The Stacks project

Lemma 81.10.3. In Situation 81.2.1 let $f : X \to Y$ be an ├ętale morphism of good algebraic spaces over $B$. If $Z \subset Y$ is an integral closed subspace, then $f^*[Z] = \sum [Z']$ where the sum is over the irreducible components (Remark 81.5.1) of $f^{-1}(Z)$.

Proof. The meaning of the lemma is that the coefficient of $[Z']$ is $1$. This follows from the fact that $f^{-1}(Z)$ is a reduced algebraic space because it is ├ętale over the integral algebraic space $Z$. $\square$


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Your email address will not be published.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