The Stacks project

Lemma 67.48.4. Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $Y \to X' \to X$ be the normalization of $X$ in $Y$.

  1. If $W \to X$ is an ├ętale morphism of algebraic spaces over $S$, then $W \times _ X X'$ is the normalization of $W$ in $W \times _ X Y$.

  2. If $Y$ and $X$ are representable, then $Y'$ is representable and is canonically isomorphic to the normalization of the scheme $X$ in the scheme $Y$ as constructed in Morphisms, Section 29.54.

Proof. It is immediate from the construction that the formation of the normalization of $X$ in $Y$ commutes with ├ętale base change, i.e., part (1) holds. On the other hand, if $X$ and $Y$ are schemes, then for $U \subset X$ affine open, $f_*\mathcal{O}_ Y(U) = \mathcal{O}_ Y(f^{-1}(U))$ and hence $\nu ^{-1}(U)$ is the spectrum of exactly the same ring as we get in the corresponding construction for schemes. $\square$

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Your email address will not be published.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