## Tag `01U1`

Chapter 28: Morphisms of Schemes > Section 28.22: Open morphisms

Lemma 28.22.2. Let $f : X \to S$ be a morphism.

- If $f$ is locally of finite presentation and generalizations lift along $f$, then $f$ is open.
- If $f$ is locally of finite presentation and generalizations lift along every base change of $f$, then $f$ is universally open.

Proof.It suffices to prove the first assertion. This reduces to the case where both $X$ and $S$ are affine. In this case the result follows from Algebra, Lemma 10.40.3 and Proposition 10.40.8. $\square$

The code snippet corresponding to this tag is a part of the file `morphisms.tex` and is located in lines 4037–4046 (see updates for more information).

```
\begin{lemma}
\label{lemma-locally-finite-presentation-universally-open}
Let $f : X \to S$ be a morphism.
\begin{enumerate}
\item If $f$ is locally of finite presentation and generalizations lift
along $f$, then $f$ is open.
\item If $f$ is locally of finite presentation and generalizations lift
along every base change of $f$, then $f$ is universally open.
\end{enumerate}
\end{lemma}
\begin{proof}
It suffices to prove the first assertion.
This reduces to the case where both $X$ and $S$ are affine.
In this case the result follows from
Algebra, Lemma \ref{algebra-lemma-going-up-down-specialization}
and Proposition \ref{algebra-proposition-fppf-open}.
\end{proof}
```

## Comments (0)

## Add a comment on tag `01U1`

Your email address will not be published.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.

There are no comments yet for this tag.