Brown representability for stable categories

This paper proves Brown Representability for stable categories. These categories are obtained from additive categories by declaring all morphisms which factor through objects from a fixed pre-enveloping class to be zero. Brown Representability for such categories is obtained by first proving that stable categories are what Brown in [2] calls ‘‘abstract homotopy categories’’, next using Brown’s theory for these categories. The result is applied to two cases: one is a category of complexes of modules over a ring. Here we recover a representability theorem for functors on derived categories, first given by Neeman in [11, Thm. 3.1]. The other case is the category of modules over an Artinian ring. Here we obtain a result on representability, Theorem 4.4, which appears to be new.