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
Comments (0)