Lemma 32.15.4. Let $f : X \to S$ be a finite type morphism of schemes. Assume $S$ is locally Noetherian. Then the following are equivalent

**Proof.**
The equivalence of (1) and (2) is a special case of Lemma 32.14.2. The equivalence of (1) and (3) is a special case of Schemes, Proposition 26.20.6. Trivially (3) implies (4). Thus all we have to do is prove that (4) implies (2). We will prove that $\mathbf{A}^ n \times X \to \mathbf{A}^ n \times S$ is closed by the criterion of Schemes, Lemma 26.19.8. Pick $n$ and a specialization $z \leadsto z'$ of points in $\mathbf{A}^ n \times S$ and a point $y \in \mathbf{A}^ n \times X$ lying over $z$. Note that $\kappa (y)$ is a finitely generated field extension of $\kappa (z)$ as $\mathbf{A}^ n \times X \to \mathbf{A}^ n \times S$ is of finite type. Hence by Properties, Lemma 28.5.10 or Algebra, Lemma 10.119.13 implies that there exists a discrete valuation ring $A \subset \kappa (y)$ with fraction field $\kappa (z)$ dominating the image of $\mathcal{O}_{\mathbf{A}^ n \times S, z'}$ in $\kappa (z)$. This gives a commutative diagram

Now property (4) implies that there exists a morphism $\mathop{\mathrm{Spec}}(A) \to X$ which fits into this diagram. Since we already have the morphism $\mathop{\mathrm{Spec}}(A) \to \mathbf{A}^ n$ from the left lower horizontal arrow we also get a morphism $\mathop{\mathrm{Spec}}(A) \to \mathbf{A}^ n \times X$ fitting into the left square. Thus the image $y' \in \mathbf{A}^ n \times X$ of the closed point is a specialization of $y$ lying over $z'$. This proves that specializations lift along $\mathbf{A}^ n \times X \to \mathbf{A}^ n \times S$ and we win. $\square$

## 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Your email address will not be published.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)