The Stacks project

Lemma 42.24.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$ are integral and $n = \dim _\delta (Y)$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ Y$-module. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Then

\[ f^*(c_1(\mathcal{L}) \cap [Y]) = c_1(f^*\mathcal{L}) \cap [X] \]

in $\mathop{\mathrm{CH}}\nolimits _{n + r - 1}(X)$.

Proof. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. We will show that actually $f^*\text{div}_\mathcal {L}(s) = \text{div}_{f^*\mathcal{L}}(f^*s)$ and hence the lemma holds. To see this let $\xi \in Y$ be a point and let $s_\xi \in \mathcal{L}_\xi $ be a generator. Write $s = gs_\xi $ with $g \in R(Y)^*$. Then there is an open neighbourhood $V \subset Y$ of $\xi $ such that $s_\xi \in \mathcal{L}(V)$ and such that $s_\xi $ generates $\mathcal{L}|_ V$. Hence we see that

\[ \text{div}_\mathcal {L}(s)|_ V = \text{div}_ Y(g)|_ V. \]

In exactly the same way, since $f^*s_\xi $ generates $f^*\mathcal{L}$ over $f^{-1}(V)$ and since $f^*s = g f^*s_\xi $ we also have

\[ \text{div}_\mathcal {L}(f^*s)|_{f^{-1}(V)} = \text{div}_ X(g)|_{f^{-1}(V)}. \]

Thus the desired equality of cycles over $f^{-1}(V)$ follows from the corresponding result for pullbacks of principal divisors, see Lemma 42.17.2. $\square$

Comments (4)

Comment #6535 by Xuande Liu on

"In exactly the same way, since f^s_eta generates L over f^{-1}(V)". It should be f^s_eta generate f^*L over f^{-1}(V).

Comment #7917 by q1k on

The line bundle L is stated twice

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Your email address will not be published.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 02SM. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 42.24.3 as pdf