## Tag `01YY`

Chapter 31: Limits of Schemes > Section 31.4: Descending properties

Lemma 31.4.6. In Situation 31.4.5.

- We have $S_{set} = \mathop{\rm lim}\nolimits_i S_{i, set}$ where $S_{set}$ indicates the underlying set of the scheme $S$.
- We have $S_{top} = \mathop{\rm lim}\nolimits_i S_{i, top}$ where $S_{top}$ indicates the underlying topological space of the scheme $S$.
- If $s, s' \in S$ and $s'$ is not a specialization of $s$ then for some $i \in I$ the image $s'_i \in S_i$ of $s'$ is not a specialization of the image $s_i \in S_i$ of $s$.
- Add more easy facts on topology of $S$ here. (Requirement: whatever is added should be easy in the affine case.)

Proof.Part (1) is a special case of Lemma 31.4.1.Part (2) is a special case of Lemma 31.4.2.

Part (3) is a special case of Lemma 31.4.4. $\square$

The code snippet corresponding to this tag is a part of the file `limits.tex` and is located in lines 391–405 (see updates for more information).

```
\begin{lemma}
\label{lemma-topology-limit}
In Situation \ref{situation-descent}.
\begin{enumerate}
\item We have $S_{set} = \lim_i S_{i, set}$ where $S_{set}$
indicates the underlying set of the scheme $S$.
\item We have $S_{top} = \lim_i S_{i, top}$ where $S_{top}$
indicates the underlying topological space of the scheme $S$.
\item If $s, s' \in S$ and $s'$ is not a specialization of $s$
then for some $i \in I$ the image $s'_i \in S_i$ of $s'$ is not
a specialization of the image $s_i \in S_i$ of $s$.
\item Add more easy facts on topology of $S$ here.
(Requirement: whatever is added should be easy in the affine case.)
\end{enumerate}
\end{lemma}
\begin{proof}
Part (1) is a special case of Lemma \ref{lemma-inverse-limit-sets}.
\medskip\noindent
Part (2) is a special case of Lemma \ref{lemma-inverse-limit-top}.
\medskip\noindent
Part (3) is a special case of Lemma \ref{lemma-inverse-limit-irreducibles}.
\end{proof}
```

## Comments (2)

## Add a comment on tag `01YY`

Your email address will not be published.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 lower-right corner).

All contributions are licensed under the GNU Free Documentation License.