The Stacks project

Proof. Assume $f : U \to X$ is an étale morphism with $U$ a scheme and $u \in U$ maps to $x$. Then $\mathcal{F}|_ U = f^*\mathcal{F}$ is a coherent $\mathcal{O}_ U$-module on the locally Noetherian scheme $U$ and in particular $(\mathcal{F}|_ U)_ u$ is a finite $\mathcal{O}_{U, u}$-module, see Cohomology of Spaces, Lemma 69.12.2 and Cohomology of Schemes, Lemma 30.9.1. Recall that the support of $\mathcal{F}|_ U$ is a closed subset of $U$ (Morphisms, Lemma 29.5.3) and that the support of $(\mathcal{F}|_ U)_ u$ is the pullback of the support of $\mathcal{F}|_ U$ by the morphism $\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u}) \to U$. Thus $u$ is a generic point of an irreducible component of $\text{Supp}(\mathcal{F}|_ U)$ if and only if the support of $(\mathcal{F}|_ U)_ u$ is equal to the maximal ideal of $\mathcal{O}_{U, u}$. Now the equivalence of (1), (2), (3) follows from by Algebra, Lemma 10.62.3.

If $X$ is decent we choose an étale morphism $f : U \to X$ and a point $u \in U$ mapping to $x$. The support of $\mathcal{F}$ pulls back to the support of $\mathcal{F}|_ U$, see Morphisms of Spaces, Lemma 67.15.2. Also, specializations $x' \leadsto x$ in $|X|$ lift to specializations $u' \leadsto u$ in $U$ and any nontrivial specialization $u' \leadsto u$ in $U$ maps to a nontrivial specialization $f(u') \leadsto f(u)$ in $|X|$, see Decent Spaces, Lemmas 68.12.2 and 68.12.1. Using that $|X|$ and $U$ are sober topological spaces (Decent Spaces, Proposition 68.12.4 and Schemes, Lemma 26.11.1) we conclude $x$ is a generic point of the support of $\mathcal{F}$ if and only if $u$ is a generic point of the support of $\mathcal{F}|_ U$. We conclude (4) is equivalent to (1).

The parenthetical statement follows from Decent Spaces, Lemma 68.14.1. $\square$