Lemma 38.15.4. In Situation 38.15.1. Let $(g : T \to S, t' \leadsto t, \xi )$ be an impurity of $\mathcal{F}$ above $s$. Assume $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ is a directed limit of affine schemes over $S$. Then for some $i$ the triple $(T_ i \to S, t'_ i \leadsto t_ i, \xi _ i)$ is an impurity of $\mathcal{F}$ above $s$.
Proof. The notation in the statement means this: Let $p_ i : T \to T_ i$ be the projection morphisms, let $t_ i = p_ i(t)$ and $t'_ i = p_ i(t')$. Finally $\xi _ i \in X_{T_ i}$ is the image of $\xi $. By Divisors, Lemma 31.7.3 it is true that $\xi _ i$ is a point of the relative assassin of $\mathcal{F}_{T_ i}$ over $T_ i$. Thus the only point is to show that $\overline{\{ \xi _ i\} } \cap X_{t_ i} = \emptyset $ for some $i$.
First proof. Let $Z_ i = \overline{\{ \xi _ i\} } \subset X_{T_ i}$ and $Z = \overline{\{ \xi \} } \subset X_ T$ endowed with the reduced induced scheme structure. Then $Z = \mathop{\mathrm{lim}}\nolimits Z_ i$ by Limits, Lemma 32.4.4. Choose a field $k$ and a morphism $\mathop{\mathrm{Spec}}(k) \to T$ whose image is $t$. Then
\[ \emptyset = Z \times _ T \mathop{\mathrm{Spec}}(k) = (\mathop{\mathrm{lim}}\nolimits Z_ i) \times _{(\mathop{\mathrm{lim}}\nolimits T_ i)} \mathop{\mathrm{Spec}}(k) = \mathop{\mathrm{lim}}\nolimits Z_ i \times _{T_ i} \mathop{\mathrm{Spec}}(k) \]
because limits commute with fibred products (limits commute with limits). Each $Z_ i \times _{T_ i} \mathop{\mathrm{Spec}}(k)$ is quasi-compact because $X_{T_ i} \to T_ i$ is of finite type and hence $Z_ i \to T_ i$ is of finite type. Hence $Z_ i \times _{T_ i} \mathop{\mathrm{Spec}}(k)$ is empty for some $i$ by Limits, Lemma 32.4.3. Since the image of the composition $\mathop{\mathrm{Spec}}(k) \to T \to T_ i$ is $t_ i$ we obtain what we want.
Second proof. Set $Z = \overline{\{ \xi \} }$. Apply Limits, Lemma 32.14.1 to this situation to obtain an open neighbourhood $V \subset T$ of $t$, a commutative diagram
\[ \xymatrix{ V \ar[d] \ar[r]_ a & T' \ar[d]^ b \\ T \ar[r]^ g & S, } \]
and a closed subscheme $Z' \subset X_{T'}$ such that
the morphism $b : T' \to S$ is locally of finite presentation,
we have $Z' \cap X_{a(t)} = \emptyset $, and
$Z \cap X_ V$ maps into $Z'$ via the morphism $X_ V \to X_{T'}$.
We may assume $V$ is an affine open of $T$, hence by Limits, Lemmas 32.4.11 and 32.4.13 we can find an $i$ and an affine open $V_ i \subset T_ i$ with $V = f_ i^{-1}(V_ i)$. By Limits, Proposition 32.6.1 after possibly increasing $i$ a bit we can find a morphism $a_ i : V_ i \to T'$ such that $a = a_ i \circ f_ i|_ V$. The induced morphism $X_{V_ i} \to X_{T'}$ maps $\xi _ i$ into $Z'$. As $Z' \cap X_{a(t)} = \emptyset $ we conclude that $(T_ i \to S, t'_ i \leadsto t_ i, \xi _ i)$ is an impurity of $\mathcal{F}$ above $s$. $\square$
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