Indization and Derivation of Abelian Categories

Abstract

In this chapter we study the derived category D^b(Ind(C)) of the category of ind-objects of the abelian category C. The main difficulty comes from the fact that, as we shall see, the category Ind(C) does not have enough injectives in general. This difficulty is partly overcome by introducing the weaker notion of “quasi-injective objects”, and these objects are sufficient to derive functors on Ind(C) which are indization of functors on C.

As a byproduct, we shall give a sufficient condition which ensures that the right derived functor of a left exact functor commutes with small filtrant inductive limits.

Finally, we study the relations between D^b(Ind(C)) and the category Ind(D^b(C)) of ind-objects of D^b(C).