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I am looking for the explicit reference to the fact that for a quasi-projective scheme a class in the étale cohomology of a sheaf of a certain degree can by computed using Cech cocycles.

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This result is due to Artin and can be found in the paper "On the Joins of Hensel Rings", Corollary 4.2. There the assumption on X is that it is Noetherian and every finite set of points is contained in an affine. If X is quasi-projective over a Noetherian ring, then this is true (as Artin points out in the paper).

There is no good definition of a quasi-projective scheme without reference to a base scheme. Maybe you mean: a quasi-compact and separated scheme X which has an ample invertible sheaf? In this case Artin's result implies it for X by standard limit techniques.

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See [Milne], Étale Cohomology, Theorem III.2.17.

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