Lemma 66.23.6. Let $S$ be a scheme. Let $f : X \to Y$, $g : Y \to Z$ be morphisms of algebraic spaces over $S$. If $g \circ f : X \to Z$ is locally of finite type, then $f : X \to Y$ is locally of finite type.

**Proof.**
We can find a diagram

\[ \xymatrix{ U \ar[r] \ar[d] & V \ar[r] \ar[d] & W \ar[d] \\ X \ar[r] & Y \ar[r] & Z } \]

where $U$, $V$, $W$ are schemes, the vertical arrows are étale and surjective, see Spaces, Lemma 64.11.6. At this point we can use Lemma 66.23.4 and Morphisms, Lemma 29.15.8 to conclude. $\square$

## 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Your email address will not be published.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)