The Stacks project

Definition 68.22.1. Let $S$ be a scheme. Let $X$ and $Y$ algebraic spaces over $S$. Assume $X$ and $Y$ are decent and that $|X|$ and $|Y|$ have finitely many irreducible components. We say a morphism $f : X \to Y$ is birational if

  1. $|f|$ induces a bijection between the set of generic points of irreducible components of $|X|$ and the set of generic points of the irreducible components of $|Y|$, and

  2. for every generic point $x \in |X|$ of an irreducible component the local ring map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is an isomorphism (see clarification below).

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Your email address will not be published.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



View Definition 68.22.1 as pdf