Lemma 38.18.2. Let f : X \to S be a morphism of schemes of finite type. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module of finite type. Let s \in S. Let (S', s') \to (S, s) be an elementary étale neighbourhood and let
\xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ S & S' \ar[l] }
be a commutative diagram of morphisms of schemes. Assume
\mathcal{F} is flat over S at all points of X_ s,
X' \to S' is of finite type,
g^*\mathcal{F} is pure along X'_{s'},
g : X' \to X is étale, and
g(X') contains \text{Ass}_{X_ s}(\mathcal{F}_ s).
In this situation \mathcal{F} is pure along X_ s if and only if the image of X' \to X \times _ S S' contains the points of \text{Ass}_{X \times _ S S'/S'}(\mathcal{F} \times _ S S') lying over points in S' which specialize to s'.
Proof.
Since the morphism S' \to S is étale, we see that if \mathcal{F} is pure along X_ s, then \mathcal{F} \times _ S S' is pure along X_ s, see Lemma 38.16.4. Since purity satisfies flat descent, see Lemma 38.16.6, we see that if \mathcal{F} \times _ S S' is pure along X_{s'}, then \mathcal{F} is pure along X_ s. Hence we may replace S by S' and assume that S = S' so that g : X' \to X is an étale morphism between schemes of finite type over S. Moreover, we may replace S by \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) and assume that S is local.
First, assume that \mathcal{F} is pure along X_ s. In this case every point of \text{Ass}_{X/S}(\mathcal{F}) specializes to a point of X_ s by purity. Hence by Lemma 38.18.1 we see that every point of \text{Ass}_{X/S}(\mathcal{F}) specializes to a point of \text{Ass}_{X_ s}(\mathcal{F}_ s). Thus every point of \text{Ass}_{X/S}(\mathcal{F}) is in the image of g (as the image is open and contains \text{Ass}_{X_ s}(\mathcal{F}_ s)).
Conversely, assume that g(X') contains \text{Ass}_{X/S}(\mathcal{F}). Let S^ h = \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}^ h) be the henselization of S at s. Denote g^ h : (X')^ h \to X^ h the base change of g by S^ h \to S, and denote \mathcal{F}^ h the pullback of \mathcal{F} to X^ h. By Divisors, Lemma 31.7.3 and Remark 31.7.4 the relative assassin \text{Ass}_{X^ h/S^ h}(\mathcal{F}^ h) is the inverse image of \text{Ass}_{X/S}(\mathcal{F}) via the projection X^ h \to X. As we have assumed that g(X') contains \text{Ass}_{X/S}(\mathcal{F}) we conclude that the base change g^ h((X')^ h) = g(X') \times _ S S^ h contains \text{Ass}_{X^ h/S^ h}(\mathcal{F}^ h). In this way we reduce to the case where S is the spectrum of a henselian local ring. Let x \in \text{Ass}_{X/S}(\mathcal{F}). To finish the proof of the lemma we have to show that x specializes to a point of X_ s, see criterion (4) for purity in discussion following Definition 38.16.1. By assumption there exists a x' \in X' such that g(x') = x. As g : X' \to X is étale, we see that x' \in \text{Ass}_{X'/S}(g^*\mathcal{F}), see Lemma 38.2.8 (applied to the morphism of fibres X'_ w \to X_ w where w \in S is the image of x'). Since g^*\mathcal{F} is pure along X'_ s we see that x' \leadsto y for some y \in X'_ s. Hence x = g(x') \leadsto g(y) and g(y) \in X_ s as desired.
\square
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