Deriving Auslander's formula

Abstract

Auslander's formula shows that any abelian category C is equivalent to the category of coherent functors on C modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to showing that the homo- topy category of injective objects of some appropriate Grothendieck abelian category (the category of ind-objects of C) is compactly gener- ated and that the full subcategory of compact objects is equivalent to the bounded derived category of C. The same approach shows for an arbitrary Grothendieck abelian category that its derived category and the homotopy category of injective objects are well-generated trian- gulated categories. For sufficiently large cardinalswe identify their �-compact objects and compare them.