Lemma 38.2.7. Let $g : T \to S$ be a finite flat morphism of schemes. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ S$-module. Let $t \in T$ be a point with image $s \in S$. Then

$t \in \text{WeakAss}(g^*\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}(\mathcal{G})$

Proof. The implication “$\Leftarrow$” follows immediately from Divisors, Lemma 31.6.4. Assume $t \in \text{WeakAss}(g^*\mathcal{G})$. Let $\mathop{\mathrm{Spec}}(A) \subset S$ be an affine open neighbourhood of $s$. Let $\mathcal{G}$ be the quasi-coherent sheaf associated to the $A$-module $M$. Let $\mathfrak p \subset A$ be the prime ideal corresponding to $s$. As $g$ is finite flat we have $g^{-1}(\mathop{\mathrm{Spec}}(A)) = \mathop{\mathrm{Spec}}(B)$ for some finite flat $A$-algebra $B$. Note that $g^*\mathcal{G}$ is the quasi-coherent $\mathcal{O}_{\mathop{\mathrm{Spec}}(B)}$-module associated to the $B$-module $M \otimes _ A B$ and $g_*g^*\mathcal{G}$ is the quasi-coherent $\mathcal{O}_{\mathop{\mathrm{Spec}}(A)}$-module associated to the $A$-module $M \otimes _ A B$. By Algebra, Lemma 10.78.5 we have $B_{\mathfrak p} \cong A_{\mathfrak p}^{\oplus n}$ for some integer $n \geq 0$. Note that $n \geq 1$ as we assumed there exists at least one point of $T$ lying over $s$. Hence we see by looking at stalks that

$s \in \text{WeakAss}(\mathcal{G}) \Leftrightarrow s \in \text{WeakAss}(g_*g^*\mathcal{G})$

Now the assumption that $t \in \text{WeakAss}(g^*\mathcal{G})$ implies that $s \in \text{WeakAss}(g_*g^*\mathcal{G})$ by Divisors, Lemma 31.6.3 and hence by the above $s \in \text{WeakAss}(\mathcal{G})$. $\square$

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