Deriving Auslander's formula

TL;DR

This paper extends Auslander's formula to derived categories, showing an equivalence between the homotopy category of injectives and the bounded derived category for certain abelian categories.

Contribution

It establishes a derived version of Auslander's formula, connecting homotopy categories of injectives with bounded derived categories in Grothendieck abelian categories.

Findings

Homotopy category of injectives is compactly generated.

Full subcategory of compact objects is equivalent to bounded derived category.

Derived and homotopy categories are well-generated triangulated categories.

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 homotopy category of injective objects of some appropriate Grothendieck abelian category (the category of ind-objects of C) is compactly generated 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 triangulated categories. For sufficiently large cardinals alpha we identify their alpha-compact objects and compare them.