The Stacks project

Lemma 70.4.5. Let $S$ be a scheme. Let $X$ be a normal integral algebraic space over $S$. For every $x \in |X|$ there exists a normal integral affine scheme $U$ and an ├ętale morphism $U \to X$ such that $x$ is in the image.

Proof. Choose an affine scheme $U$ and an ├ętale morphism $U \to X$ such that $x$ is in the image. Let $u_ i$, $i \in I$ be the generic points of irreducible components of $U$. Then each $u_ i$ maps to the generic point of $X$ (Decent Spaces, Lemma 66.20.1). By our definition of a decent space (Decent Spaces, Definition 66.6.1), we see that $I$ is finite. Hence $U = \mathop{\mathrm{Spec}}(A)$ where $A$ is a normal ring with finitely many minimal primes. Thus $A = \prod _{i \in I} A_ i$ is a product of normal domains by Algebra, Lemma 10.36.16. Then $U = \coprod U_ i$ with $U_ i = \mathop{\mathrm{Spec}}(A_ i)$ and $x$ is in the image of $U_ i \to X$ for some $i$. This proves the lemma. $\square$

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Your email address will not be published.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