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.
$\square$

## 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Your email address will not be published.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.

## Comments (0)