Cotorsion pairs and model categories

Mark Hovey 
Wesleyan University   
mhovey@wesleyan.edu   

This paper is an expanded version of two talks given by the author at
the Summer School on the Interactions between Homotopy Theory and
Algebra at the University of Chicago, July 26 to August 6, 2004.  It
concerns the relationship between Quillen model structures on abelian
categories and cotorsion pairs, an algebraic notion that
simultaneously generalizes the notion of projective and injective
objects.  In brief, a model category structure on an abelian category
A that respects the abelian structure in a simple way
is equivalent to two compatible complete cotorsion pairs on A.  

We describe the author's own work on this and also that of Jim
Gillespie.  Gillespie has proved a general theorem about promoting a
cotorsion pair on an abelian category to a model structure on chain
complexes over that category.  When applied to quasi-coherent sheaves,
it produces a model structure compatible with the tensor product of
chain complexes of sheaves.  The existence of the derived tensor product
on a nice enough scheme and its expected properties now follow formally
from this model structure.