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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