The Stacks project

Lemma 38.16.6. Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$. Let $(S', s') \to (S, s)$ be a morphism of pointed schemes. Assume $S' \to S$ is flat at $s'$.

  1. If $\mathcal{F}_{S'}$ is pure along $X_{s'}$, then $\mathcal{F}$ is pure along $X_ s$.

  2. If $\mathcal{F}_{S'}$ is universally pure along $X_{s'}$, then $\mathcal{F}$ is universally pure along $X_ s$.

Proof. Let $(T \to S, t' \leadsto t, \xi )$ be an impurity of $\mathcal{F}$ above $s$. Set $T_1 = T \times _ S S'$, and let $t_1$ be the unique point of $T_1$ mapping to $t$ and $s'$. Since $T_1 \to T$ is flat at $t_1$, see Morphisms, Lemma 29.25.8, there exists a specialization $t'_1 \leadsto t_1$ lying over $t' \leadsto t$, see Algebra, Section 10.41. Choose a point $\xi _1 \in X_{t'_1}$ which corresponds to a generic point of $\mathop{\mathrm{Spec}}(\kappa (t'_1) \otimes _{\kappa (t')} \kappa (\xi ))$, see Schemes, Lemma 26.17.5. By Divisors, Lemma 31.7.3 we see that $\xi _1 \in \text{Ass}_{X_{T_1}/T_1}(\mathcal{F}_{T_1})$. As the Zariski closure of $\{ \xi _1\} $ in $X_{T_1}$ maps into the Zariski closure of $\{ \xi \} $ in $X_ T$ we conclude that this closure is disjoint from $X_{t_1}$. Hence $(T_1 \to S', t'_1 \leadsto t_1, \xi _1)$ is an impurity of $\mathcal{F}_{S'}$ above $s'$. In other words we have proved the contrapositive to part (2) of the lemma. Finally, if $(T, t) \to (S, s)$ is an elementary ├ętale neighbourhood, then $(T_1, t_1) \to (S', s')$ is an elementary ├ętale neighbourhood too, and in this way we see that (1) holds. $\square$

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