Algebraic Geometry II: Cohomology of Algebraic Varieties. by V. I. Danilov (auth.), I. R. Shafarevich (eds.)

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.

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