Cohomology of Coherent Sheaves

Abstract

In this chapter, we continue the study of coherent sheaves, by studying their cohomology. The first key result is that the higher cohomology groups for coherent sheaves vanish for affine schemes. Using this we can compute cohomology for projective spaces using the Čech complex for the standard open affine cover, and establish finite-dimensionality and other basic results. We also consider analogous statements for complex manifolds. With these results in hand, we complete our discussion of the GAGA theorems. The second basic result is that if 蒖 is a coherent algebraic sheaf on ℙⁿ _ℂ, its cohomology is isomorphic to the cohomology of 蒖_an. Thus the calculation of the latter reduces to a purely algebraic problem. This cohomological result is also needed for the proof of the first GAGA theorem stated in the previous chapter.