Lemma 69.13.3. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Assume $X$ Noetherian and $f$ of finite presentation. Then there exists a dense open $V \subset Y$ and an immersion $V \to \mathbf{A}^ n_ X$.

**Proof.**
The assumptions imply that $Y$ is Noetherian (Morphisms of Spaces, Lemma 66.28.6). Then $Y$ is quasi-separated, hence has a dense open subscheme (Properties of Spaces, Proposition 65.13.3). Thus we may assume that $Y$ is a Noetherian scheme. By removing intersections of irreducible components of $Y$ (use Topology, Lemma 5.9.2 and Properties, Lemma 28.5.5) we may assume that $Y$ is a disjoint union of irreducible Noetherian schemes. Since there is an immersion

(details omitted) we see that it suffices to prove the result in case $Y$ is irreducible.

Assume $Y$ is an irreducible scheme. Let $T \subset |X|$ be the closure of the image of $f : Y \to X$. Note that since $|Y|$ and $|X|$ are sober topological spaces (Properties of Spaces, Lemma 65.15.1) $T$ is irreducible with a unique generic point $\xi $ which is the image of the generic point $\eta $ of $Y$. Let $\mathcal{I} \subset X$ be a quasi-coherent sheaf of ideals cutting out the reduced induced space structure on $T$ (Properties of Spaces, Definition 65.12.5). Since $\mathcal{O}_{Y, \eta }$ is an Artinian local ring we see that for some $n > 0$ we have $f^{-1}\mathcal{I}^ n \mathcal{O}_{Y, \eta } = 0$. As $f^{-1}\mathcal{I}\mathcal{O}_ Y$ is a finite type quasi-coherent ideal we conclude that $f^{-1}\mathcal{I}^ n\mathcal{O}_ V = 0$ for some nonempty open $V \subset Y$. Let $Z \subset X$ be the closed subspace cut out by $\mathcal{I}^ n$. By construction $V \to Y \to X$ factors through $Z$. Because $\mathbf{A}^ n_ Z \to \mathbf{A}^ n_ X$ is an immersion, we may replace $X$ by $Z$ and $Y$ by $V$. Hence we reach the situation where $Y$ and $X$ are irreducible and $Y \to X$ maps the generic point of $Y$ onto the generic point of $X$.

Assume $Y$ and $X$ are irreducible, $Y$ is a scheme, and $Y \to X$ maps the generic point of $Y$ onto the generic point of $X$. By Properties of Spaces, Proposition 65.13.3 $X$ has a dense open subscheme $U \subset X$. Choose a nonempty affine open $V \subset Y$ whose image in $X$ is contained in $U$. By Morphisms, Lemma 29.39.2 we may factor $V \to U$ as $V \to \mathbf{A}^ n_ U \to U$. Composing with $\mathbf{A}^ n_ U \to \mathbf{A}^ n_ X$ we obtain the desired immersion. $\square$

## 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)