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 $r$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. 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 (2)

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).

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