Deriving Auslander's formula

Abstract

Auslander's formula shows that any abelian category $\mathsf{C}$ is equivalent to the category of coherent functors on $\mathsf{C}$ modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to showing that the homotopy category of injective objects of some appropriate Grothendieck abelian category (the category of ind-objects of $\mathsf{C}$) is compactly generated and that the full subcategory of compact objects is equivalent to the bounded derived category of $\mathsf{C}$. The same approach shows for an arbitrary Grothendieck abelian category that its derived category and the homotopy category of injective objects are well-generated triangulated categories. For sufficiently large cardinals $\alpha$ we identify their $\alpha$-compact objects and compare them.