The Stacks project

Proof. Part (2) is immediate from the definitions (this is one of the advantages of the flat-fppf site over the lisse-étale site). The implication (1)(a) $\Rightarrow $ (1)(b) is immediate as well. To see (1)(b) $\Rightarrow $ (1)(a) let $U$ be a scheme and let $x : U \to \mathcal{X}$ be a surjective smooth morphism. Then $x$ is an object of the lisse-étale site of $\mathcal{X}$. Hence we see that (1)(b) implies that $\mathcal{F}|_{U_{\acute{e}tale}} = 0$. Let $V \to \mathcal{X}$ be an flat morphism where $V$ is a scheme. Set $W = U \times _\mathcal {X} V$ and consider the diagram

Note that the projection $p : W \to U$ is flat and the projection $q : W \to V$ is smooth and surjective. This implies that $q_{small}^*$ is a faithful functor on quasi-coherent modules. By assumption $\mathcal{F}$ has the flat base change property so that we obtain $p_{small}^*\mathcal{F}|_{U_{\acute{e}tale}} \cong q_{small}^*\mathcal{F}|_{V_{\acute{e}tale}}$. Thus if $\mathcal{F}$ is in the kernel of $g^*$, then $\mathcal{F}|_{V_{\acute{e}tale}} = 0$ as desired. $\square$