The Stacks project

Lemma 82.6.3. In Situation 82.2.1 let $X/B$ be good. Let $Y \subset X$ be a closed subspace. If $\dim _\delta (Y) \leq k$, then $[Y]_ k = [i_*\mathcal{O}_ Y]_ k$ where $i : Y \to X$ is the inclusion morphism.

Proof. Let $Z$ be an integral closed subspace of $X$ with $\dim _\delta (Z) = k$. If $Z \not\subset Y$ the $Z$ has coefficient zero in both $[Y]_ k$ and $[i_*\mathcal{O}_ Y]_ k$. If $Z \subset Y$, then the generic point of $Z$ may be viewed as a point $y \in |Y|$ whose image $x \in |X|$. Then the coefficient of $Z$ in $[Y]_ k$ is the length of $\mathcal{O}_ Y$ at $y$ and the coefficient of $Z$ in $[i_*\mathcal{O}_ Y]_ k$ is the length of $i_*\mathcal{O}_ Y$ at $x$. Thus the equality of the coefficients follows from Lemma 82.4.3. $\square$


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Your email address will not be published.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