The Stacks project

Lemma 38.15.6. In Situation 38.15.1. Assume that $S$ is locally Noetherian. If there exists an impurity of $\mathcal{F}$ above $s$, then there exists an impurity $(g : T \to S, t' \leadsto t, \xi )$ of $\mathcal{F}$ above $s$ such that $g$ is quasi-finite at $t$.

Proof. We may replace $S$ by an affine neighbourhood of $s$. By Lemma 38.15.3 we may assume that we have an impurity $(g : T \to S, t' \leadsto t, \xi )$ of such that $g$ is locally of finite type and $t$ a closed point of the fibre of $g$ above $s$. We may replace $T$ by the reduced induced scheme structure on $\overline{\{ t'\} }$. Let $Z = \overline{\{ \xi \} } \subset X_ T$. By assumption $Z_ t = \emptyset $ and the image of $Z \to T$ contains $t'$. By More on Morphisms, Lemma 37.25.1 there exists a nonempty open $V \subset Z$ such that for any $w \in f(V)$ any generic point $\xi '$ of $V_ w$ is in $\text{Ass}_{X_ T/T}(\mathcal{F}_ T)$. By More on Morphisms, Lemma 37.24.2 there exists a nonempty open $W \subset T$ with $W \subset f(V)$. By More on Morphisms, Lemma 37.52.7 there exists a closed subscheme $T' \subset T$ such that $t \in T'$, $T' \to S$ is quasi-finite at $t$, and there exists a point $z \in T' \cap W$, $z \leadsto t$ which does not map to $s$. Choose any generic point $\xi '$ of the nonempty scheme $V_ z$. Then $(T' \to S, z \leadsto t, \xi ')$ is the desired impurity. $\square$

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