In this paper, we try to realize the unbounded derived category of an
abelian category as the homotopy category of a Quillen model structure
on the category of unbounded chain complexes. We construct such a model
structure based on injective resolutions for an arbitrary Grothendieck
category, as has apparently also been done by Morel. In particular,
this works for sheaves on a ringed space, and for quasi-coherent sheaves
on a quasi-compact, quasi-separated scheme. However, this injective
model structure is not well suited to studying the derived tensor
product, so we investigate other model structures. The most successful
of these is the flat model structure on complexes of sheaves over a
ringed space. This is based on flat resolutions, and is compatible with
the tensor product. As a corollary, we get model categories of
differential graded algebras of sheaves and differential graded modules
over a given differential graded algebra of sheaves.
This is the author's first attempt to understand sheaves, so comments
from those more experienced with the subject are welcome.
This is the author's first attempt to understand sheaves, so comments from those more experienced with the subject are welcome.