The Stacks project

Lemma 82.22.5. In Situation 82.2.1. Let $f : X' \to X$ be a proper morphism of good algebraic spaces over $B$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 82.22.1. Form the diagram

\[ \xymatrix{ D' \ar[d]_ g \ar[r]_{i'} & X' \ar[d]^ f \\ D \ar[r]^ i & X } \]

as in Remark 82.22.3. For any $(k + 1)$-cycle $\alpha '$ on $X'$ we have $i^*f_*\alpha ' = g_*(i')^*\alpha '$ in $\mathop{\mathrm{CH}}\nolimits _ k(D)$ (this makes sense as $f_*$ is defined on the level of cycles).

Proof. Suppose $\alpha = [W']$ for some integral closed subspace $W' \subset X'$. Let $W \subset X$ be the “image” of $W'$ as in Lemma 82.7.1. In case $W' \not\subset D'$, then $W \not\subset D$ and we see that

\[ [W' \cap D']_ k = \text{div}_{\mathcal{L}'|_{W'}}({s'|_{W'}}) \quad \text{and}\quad [W \cap D]_ k = \text{div}_{\mathcal{L}|_ W}(s|_ W) \]

and hence $f_*$ of the first cycle equals the second cycle by Lemma 82.19.3. Hence the equality holds as cycles. In case $W' \subset D'$, then $W \subset D$ and $f_*(c_1(\mathcal{L}|_{W'}) \cap [W'])$ is equal to $c_1(\mathcal{L}|_ W) \cap [W]$ in $\mathop{\mathrm{CH}}\nolimits _ k(W)$ by the second assertion of Lemma 82.19.3. By Remark 82.15.3 the result follows for general $\alpha '$. $\square$


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Your email address will not be published.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 0ER5. Beware of the difference between the letter 'O' and the digit '0'.