follows that the category \({\mathcal {D}}(G)\) is always a well-generated triangulated category. It is compactly generated whenever the generating set \(\{G_i\}\) has each \(G_i\) finitely presented, and in this case, we show that two recollement situations hold. The first is when passing from the homotopy category \(K({\mathcal {G}})\) to \({\mathcal {D}}(G)\). The second is a \(G\)-derived analog of a recollement due to Krause. We describe several examples ranging from pure and clean derived categories to quasi-coherent sheaves on the projective line.