Situation 76.2.1. Let $S$ be a scheme. Let $f : X \to Y$ be a finite type, decent1 morphism of algebraic spaces over $S$. Also, $\mathcal{F}$ is a finite type quasi-coherent $\mathcal{O}_ X$-module. Finally $y \in |Y|$ is a point of $Y$.
76.2 Impurities
The section is the analogue of More on Flatness, Section 38.15.
In this situation consider a scheme $T$, a morphism $g : T \to Y$, a point $t \in T$ with $g(t) = y$, a specialization $t' \leadsto t$ in $T$, and a point $\xi \in |X_ T|$ lying over $t'$. Here $X_ T = T \times _ Y X$. Picture
76.2.1.1
\begin{equation} \label{spaces-flat-equation-impurity} \vcenter { \xymatrix{ \xi \ar@{|->}[d] & \\ t' \ar@{~>}[r] & t \ar@{|->}[r] \ar[r] & y } } \quad \quad \vcenter { \xymatrix{ X_ T \ar[d]_{f_ T} \ar[r] & X \ar[d]^ f \\ T \ar[r]^ g & Y } } \end{equation}
Moreover, denote $\mathcal{F}_ T$ the pullback of $\mathcal{F}$ to $X_ T$.
Definition 76.2.2. In Situation 76.2.1 we say a diagram (76.2.1.1) defines an impurity of $\mathcal{F}$ above $y$ if $\xi \in \text{Ass}_{X_ T/T}(\mathcal{F}_ T)$ and $t \not\in f_ T(\overline{\{ \xi \} })$. We will indicate this by saying “let $(g : T \to Y, t' \leadsto t, \xi )$ be an impurity of $\mathcal{F}$ above $y$”.
Another way to say this is: $(g : T \to Y, t' \leadsto t, \xi )$ is an impurity of $\mathcal{F}$ above $y$ if there exists no specialization $\xi \leadsto \theta $ in the topological space $|X_ T|$ with $f_ T(\theta ) = t$. Specializations in non-decent algebraic spaces do not behave well. If the morphism $f$ is decent, then $X_ T$ is a decent algebraic space for all morphisms $g : T \to Y$ as above, see Decent Spaces, Definition 67.17.1.
Lemma 76.2.3. In Situation 76.2.1. Let $(g : T \to S, t' \leadsto t, \xi )$ be an impurity of $\mathcal{F}$ above $y$. Assume $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ is a directed limit of affine schemes over $Y$. Then for some $i$ the triple $(T_ i \to Y, t'_ i \leadsto t_ i, \xi _ i)$ is an impurity of $\mathcal{F}$ above $y$.
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 on Spaces, Lemma 70.4.7 we have $\xi _ i \in \text{Ass}_{X_{T_ i}/T_ i}(\mathcal{F}_{T_ i})$. Thus the only point is to show that $t_ i \not\in f_{T_ i}(\overline{\{ \xi _ i\} })$ for some $i$.
Let $Z_ i \subset X_{T_ i}$ be the reduced induced scheme structure on $\overline{\{ \xi _ i\} } \subset |X_{T_ i}|$ and let $Z \subset X_ T$ be the reduced induced scheme structure on $\overline{\{ \xi \} } \subset |X_ T|$. Then $Z = \mathop{\mathrm{lim}}\nolimits Z_ i$ by Limits of Spaces, Lemma 69.5.4 (the lemma applies because each $X_{T_ i}$ is decent). 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 of Spaces, Lemma 69.5.3. Since the image of the composition $\mathop{\mathrm{Spec}}(k) \to T \to T_ i$ is $t_ i$ we obtain what we want. $\square$
Impurities go up along flat base change.
Lemma 76.2.4. In Situation 76.2.1. Let $(Y_1, y_1) \to (Y, y)$ be a morphism of pointed algebraic spaces over $S$. Assume $Y_1 \to Y$ is flat at $y_1$. If $(T \to Y, t' \leadsto t, \xi )$ is an impurity of $\mathcal{F}$ above $y$, then there exists an impurity $(T_1 \to Y_1, t_1' \leadsto t_1, \xi _1)$ of the pullback $\mathcal{F}_1$ of $\mathcal{F}$ to $X_1 = Y_1 \times _ Y X$ over $y_1$ such that $T_1$ is étale over $Y_1 \times _ Y T$.
Proof. Choose an étale morphism $T_1 \to Y_1 \times _ Y T$ where $T_1$ is a scheme and let $t_1 \in T_1$ be a point mapping to $y_1$ and $t$. It is possible to find a pair $(T_1, t_1)$ like this by Properties of Spaces, Lemma 65.4.3. The morphism of schemes $T_1 \to T$ is flat at $t_1$ (use Morphisms of Spaces, Lemma 66.30.4 and the definition of flat morphisms of algebraic spaces) there exists a specialization $t'_1 \leadsto t_1$ lying over $t' \leadsto t$, see Morphisms, Lemma 29.25.9. Choose a point $\xi _1 \in |X_{T_1}|$ mapping to $t'_1$ and $\xi $ with $\xi _1 \in \text{Ass}_{X_{T_1}/T_1}(\mathcal{F}_{T_1})$. point of $\mathop{\mathrm{Spec}}(\kappa (t'_1) \otimes _{\kappa (t')} \kappa (\xi ))$. This is possible by Divisors on Spaces, Lemma 70.4.7. As the closure $Z_1$ of $\{ \xi _1\} $ in $|X_{T_1}|$ maps into the closure of $\{ \xi \} $ in $|X_ T|$ we conclude that the image of $Z_1$ in $|T_1|$ cannot contain $t_1$. Hence $(T_1 \to Y_1, t'_1 \leadsto t_1, \xi _1)$ is an impurity of $\mathcal{F}_1$ above $Y_1$. $\square$
Lemma 76.2.5. In Situation 76.2.1. Let $\overline{y}$ be a geometric point lying over $y$. Let $\mathcal{O} = \mathcal{O}_{Y, \overline{y}}$ be the étale local ring of $Y$ at $\overline{y}$. Denote $Y^{sh} = \mathop{\mathrm{Spec}}(\mathcal{O})$, $X^{sh} = X \times _ Y Y^{sh}$, and $\mathcal{F}^{sh}$ the pullback of $\mathcal{F}$ to $X^{sh}$. The following are equivalent
there exists an impurity $(Y^{sh} \to Y, y' \leadsto \overline{y}, \xi )$ of $\mathcal{F}$ above $y$,
every point of $\text{Ass}_{X^{sh}/Y^{sh}}(\mathcal{F}^{sh})$ specializes to a point of the closed fibre $X_{\overline{y}}$,
there exists an impurity $(T \to Y, t' \leadsto t, \xi )$ of $\mathcal{F}$ above $y$ such that $(T, t) \to (Y, y)$ is an étale neighbourhood, and
there exists an impurity $(T \to Y, t' \leadsto t, \xi )$ of $\mathcal{F}$ above $y$ such that $T \to Y$ is quasi-finite at $t$.
Proof. That parts (1) and (2) are equivalent is immediate from the definition.
Recall that $\mathcal{O} = \mathcal{O}_{Y, \overline{y}}$ is the filtered colimit of $\mathcal{O}(V)$ over the category of étale neighbourhoods $(V, \overline{v}) \to (Y, \overline{y})$ (Properties of Spaces, Lemma 65.19.3). Moreover, it suffices to consider affine étale neighbourhoods $V$. Hence $Y^{sh} = \mathop{\mathrm{Spec}}(\mathcal{O}) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Spec}}(\mathcal{O}(V)) = \mathop{\mathrm{lim}}\nolimits V$. Thus we see that (1) implies (3) by Lemma 76.2.3.
Since an étale morphism is locally quasi-finite (Morphisms of Spaces, Lemma 66.39.5) we see that (3) implies (4).
Finally, assume (4). After replacing $T$ by an open neighbourhood of $t$ we may assume $T \to Y$ is locally quasi-finite. By Lemma 76.2.4 we find an impurity $(T_1 \to Y^{sh}, t_1' \leadsto t_1, \xi _1)$ with $T_1 \to T \times _ Y Y^{sh}$ étale. Since an étale morphism is locally quasi-finite and using Morphisms of Spaces, Lemma 66.27.4 and Morphisms, Lemma 29.20.12 we see that $T_1 \to Y^{sh}$ is locally quasi-finite. As $\mathcal{O}$ is strictly henselian, we can apply More on Morphisms, Lemma 37.41.1 to see that after replacing $T_1$ by an open and closed neighbourhood of $t_1$ we may assume that $T_1 \to Y^{sh} = \mathop{\mathrm{Spec}}(\mathcal{O})$ is finite. Let $\theta \in |X^{sh}|$ be the image of $\xi _1$ and let $y' \in \mathop{\mathrm{Spec}}(\mathcal{O})$ be the image of $t_1'$. By Divisors on Spaces, Lemma 70.4.7 we see that $\theta \in \text{Ass}_{X^{sh}/Y^{sh}}(\mathcal{F}^{sh})$. Since $\pi : X_{T_1} \to X^{sh}$ is finite, it induces a closed map $|X_{T_1}| \to |X^{sh}|$. Hence the image of $\overline{\{ \xi _1\} }$ is $\overline{\{ \theta \} }$. It follows that $(Y^{sh} \to Y, y' \leadsto \overline{y}, \theta )$ is an impurity of $\mathcal{F}$ above $y$ and the proof is complete. $\square$
[1] Quasi-separated morphisms are decent, see Decent Spaces, Lemma 67.17.2. For any morphism $\mathop{\mathrm{Spec}}(k) \to Y$ where $k$ is a field, the algebraic space $X_ k$ is of finite presentation over $k$ because it is of finite type over $k$ and quasi-separated by Decent Spaces, Lemma 67.14.1.
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