In this chapter we write about cohomology of algebraic spaces. Although we prove some results on cohomology of abelian sheaves, we focus mainly on cohomology of quasi-coherent sheaves, i.e., we prove analogues of the results in the chapter “Cohomology of Schemes”. Some of the results in this chapter can be found in [Kn].
An important missing ingredient in this chapter is the induction principle, i.e., the analogue for quasi-compact and quasi-separated algebraic spaces of Cohomology of Schemes, Lemma 30.4.1. This is formulated precisely and proved in detail in Derived Categories of Spaces, Section 74.9. Instead of the induction principle, in this chapter we use the alternating Čech complex, see Section 68.6. It is designed to prove vanishing statements such as Proposition 68.7.2, but in some cases the induction principle is a more powerful and perhaps more “standard” tool. We encourage the reader to take a look at the induction principle after reading some of the material in this section.
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