The Stacks project

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

\[ \emptyset = Z \times _ T \mathop{\mathrm{Spec}}(k) = (\mathop{\mathrm{lim}}\nolimits Z_ i) \times _{(\mathop{\mathrm{lim}}\nolimits T_ i)} \mathop{\mathrm{Spec}}(k) = \mathop{\mathrm{lim}}\nolimits Z_ i \times _{T_ i} \mathop{\mathrm{Spec}}(k) \]

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 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. $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CV9. Beware of the difference between the letter 'O' and the digit '0'.