The Stacks project

Proof. Let $(g : T \to S, t' \leadsto t, \xi )$ be any impurity of $\mathcal{F}$ above $s$. We apply Limits, Lemma 32.14.1 to $t \in T$ and $Z = \overline{\{ \xi \} }$ to obtain an open neighbourhood $V \subset T$ of $t$, a commutative diagram

and a closed subscheme $Z' \subset X_{T'}$ such that

1. the morphism $b : T' \to S$ is locally of finite presentation,

2. we have $Z' \cap X_{a(t)} = \emptyset $, and

3. $Z \cap X_ V$ maps into $Z'$ via the morphism $X_ V \to X_{T'}$.

As $t'$ specializes to $t$ we may replace $T$ by the open neighbourhood $V$ of $t$. Thus we have a commutative diagram

where $b \circ a = g$. Let $\xi ' \in X_{T'}$ denote the image of $\xi $. By Divisors, Lemma 31.7.3 we see that $\xi ' \in \text{Ass}_{X_{T'}/T'}(\mathcal{F}_{T'})$. Moreover, by construction the closure of $\overline{\{ \xi '\} }$ is contained in the closed subset $Z'$ which avoids the fibre $X_{a(t)}$. In this way we see that $(T' \to S, a(t') \leadsto a(t), \xi ')$ is an impurity of $\mathcal{F}$ above $s$.

Thus we may assume that $g : T \to S$ is locally of finite presentation. Let $Z = \overline{\{ \xi \} }$. By assumption $Z_ t = \emptyset $. By More on Morphisms, Lemma 37.24.1 this means that $Z_{t''} = \emptyset $ for $t''$ in an open subset of $\overline{\{ t\} }$. Since the fibre of $T \to S$ over $s$ is a Jacobson scheme, see Morphisms, Lemma 29.16.10 we find that there exist a closed point $t'' \in \overline{\{ t\} }$ such that $Z_{t''} = \emptyset $. Then $(g : T \to S, t' \leadsto t'', \xi )$ is the desired impurity. $\square$

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

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

and a closed subscheme $Z' \subset X_{T'}$ such that

1. the morphism $b : T' \to S$ is locally of finite presentation,

2. we have $Z' \cap X_{a(t)} = \emptyset $, and

3. $Z \cap X_ V$ 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 $s$. $\square$

Proof. Let $(g : T \to S, t' \leadsto t, \xi )$ be an impurity of $\mathcal{F}$ above $s$ such that $g$ is quasi-finite at $t$. After shrinking $T$ we may assume that $g$ is locally of finite type. Apply More on Morphisms, Lemma 37.41.1 to $T \to S$ and $t \mapsto s$. This gives us a diagram

where $(U, u) \to (S, s)$ is an elementary étale neighbourhood and $V \subset T \times _ S U$ is an open neighbourhood of $v = (t, u)$ such that $V \to U$ is finite and such that $v$ is the unique point of $V$ lying over $u$. Since the morphism $V \to T$ is étale hence flat we see that there exists a specialization $v' \leadsto v$ such that $v' \mapsto t'$. Note that $\kappa (t') \subset \kappa (v')$ is finite separable. Pick any point $\zeta \in X_{v'}$ mapping to $\xi \in X_{t'}$. By Divisors, Lemma 31.7.3 we see that $\zeta \in \text{Ass}_{X_ V/V}(\mathcal{F}_ V)$. Moreover, the closure $\overline{\{ \zeta \} }$ does not meet the fibre $X_ v$ as by assumption the closure $\overline{\{ \xi \} }$ does not meet $X_ t$. In other words $(V \to S, v' \leadsto v, \zeta )$ is an impurity of $\mathcal{F}$ above $S$.

Next, let $u' \in U'$ be the image of $v'$ and let $\theta \in X_ U$ be the image of $\zeta $. Then $\theta \mapsto u'$ and $u' \leadsto u$. By Divisors, Lemma 31.7.3 we see that $\theta \in \text{Ass}_{X_ U/U}(\mathcal{F})$. Moreover, as $\pi : X_ V \to X_ U$ is finite we see that $\pi \big (\overline{\{ \zeta \} }\big ) = \overline{\{ \pi (\zeta )\} }$. Since $v$ is the unique point of $V$ lying over $u$ we see that $X_ u \cap \overline{\{ \pi (\zeta )\} } = \emptyset $ because $X_ v \cap \overline{\{ \zeta \} } = \emptyset $. In this way we conclude that $(U \to S, u' \leadsto u, \theta )$ is an impurity of $\mathcal{F}$ above $s$ and we win. $\square$

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. $\square$

Proof. We may replace $S$ by an affine neighbourhood of $s$. Say $S = \mathop{\mathrm{Spec}}(A)$ and $s$ corresponds to the prime $\mathfrak p \subset A$. Then $\mathcal{O}_{S, s}^ h = \mathop{\mathrm{colim}}\nolimits _{(T, t)} \Gamma (T, \mathcal{O}_ T)$ where the limit is over the opposite of the cofiltered category of affine elementary étale neighbourhoods $(T, t)$ of $(S, s)$, see More on Morphisms, Lemma 37.35.5 and its proof. Hence $S^ h = \mathop{\mathrm{lim}}\nolimits _ i T_ i$ and we win by Lemma 38.15.4. $\square$

Proof. As an étale morphism is locally quasi-finite it is clear that (2) implies (3). We have seen that (3) implies (2) in Lemma 38.15.5. We have seen that (1) implies (2) in Lemma 38.15.7. Finally, if $(T \to S, t' \leadsto t, \xi )$ is an impurity of $\mathcal{F}$ above $s$ such that $(T, t) \to (S, s)$ is an elementary étale neighbourhood, then we can choose a factorization $S^ h \to T \to S$ of the structure morphism $S^ h \to S$. Choose any point $s' \in S^ h$ mapping to $t'$ and choose any $\xi ' \in X_{s'}$ mapping to $\xi \in X_{t'}$. Then $(S^ h \to S, s' \leadsto s, \xi ')$ is an impurity of $\mathcal{F}$ above $s$. We omit the details. $\square$