Lemma 31.7.2. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $U \subset X$ and $V \subset S$ be affine opens with $f(U) \subset V$. Write $U = \mathop{\mathrm{Spec}}(A)$, $V = \mathop{\mathrm{Spec}}(R)$, and set $M = \Gamma (U, \mathcal{F})$. Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime. Then
\[ \mathfrak p \in \text{Ass}_{A/R}(M) \Rightarrow x \in \text{Ass}_{X/S}(\mathcal{F}) \]
If all fibres $X_ s$ of $f$ are locally Noetherian, then $\mathfrak p \in \text{Ass}_{A/R}(M) \Leftrightarrow x \in \text{Ass}_{X/S}(\mathcal{F})$ for all pairs $(\mathfrak p, x)$ as above.
Proof.
The set $\text{Ass}_{A/R}(M)$ is defined in Algebra, Definition 10.65.2. Choose a pair $(\mathfrak p, x)$. Let $s = f(x)$. Let $\mathfrak r \subset R$ be the prime lying under $\mathfrak p$, i.e., the prime corresponding to $s$. Let $\mathfrak p' \subset A \otimes _ R \kappa (\mathfrak r)$ be the prime whose inverse image is $\mathfrak p$, i.e., the prime corresponding to $x$ viewed as a point of its fibre $X_ s$. Then $\mathfrak p \in \text{Ass}_{A/R}(M)$ if and only if $\mathfrak p'$ is an associated prime of $M \otimes _ R \kappa (\mathfrak r)$, see Algebra, Lemma 10.65.1. Note that the ring $A \otimes _ R \kappa (\mathfrak r)$ corresponds to $U_ s$ and the module $M \otimes _ R \kappa (\mathfrak r)$ corresponds to the quasi-coherent sheaf $\mathcal{F}_ s|_{U_ s}$. Hence $x$ is an associated point of $\mathcal{F}_ s$ by Lemma 31.2.2. The reverse implication holds if $\mathfrak p'$ is finitely generated which is how the last sentence is seen to be true.
$\square$
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