Faithfully flat covering

We also have a good many coverings which are not Zariski coverings— consider the case R = k. But the analogy is close enough that we say a functor F satisfying (a) and (b) is a sheaf in the faithfully flat or fpqc topology. 

We prove the if part. We may assume that Y is affine. So X is quasicompact, and has a finite affine open covering (f/j). Replacing X by we may assume that X is also affine. Thus / is faithfully flat concentrated. If f M is quasi-coherent, then = L M satisfies the assumption of 3 of the proposition, as can be seen easily. So A4 = Rf)- is quasi-coherent. ... 

Next consider the general case. As X is quasi-compact, there is a finite affine open covering Ui) of X. Set Y = Llj lA, and let p T —> X be the canonical map. Note that p is locally an open immersion and faithfully flat. Let X, = Nerve(p) AM-... 

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