Lemma 71.3.4. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Let \mathcal{G} be a quasi-coherent \mathcal{O}_ Y-module. Let x \in |X| and y = f(x) \in |Y|. If
y \in \text{WeakAss}_ S(\mathcal{G}),
f is flat at x, and
the dimension of the local ring of the fibre of f at x is zero (Morphisms of Spaces, Definition 67.33.1),
then x \in \text{WeakAss}(f^*\mathcal{G}).
Proof.
Choose a scheme V, a point v \in V, and an étale morphism V \to Y mapping v to y. Choose a scheme U, a point u \in U, and an étale morphism U \to V \times _ Y X mapping v to a point lying over v and x. This is possible because there is a t \in |V \times _ Y X| mapping to (v, y) by Properties of Spaces, Lemma 66.4.3. By definition we see that the dimension of \mathcal{O}_{U_ v, u} is zero. Hence u is a generic point of the fiber U_ v. By our definition of weakly associated points the problem is reduced to the morphism of schemes U \to V. This case is treated in Divisors, Lemma 31.6.4.
\square
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