The Stacks project

Lemma 33.7.8. Let $k$ be a field. Let $X$ be a scheme over $k$. Let $A$ be a $k$-algebra. Let $V \subset X_ A$ be a quasi-compact open. Then there exists a finitely generated $k$-subalgebra $A' \subset A$ and a quasi-compact open $V' \subset X_{A'}$ such that $V = V'_ A$.

Proof. We remark that if $X$ is also quasi-separated this follows from Limits, Lemma 32.4.11. Let $U_1, \ldots , U_ n$ be finitely many affine opens of $X$ such that $V \subset \bigcup U_{i, A}$. Say $U_ i = \mathop{\mathrm{Spec}}(R_ i)$. Since $V$ is quasi-compact we can find finitely many $f_{ij} \in R_ i \otimes _ k A$, $j = 1, \ldots , n_ i$ such that $V = \bigcup _ i \bigcup _{j = 1, \ldots , n_ i} D(f_{ij})$ where $D(f_{ij}) \subset U_{i, A}$ is the corresponding standard open. (We do not claim that $V \cap U_{i, A}$ is the union of the $D(f_{ij})$, $j = 1, \ldots , n_ i$.) It is clear that we can find a finitely generated $k$-subalgebra $A' \subset A$ such that $f_{ij}$ is the image of some $f_{ij}' \in R_ i \otimes _ k A'$. Set $V' = \bigcup D(f_{ij}')$ which is a quasi-compact open of $X_{A'}$. Denote $\pi : X_ A \to X_{A'}$ the canonical morphism. We have $\pi (V) \subset V'$ as $\pi (D(f_{ij})) \subset D(f_{ij}')$. If $x \in X_ A$ with $\pi (x) \in V'$, then $\pi (x) \in D(f_{ij}')$ for some $i, j$ and we see that $x \in D(f_{ij})$ as $f_{ij}'$ maps to $f_{ij}$. Thus we see that $V = \pi ^{-1}(V')$ as desired. $\square$

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Your email address will not be published.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 0388. Beware of the difference between the letter 'O' and the digit '0'.