The Stacks project

Lemma 31.3.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$ which is flat over $S$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $S$. Then we have

\[ \text{Ass}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}) \supset \bigcup \nolimits _{s \in \text{Ass}_ S(\mathcal{G})} \text{Ass}_{X_ s}(\mathcal{F}_ s) \]

and equality holds if $S$ is locally Noetherian (for the notation $\mathcal{F}_ s$ see above).

Proof. Let $x \in X$ and let $s = f(x) \in S$. Set $B = \mathcal{O}_{X, x}$, $A = \mathcal{O}_{S, s}$, $N = \mathcal{F}_ x$, and $M = \mathcal{G}_ s$. Note that the stalk of $\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}$ at $x$ is equal to the $B$-module $M \otimes _ A N$. Hence $x \in \text{Ass}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G})$ if and only if $\mathfrak m_ B$ is in $\text{Ass}_ B(M \otimes _ A N)$. Similarly $s \in \text{Ass}_ S(\mathcal{G})$ and $x \in \text{Ass}_{X_ s}(\mathcal{F}_ s)$ if and only if $\mathfrak m_ A \in \text{Ass}_ A(M)$ and $\mathfrak m_ B/\mathfrak m_ A B \in \text{Ass}_{B \otimes \kappa (\mathfrak m_ A)}(N \otimes \kappa (\mathfrak m_ A))$. Thus the lemma follows from Algebra, Lemma 10.65.5. $\square$

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