The Stacks project

Lemma 38.18.6. Let $f : X \to S$ be a morphism of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite presentation flat over $S$. Then the set

\[ U = \{ s \in S \mid \mathcal{F}\text{ is pure along }X_ s\} \]

is open in $S$.

Proof. Let $s \in U$. Using Lemma 38.12.5 we can find an elementary étale neighbourhood $(S', s') \to (S, s)$ and a commutative diagram

\[ \xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ S & S' \ar[l] } \]

such that $g$ is étale, $X_ s \subset g(X')$, the schemes $X'$, $S'$ are affine, and such that $\Gamma (X', g^*\mathcal{F})$ is a projective $\Gamma (S', \mathcal{O}_{S'})$-module. Note that $g^*\mathcal{F}$ is universally pure over $S'$, see Lemma 38.17.4. Set $W' \subset X \times _ S S'$ equal to the image of the étale morphism $X' \to X \times _ S S'$. Note that $W$ is open and quasi-compact over $S'$. Set

\[ E = \{ t \in S' \mid \text{Ass}_{X_ t}(\mathcal{F}_ t) \subset W' \} . \]

By More on Morphisms, Lemma 37.25.5 $E$ is a constructible subset of $S'$. By Lemma 38.18.2 we see that $\mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \subset E$. By Morphisms, Lemma 29.22.4 we see that $E$ contains an open neighbourhood $V'$ of $s'$. Applying Lemma 38.18.2 once more we see that for any point $s_1$ in the image of $V'$ in $S$ the sheaf $\mathcal{F}$ is pure along $X_{s_1}$. Since $S' \to S$ is étale the image of $V'$ in $S$ is open and we win. $\square$

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