The Stacks project

Lemma 31.6.4. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ S$-module. Let $x \in X$ with $s = f(x)$. If $f$ is flat at $x$, the point $x$ is a generic point of the fibre $X_ s$, and $s \in \text{WeakAss}_ S(\mathcal{G})$, then $x \in \text{WeakAss}(f^*\mathcal{G})$.

Proof. Let $A = \mathcal{O}_{S, s}$, $B = \mathcal{O}_{X, x}$, and $M = \mathcal{G}_ s$. Let $m \in M$ be an element whose annihilator $I = \{ a \in A \mid am = 0\} $ has radical $\mathfrak m_ A$. Then $m \otimes 1$ has annihilator $I B$ as $A \to B$ is faithfully flat. Thus it suffices to see that $\sqrt{I B} = \mathfrak m_ B$. This follows from the fact that the maximal ideal of $B/\mathfrak m_ AB$ is locally nilpotent (see Algebra, Lemma 10.25.1) and the assumption that $\sqrt{I} = \mathfrak m_ A$. Some details omitted. $\square$

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