Cotorsion pairs and model categories

The purpose of this paper is to describe a connection between model categories, a structure invented by algebraic topologists that allows one to introduce the ideas of homotopy theory to situations far removed from topological spaces, 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. This connection enables one to interpret results about cotorsion pairs in terms of homotopy theory (for example, the flat cover conjecture [4] can be thought of as the search for a suitable cofibrant replacement), and vice versa. Besides describing this connection, we also indicate some applications. The stable module category of a finite group G over a field k is a basic object of study in modular representation theory; it is a triangulated category because injective and projective k[G]-modules coincide. Cotorsion pairs can be used to construct two different model structures on the category of K[G]-modules where K is a commutative Gorenstein ring (such as Z, for