The Stacks project

Proof. In this paragraph we reduce to $f$ being of finite presentation. The question is local on $X$ and $S$, hence we may assume $X$ and $S$ are affine. Write $X = \mathop{\mathrm{Spec}}(B)$, $S = \mathop{\mathrm{Spec}}(A)$ and write $B = A[x_1, \ldots , x_ n]/I$. In other words we obtain a closed immersion $i : X \to \mathbf{A}^ n_ S$ over $S$. Denote $t = i(x) \in \mathbf{A}^ n_ S$. Note that $i_*\mathcal{F}$ is a finite type quasi-coherent sheaf on $\mathbf{A}^ n_ S$ which is flat at $t$ over $S$ and note that

where $p : \mathbf{A}^ n_ S \to S$ is the projection. Note that $t$ is a weakly associated point of $i_*(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G})$ if and only if $x$ is a weakly associated point of $\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}$, see Divisors, Lemma 31.6.3. Similarly $x \in \text{Ass}_{X_ s}(\mathcal{F}_ s)$ if and only if $t \in \text{Ass}_{\mathbf{A}^ n_ s}((i_*\mathcal{F})_ s)$ (see Algebra, Lemma 10.63.14). Hence it suffices to prove the lemma in case $X = \mathbf{A}^ n_ S$. Thus we may assume that $X \to S$ is of finite presentation.

In this paragraph we reduce to $\mathcal{F}$ being of finite presentation and flat over $S$. Choose an elementary étale neighbourhood $e : (S', s') \to (S, s)$ and an open $V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ as in Proposition 38.10.3. Let $x' \in X' = X \times _ S S'$ be the unique point mapping to $x$ and $s'$. Then it suffices to prove the statement for $X' \to S'$, $x'$, $s'$, $(X' \to X)^*\mathcal{F}$, and $e^*\mathcal{G}$, see Lemma 38.2.8. Let $v \in V$ the unique point mapping to $x'$ and let $s' \in \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ be the closed point. Then $\mathcal{O}_{V, v} = \mathcal{O}_{X', x'}$ and $\mathcal{O}_{\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}), s'} = \mathcal{O}_{S', s'}$ and similarly for the stalks of pullbacks of $\mathcal{F}$ and $\mathcal{G}$. Also $V_{s'} \subset X'_{s'}$ is an open subscheme. Since the condition of being a weakly associated point depend only on the stalk of the sheaf, we may replace $X' \to S'$, $x'$, $s'$, $(X' \to X)^*\mathcal{F}$, and $e^*\mathcal{G}$ by $V \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$, $v$, $s'$, $(V \to X)^*\mathcal{F}$, and $(\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \to S)^*\mathcal{G}$. Thus we may assume that $f$ is of finite presentation and $\mathcal{F}$ of finite presentation and flat over $S$.

Assume $f$ is of finite presentation and $\mathcal{F}$ of finite presentation and flat over $S$. After shrinking $X$ and $S$ to affine neighbourhoods of $x$ and $s$, this case is handled by Lemma 38.13.2. $\square$