The Stacks project

Lemma 32.14.2. Let $f : X \to S$ be a quasi-compact morphism of schemes. The following are equivalent

  1. $f$ is universally closed,

  2. for every morphism $S' \to S$ which is locally of finite presentation the base change $X_{S'} \to S'$ is closed, and

  3. for every $n$ the morphism $\mathbf{A}^ n \times X \to \mathbf{A}^ n \times S$ is closed.

Proof. It is clear that (1) implies (2). Let us prove that (2) implies (1). Suppose that the base change $X_ T \to T$ is not closed for some scheme $T$ over $S$. By Schemes, Lemma 26.19.8 this means that there exists some specialization $t_1 \leadsto t$ in $T$ and a point $\xi \in X_ T$ mapping to $t_1$ such that $\xi $ does not specialize to a point in the fibre over $t$. Set $Z = \overline{\{ \xi \} } \subset X_ T$. Then $Z \cap X_ t = \emptyset $. Apply Lemma 32.14.1. We find an open neighbourhood $V \subset T$ of $t$, a commutative diagram

\[ \xymatrix{ V \ar[d] \ar[r]_ a & T' \ar[d]^ b \\ T \ar[r]^ g & S, } \]

and a closed subscheme $Z' \subset X_{T'}$ such that

  1. the morphism $b : T' \to S$ is locally of finite presentation,

  2. with $t' = a(t)$ we have $Z' \cap X_{t'} = \emptyset $, and

  3. $Z \cap X_ V$ maps into $Z'$ via the morphism $X_ V \to X_{T'}$.

Clearly this means that $X_{T'} \to T'$ maps the closed subset $Z'$ to a subset of $T'$ which contains $a(t_1)$ but not $t' = a(t)$. Since $a(t_1) \leadsto a(t) = t'$ we conclude that $X_{T'} \to T'$ is not closed. Hence we have shown that $X \to S$ not universally closed implies that $X_{T'} \to T'$ is not closed for some $T' \to S$ which is locally of finite presentation. In order words (2) implies (1).

Assume that $\mathbf{A}^ n \times X \to \mathbf{A}^ n \times S$ is closed for every integer $n$. We want to prove that $X_ T \to T$ is closed for every scheme $T$ which is locally of finite presentation over $S$. We may of course assume that $T$ is affine and maps into an affine open $V$ of $S$ (since $X_ T \to T$ being a closed is local on $T$). In this case there exists a closed immersion $T \to \mathbf{A}^ n \times V$ because $\mathcal{O}_ T(T)$ is a finitely presented $\mathcal{O}_ S(V)$-algebra, see Morphisms, Lemma 29.21.2. Then $T \to \mathbf{A}^ n \times S$ is a locally closed immersion. Hence we get a cartesian diagram

\[ \xymatrix{ X_ T \ar[d]_{f_ T} \ar[r] & \mathbf{A}^ n \times X \ar[d]^{f_ n} \\ T \ar[r] & \mathbf{A}^ n \times S } \]

of schemes where the horizontal arrows are locally closed immersions. Hence any closed subset $Z \subset X_ T$ can be written as $X_ T \cap Z'$ for some closed subset $Z' \subset \mathbf{A}^ n \times X$. Then $f_ T(Z) = T \cap f_ n(Z')$ and we see that if $f_ n$ is closed, then also $f_ T$ is closed. $\square$


Comments (0)


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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05JX. Beware of the difference between the letter 'O' and the digit '0'.