42.31 Relative effective Cartier divisors
Relative effective Cartier divisors are defined and studied in Divisors, Section 31.18. To develop the basic results on Chern classes of vector bundles we only need the case where both the ambient scheme and the effective Cartier divisor are flat over the base.
Lemma 42.31.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $p : X \to Y$ be a flat morphism of relative dimension $r$. Let $i : D \to X$ be a relative effective Cartier divisor (Divisors, Definition 31.18.2). Let $\mathcal{L} = \mathcal{O}_ X(D)$. For any $\alpha \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(Y)$ we have
\[ i^*p^*\alpha = (p|_ D)^*\alpha \]
in $\mathop{\mathrm{CH}}\nolimits _{k + r}(D)$ and
\[ c_1(\mathcal{L}) \cap p^*\alpha = i_* ((p|_ D)^*\alpha ) \]
in $\mathop{\mathrm{CH}}\nolimits _{k + r}(X)$.
Proof.
Let $W \subset Y$ be an integral closed subscheme of $\delta $-dimension $k + 1$. By Divisors, Lemma 31.18.1 we see that $D \cap p^{-1}W$ is an effective Cartier divisor on $p^{-1}W$. By Lemma 42.29.5 we get the first equality in
\[ i^*[p^{-1}W]_{k + r + 1} = [D \cap p^{-1}W]_{k + r} = [(p|_ D)^{-1}(W)]_{k + r}. \]
and the second because $D \cap p^{-1}(W) = (p|_ D)^{-1}(W)$ as schemes. Since by definition $p^*[W] = [p^{-1}W]_{k + r + 1}$ we see that $i^*p^*[W] = (p|_ D)^*[W]$ as cycles. If $\alpha = \sum m_ j[W_ j]$ is a general $k + 1$ cycle, then we get $i^*\alpha = \sum m_ j i^*p^*[W_ j] = \sum m_ j(p|_ D)^*[W_ j]$ as cycles. This proves then first equality. To deduce the second from the first apply Lemma 42.29.4.
$\square$
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