The Stacks project

Lemma 70.3.4. 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

  1. $y \in \text{WeakAss}_ S(\mathcal{G})$,

  2. $f$ is flat at $x$, and

  3. the dimension of the local ring of the fibre of $f$ at $x$ is zero (Morphisms of Spaces, Definition 66.33.1),

then $x \in \text{WeakAss}(f^*\mathcal{G})$.

Proof. Choose a scheme $V$, a point $v \in V$, and an ├ętale morphism $V \to Y$ mapping $v$ to $y$. Choose a scheme $U$, a point $u \in U$, and an ├ętale 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 65.4.3. 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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