Derived Categories of Stacks

In this chapter we write about derived categories associated to algebraic stacks. This means in particular derived categories of quasi-coherent sheaves, i.e., we prove analogues of the results on schemes (see Derived Categories of Schemes, Section 1) and algebraic spaces (see Derived Categories of Spaces, Section 1). The results in this chapter are different from those in [LMB00] mainly because we consistently use the “big sites”. Before reading this chapter please take a quick look at the chapters “Sheaves on Algebraic Stacks” and “Cohomology of Algebraic Stacks” where the terminology we use here is introduced.