Kaplansky classes and derived categories

Abstract

We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semi-separated scheme X. The approach generalizes and simplifies the method used by the author in (Trans Am Math Soc 356(8) 3369–3390, 2004) and (Trans Am Math Soc 358(7), 2855–2874, 2006) to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in any Grothendieck category $\mathcal{G}$, any nice enough class of objects $\mathcal{F}$ induces a model structure on the category Ch($\mathcal{G}$) of chain complexes. The main technical requirement on $\mathcal{F}$ is the existence of a regular cardinal κ such that every object $F \in \mathcal{F}$ satisfies the following property: Each κ-generated subobject of F is contained in another κ-generated subobject S for which $S, F/S \in \mathcal{F}$. Such a class $\mathcal{F}$ is called a Kaplansky class. Kaplansky classes first appeared in Enochs and López-Ramos (Rend Sem Mat Univ Padova 107, 67–79, 2002) in the context of modules over a ring R. We study in detail the connection between Kaplansky classes and model categories. We also find simple conditions to put on $\mathcal{F}$ which will guarantee that our model structure is monoidal. We will see that in several categories the class of flat objects form such Kaplansky classes, and hence induce monoidal model structures on the associated chain complex categories. We will also see that in any Grothendieck category $\mathcal{G}$, the class of all objects is a Kaplansky class which induces the usual (non-monoidal) injective model structure on Ch($\mathcal{G}$).