The Stacks project

Lemma 42.25.2. 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 $Y$. Let $\alpha $ be a $k$-cycle on $Y$. Then

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

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

Proof. Write $\alpha = \sum n_ i[W_ i]$. We will show that

\[ f^*(c_1(\mathcal{L}) \cap [W_ i]) = c_1(f^*\mathcal{L}) \cap f^*[W_ i] \]

in $\mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X)$ by producing a rational equivalence on the closed subscheme $f^{-1}(W_ i)$ of $X$. By the discussion in Remark 42.19.5 this will prove the equality of the lemma is true.

Let $W \subset Y$ be an integral closed subscheme of $\delta $-dimension $k$. Consider the closed subscheme $W' = f^{-1}(W) = W \times _ Y X$ so that we have the fibre product diagram

\[ \xymatrix{ W' \ar[r] \ar[d]_ h & X \ar[d]^ f \\ W \ar[r] & Y } \]

We have to show that $f^*(c_1(\mathcal{L}) \cap [W]) = c_1(f^*\mathcal{L}) \cap f^*[W]$. Choose a nonzero meromorphic section $s$ of $\mathcal{L}|_ W$. Let $W'_ i \subset W'$ be the irreducible components of $\delta $-dimension $k + r$. Write $[W']_{k + r} = \sum n_ i[W'_ i]$ with $n_ i$ the multiplicity of $W'_ i$ in $W'$ as per definition. So $f^*[W] = \sum n_ i[W'_ i]$ in $Z_{k + r}(X)$. Since each $W'_ i \to W$ is dominant we see that $s_ i = s|_{W'_ i}$ is a nonzero meromorphic section for each $i$. By Lemma 42.25.1 we have the following equality of cycles

\[ h^*\text{div}_{\mathcal{L}|_ W}(s) = \sum n_ i\text{div}_{f^*\mathcal{L}|_{W'_ i}}(s_ i) \]

in $Z_{k + r - 1}(W')$. This finishes the proof since the left hand side is a cycle on $W'$ which pushes to $f^*(c_1(\mathcal{L}) \cap [W])$ in $\mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X)$ and the right hand side is a cycle on $W'$ which pushes to $c_1(f^*\mathcal{L}) \cap f^*[W]$ in $\mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X)$. $\square$


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