Lemma 77.6.3. Let Y be an algebraic space over a scheme S. Let g : X' \to X be an étale morphism of algebraic spaces over Y. Assume the structure morphisms X' \to Y and X \to Y are decent and of finite type. Let \mathcal{F} be a finite type, quasi-coherent \mathcal{O}_ X-module. Let y \in |Y|. Set F = f^{-1}(\{ y\} ) \subset |X|.
If \text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|) and g^*\mathcal{F} is (universally) pure above y, then \mathcal{F} is (universally) pure above y.
If \mathcal{F} is pure above y, g(|X'|) contains F, and Y is affine local with closed point y, then \text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|).
If \mathcal{F} is pure above y, \mathcal{F} is flat at all points of F, g(|X'|) contains \text{Ass}_{X/Y}(\mathcal{F}) \cap F, and Y is affine local with closed point y, then \text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|).
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Proof.
The assumptions on X \to Y and X' \to Y guarantee that we may apply the material in Sections 77.2 and 77.3 to these morphisms and the sheaves \mathcal{F} and g^*\mathcal{F}. Since g is étale we see that \text{Ass}_{X'/Y}(g^*\mathcal{F}) is the inverse image of \text{Ass}_{X/Y}(\mathcal{F}) and the same remains true after base change.
Proof of (1). Assume \text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|). Suppose that (T \to Y, t' \leadsto t, \xi ) is an impurity of \mathcal{F} above y. Since \text{Ass}_{X_ T/T}(\mathcal{F}_ T) \subset g_ T(|X'_ T|) by Lemma 77.6.2 we can choose a point \xi ' \in |X'_ T| mapping to \xi . By the above we see that (T \to Y, t' \leadsto t, \xi ') is an impurity of g^*\mathcal{F} above y'. This implies (1) is true.
Proof of (2). This follows from the fact that g(|X'|) is open in |X| and the fact that by purity every point of \text{Ass}_{X/Y}(\mathcal{F}) specializes to a point of F.
Proof of (3). This follows from the fact that g(|X'|) is open in |X| and the fact that by purity combined with Lemma 77.6.1 every point of \text{Ass}_{X/Y}(\mathcal{F}) specializes to a point of \text{Ass}_{X/Y}(\mathcal{F}) \cap F.
\square
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