Lemma 42.26.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Assume $Y$ is integral and $n = \dim _\delta (Y)$. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. Then we have

$f^*\text{div}_\mathcal {L}(s) = \sum n_ i\text{div}_{f^*\mathcal{L}|_{X_ i}}(s_ i)$

in $Z_{n + r - 1}(X)$. Here the sum is over the irreducible components $X_ i \subset X$ of $\delta$-dimension $n + r$, the section $s_ i = f|_{X_ i}^*(s)$ is the pullback of $s$, and $n_ i = m_{X_ i, X}$ is the multiplicity of $X_ i$ in $X$.

Proof. To prove this equality of cycles, we may work locally on $Y$. Hence we may assume $Y$ is affine and $s = p/q$ for some nonzero sections $p \in \Gamma (Y, \mathcal{L})$ and $q \in \Gamma (Y, \mathcal{O})$. If we can show both

$f^*\text{div}_\mathcal {L}(p) = \sum n_ i\text{div}_{f^*\mathcal{L}|_{X_ i}}(p_ i) \quad \text{and}\quad f^*\text{div}_\mathcal {O}(q) = \sum n_ i\text{div}_{\mathcal{O}_{X_ i}}(q_ i)$

(with obvious notations) then we win by the additivity, see Divisors, Lemma 31.27.5. Thus we may assume that $s \in \Gamma (Y, \mathcal{L})$. In this case we may apply the equality (42.25.3.1) to see that

$[Z(f^*(s))]_{k + r - 1} = \sum n_ i\text{div}_{f^*\mathcal{L}|_{X_ i}}(s_ i)$

where $f^*(s) \in f^*\mathcal{L}$ denotes the pullback of $s$ to $X$. On the other hand we have

$f^*\text{div}_\mathcal {L}(s) = f^*[Z(s)]_{k - 1} = [f^{-1}(Z(s))]_{k + r - 1},$

by Lemmas 42.24.2 and 42.14.4. Since $Z(f^*(s)) = f^{-1}(Z(s))$ we win. $\square$

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