Deriving Auslander's formula

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.

2010 Mathematics Subject Classification: Primary 18E30; Secondary 16E35, 18C35, 18E15.

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