Y. The coherence, now, follows from Sect. 6. The case of an arbitrary proper morphism is reduced to a projective morphism by the following trick.
2, Sect. 6). 4. A Theorem on Affine Coverings. The most important consequence of Serre's theorem is that it enables us to calculate the cohomology of quasi-coherent sheaves with the help of arbitrary affine coverings. Theorem. Let X be a separated scheme, U = (Ui) an openaffine covering of X, and F a quasi-coherent sheal on X. Then H*(X, F) = H*(U, F). Indeed, since X is separated, all the intersections are affine. By Serre's theorem, the covering U is F-acyclic, and the theorem follows from (Chap.