The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B7A. Beware of the difference between the letter 'O' and the digit '0'.