Very weak form of [Theorem 17.1, F]
Lemma 42.35.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$. Let $p \in \mathbf{Z}$. Suppose given a rule which assigns to every locally of finite type morphism $Y' \to Y$ and every $k$ a map
\[ c \cap - : Z_ k(Y') \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k - p}(X') \]
where $Y' = X' \times _ X Y$, satisfying condition (3) of Definition 42.33.1 whenever $\mathcal{L}'|_{D'} \cong \mathcal{O}_{D'}$. Then $c \cap -$ factors through rational equivalence.
Proof.
The statement makes sense because given a triple $(\mathcal{L}, s, i : D \to X)$ as in Definition 42.29.1 such that $\mathcal{L}|_ D \cong \mathcal{O}_ D$, then the operation $i^*$ is defined on the level of cycles, see Remark 42.29.6. Let $\alpha \in Z_ k(X')$ be a cycle which is rationally equivalent to zero. We have to show that $c \cap \alpha = 0$. By Lemma 42.21.1 there exists a cycle $\beta \in Z_{k + 1}(X' \times \mathbf{P}^1)$ such that $\alpha = i_0^*\beta - i_\infty ^*\beta $ where $i_0, i_\infty : X' \to X' \times \mathbf{P}^1$ are the closed immersions of $X'$ over $0, \infty $. Since these are examples of effective Cartier divisors with trivial normal bundles, we see that $c \cap i_0^*\beta = j_0^*(c \cap \beta )$ and $c \cap i_\infty ^*\beta = j_\infty ^*(c \cap \beta )$ where $j_0, j_\infty : Y' \to Y' \times \mathbf{P}^1$ are closed immersions as before. Since $j_0^*(c \cap \beta ) \sim _{rat} j_\infty ^*(c \cap \beta )$ (follows from Lemma 42.21.1) we conclude.
$\square$
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