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Dec 4, 2017 at 20:38 comment added Kestutis Cesnavicius These matters are discussed in SGA3, Exposé XXIV, section 8. Proposition 8.4 there is the statement you want and is specific to the étale topology, so you may want to assume that $F$ is smooth.
Dec 4, 2017 at 19:59 answer added Martin Bright timeline score: 3
Dec 4, 2017 at 19:28 history asked Owen Biesel CC BY-SA 3.0