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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 étale neighbourhood, then (T_1, t_1) \to (S', s') is an elementary étale neighbourhood too, and in this way we see that (1) holds. \square


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