Lemma 42.59.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $p : P \to X$ be a smooth morphism of schemes locally of finite type over $S$ and let $s : X \to P$ be a section. Then $s$ is a regular immersion and $1 = s^! \circ p^*$ in $A^*(X)^\wedge $ where $p^* \in A^*(P \to X)^\wedge $ is the bivariant class of Lemma 42.33.2.
Proof. The first statement is Divisors, Lemma 31.22.8. It suffices to show that $s^! \cap p^*[Z] = [Z]$ in $\mathop{\mathrm{CH}}\nolimits _*(X)$ for any integral closed subscheme $Z \subset X$ as the assumptions are preserved by base change by $X' \to X$ locally of finite type. After replacing $P$ by an open neighbourhood of $s(Z)$ we may assume $P \to X$ is smooth of fixed relative dimension $r$. Say $\dim _\delta (Z) = n$. Then every irreducible component of $p^{-1}(Z)$ has dimension $r + n$ and $p^*[Z]$ is given by $[p^{-1}(Z)]_{n + r}$. Observe that $s(X) \cap p^{-1}(Z) = s(Z)$ scheme theoretically. Hence by the same reference as used above $s(X) \cap p^{-1}(Z)$ is a closed subscheme regularly embedded in $\overline{p}^{-1}(Z)$ of codimension $r$. We conclude by Lemma 42.54.5. $\square$
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