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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