The Stacks project

Lemma 82.18.3. In Situation 82.2.1 let $Y/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ Y$-module. Let $s \in \Gamma (Y, \mathcal{L})$ be a regular section and assume $\dim _\delta (Y) \leq k + 1$. Write $[Y]_{k + 1} = \sum n_ i[Y_ i]$ where $Y_ i \subset Y$ are the irreducible components of $Y$ of $\delta $-dimension $k + 1$. Set $s_ i = s|_{Y_ i} \in \Gamma (Y_ i, \mathcal{L}|_{Y_ i})$. Then

82.18.3.1
\begin{equation} \label{spaces-chow-equation-equal-as-cycles} [Z(s)]_ k = \sum n_ i[Z(s_ i)]_ k \end{equation}

as $k$-cycles on $Y$.

Proof. Let $\varphi : V \to Y$ be a surjective étale morphism where $V$ is a scheme. It suffices to prove the equality after pulling back by $\varphi $, see Lemma 82.10.3. That same lemma tells us that $\varphi ^*[Y_ i] = [\varphi ^{-1}(Y_ i)] = \sum [V_{i, j}]$ where $V_{i, j}$ are the irreducible components of $V$ lying over $Y_ i$. Hence if we first apply the case of schemes (Chow Homology, Lemma 42.25.3) to $\varphi ^*s_ i$ on $Y_ i \times _ Y V$ we find that $\varphi ^*[Z(s_ i)]_ k = [Z(\varphi ^*s_ i)] = \sum [Z(s_{i, j})]_ k$ where $s_{i, j}$ is the pullback of $s$ to $V_{i, j}$. Applying the case of schemes to $\varphi ^*s$ we get

\[ \varphi ^*[Z(s)]_ k = [Z(\varphi ^*s)]_ k = \sum n_ i[Z(s_{i, j})]_ k \]

by our remark on multiplicities above. Combining all of the above the proof is complete. $\square$


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Your email address will not be published.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