Lemma 26.2.2. Let $X$, $Y$ be locally ringed spaces. If $f : X \to Y$ is an isomorphism of ringed spaces, then $f$ is an isomorphism of locally ringed spaces.

** An isomorphism of ringed spaces between locally ringed spaces is an isomorphism of locally ringed spaces. **

**Proof.**
This follows trivially from the corresponding fact in algebra: Suppose $A$, $B$ are local rings. Any isomorphism of rings $A \to B$ is a local ring homomorphism.
$\square$

## 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Your email address will not be published.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 (2)

Comment #1277 by Johan Commelin on

Comment #8546 by Shizhang on