Étale Cohomology Seminar Lecture 4

In general, the sheaf criterion on the étale topology may be difficult to verify directly, as a scheme will in general have many étale covers. It is clear that a necessary condition for a presheaf F to be a sheaf on Xet is that it be a sheaf with respect to Zariski covers (i.e., its restriction to Xzar is a sheaf), and that it be a sheaf with respect to one-piece étale covers (V → U) such that V and U are affine. We have formulated this awkward necessary condition because, in fact, it is also sufficient. To illustrate the flexibility of this result will show this in a slightly more general setting, the fpqc topology. The objects of the site Xfpqc are maps U → X that are flat and locally quasicompact, and covers are flat, locally quasicompact, and jointly surjective families of morphisms. The proof we give will immediately imply the result for the étale site.