A Quillen Approach to Derived Categories and Tensor Products

We put a monoidal model category structure (in the sense of Quillen) on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semi-separated scheme X. The approach generalizes and simpli es the method used by the author in [Gil04] and [Gil06] 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 G, a nice enough class of objects F (which we call a Kaplansky class) induces a model structure on the category Ch(G) of chain complexes. We also nd simple conditions to put on F which will guarantee that our model structure in monoidal. We see that the common model structures used in practice are all induced by such Kaplansky classes.