Situation 77.2.1. Let $S$ be a scheme. Let $f : X \to Y$ be a finite type, decent^{1} 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 $Y$.

## 77.2 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

Moreover, denote $\mathcal{F}_ T$ the pullback of $\mathcal{F}$ to $X_ T$.

Definition 77.2.2. In Situation 77.2.1 we say a diagram (77.2.1.1) 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 68.17.1.

Lemma 77.2.3. In Situation 77.2.1. 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 71.4.7 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 70.5.4 (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

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 70.5.3. 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 77.2.4. In Situation 77.2.1. 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 66.4.3. The morphism of schemes $T_1 \to T$ is flat at $t_1$ (use Morphisms of Spaces, Lemma 67.30.4 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 71.4.7. 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 77.2.5. In Situation 77.2.1. 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 66.19.3). 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 77.2.3.

Since an étale morphism is locally quasi-finite (Morphisms of Spaces, Lemma 67.39.5) 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 77.2.4 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 67.27.4 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 71.4.7 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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)