The Stacks project

Lemma 57.17.4. Let $K$ be an algebraically closed field. Let $S$ be a finite type scheme over $K$. Let $X \to S$ and $Y \to S$ be finite type morphisms. There exists a countable set $I$ and for $i \in I$ a pair $(S_ i \to S, h_ i)$ with the following properties

  1. $S_ i \to S$ is a morphism of finite type, set $X_ i = X \times _ S S_ i$ and $Y_ i = Y \times _ S S_ i$,

  2. $h_ i : X_ i \to Y_ i$ is an isomorphism over $S_ i$, and

  3. for any closed point $s \in S(K)$ if $X_ s \cong Y_ s$ over $K = \kappa (s)$ then $s$ is in the image of $S_ i \to S$ for some $i$.

Proof. The field $K$ is the filtered union of its countable subfields. Dually, $\mathop{\mathrm{Spec}}(K)$ is the cofiltered limit of the spectra of the countable subfields of $K$. Hence Limits, Lemma 32.10.1 guarantees that we can find a countable subfield $k$ and morphisms $X_0 \to S_0$ and $Y_0 \to S_0$ of schemes of finite type over $k$ such that $X \to S$ and $Y \to S$ are the base changes of these.

By Lemma 57.17.1 there is a countable set $I$ and pairs $(S_{0, i} \to S_0, h_{0, i})$ such that

  1. $S_{0, i} \to S_0$ is a morphism of finite type, set $X_{0, i} = X_0 \times _{S_0} S_{0, i}$ and $Y_{0, i} = Y_0 \times _{S_0} S_{0, i}$,

  2. $h_{0, i} : X_{0, i} \to Y_{0, i}$ is an isomorphism over $S_{0, i}$.

such that every pair $(T \to S_0, h_ T)$ with $T \to S_0$ of finite type and $h_ T : X_0 \times _{S_0} T \to Y_0 \times _{S_0} T$ an isomorphism is isomorphic to one of these. Denote $(S_ i \to S, h_ i)$ the base change of $(S_{0, i} \to S_0, h_{0, i})$ by $\mathop{\mathrm{Spec}}(K) \to \mathop{\mathrm{Spec}}(k)$. We claim this works.

Let $s \in S(K)$ and let $h_ s : X_ s \to Y_ s$ be an isomorphism over $K = \kappa (s)$. We can write $K$ as the filtered union of its finitely generated $k$-subalgebras. Hence by Limits, Proposition 32.6.1 and Lemma 32.10.1 we can find such a finitely generated $k$-subalgebra $K \supset A \supset k$ such that

  1. there is a commutative diagram

    \[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[d]_ s \ar[r] & \mathop{\mathrm{Spec}}(A) \ar[d]^{s'} \\ S \ar[r] & S_0} \]

    for some morphism $s' : \mathop{\mathrm{Spec}}(A) \to S_0$ over $k$,

  2. $h_ s$ is the base change of an isomorphism $h_{s'} : X_0 \times _{S_0, s'} \mathop{\mathrm{Spec}}(A) \to X_0 \times _{S_0, s'} \mathop{\mathrm{Spec}}(A)$ over $A$.

Of course, then $(s' : \mathop{\mathrm{Spec}}(A) \to S_0, h_{s'})$ is isomorphic to the pair $(S_{0, i} \to S_0, h_{0, i})$ for some $i \in I$. This concludes the proof because the commutative diagram in (1) shows that $s$ is in the image of the base change of $s'$ to $\mathop{\mathrm{Spec}}(K)$. $\square$

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 0G0X. Beware of the difference between the letter 'O' and the digit '0'.