The Stacks project

Proof. Assume (2). Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Choose a point $u \in U$ mapping to $x$. Let $v \in V$ be the image of $u$. We will deduce the result from the corresponding result for $\mathcal{F}|_ U = (U \to X)^*\mathcal{F}$ and the point $u$. $U_ v$. This works because $\text{Ass}_{U/V}(\mathcal{F}|_ U) \cap |U_ v|$ is equal to $\text{Ass}_{U_ v}(\mathcal{F}|_{U_ v})$ and equal to the inverse image of $F \cap \text{Ass}_{X/Y}(\mathcal{F})$. Since the map $|U_ v| \to F$ is continuous we see that specializations in $|U_ v|$ map to specializations in $F$, hence condition (2) is inherited by $U \to V$, $\mathcal{F}|_ U$, and the point $u$. Thus More on Flatness, Theorem 38.26.1 applies and we conclude that (1) holds.

If $Y$ is decent, then we can represent $y$ by a quasi-compact monomorphism $\mathop{\mathrm{Spec}}(k) \to Y$ (by definition of decent spaces, see Decent Spaces, Definition 68.6.1). Then $F = |X_ k|$, see Decent Spaces, Lemma 68.18.6. If in addition $X$ is decent (or more generally if $f$ is decent, see Decent Spaces, Definition 68.17.1 and Decent Spaces, Lemma 68.17.3), then $X_ y$ is a decent space too. Furthermore, specializations in $F$ can be lifted to specializations in $U_ v \to X_ y$, see Decent Spaces, Lemma 68.12.2. Having said this it is clear that the reverse implication holds, because it holds in the case of schemes. $\square$