The flat model structure on complexes of sheaves

Abstract

Let C h ( O ) \mathbf {Ch}(\mathcal {O}) be the category of chain complexes of O \mathcal {O} -modules on a topological space T T (where O \mathcal {O} is a sheaf of rings on T T ). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on C h ( O ) \mathbf {Ch}(\mathcal {O}) . As a corollary, we have a general framework for doing homological algebra in the category S h ( O ) \mathbf {Sh}(\mathcal {O}) of O \mathcal {O} -modules. I.e., we have a natural way to define the functors Ext \operatorname {Ext} and Tor \operatorname {Tor} in S h ( O ) \mathbf {Sh}(\mathcal {O}) .