The Stacks project

Lemma 42.59.13. 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 $S$ such that both $X$ and $Y$ are quasi-compact, regular, have affine diagonal, and finite dimension. Then $f$ is a local complete intersection morphism. Assume moreover the gysin map exists for $f$ and that $f$ is proper. Then

\[ f_*(\alpha \cdot f^!\beta ) = f_*\alpha \cdot \beta \]

in $\mathop{\mathrm{CH}}\nolimits ^*(Y) \otimes \mathbf{Q}$ where the intersection product is as in Section 42.58.

Proof. The first statement follows from More on Morphisms, Lemma 37.62.11. Observe that $f^![Y] = [X]$, see Lemma 42.59.8. Write $\alpha = ch(\alpha ') \cap [X]$ and $\beta = ch(\beta ') \cap [Y]$ $\alpha ' \in K_0(\textit{Vect}(X)) \otimes \mathbf{Q}$ and $\beta ' \in K_0(\textit{Vect}(Y)) \otimes \mathbf{Q}$ as in Section 42.58. Set $c = ch(\alpha ')$ and $c' = ch(\beta ')$. We have

\begin{align*} f_*(\alpha \cdot f^!\beta ) & = f_*(c \cap f^!(c' \cap [Y]_ e)) \\ & = f_*(c \cap c' \cap f^![Y]_ e) \\ & = f_*(c \cap c' \cap [X]_ d) \\ & = f_*(c' \cap c \cap [X]_ d) \\ & = c' \cap f_*(c \cap [X]_ d) \\ & = \beta \cdot f_*(\alpha ) \end{align*}

The first equality by the construction of the intersection product. By Lemma 42.59.7 we know that $f^!$ commutes with $c'$. The fact that Chern classes are in the center of the bivariant ring justifies switching the order of capping $[X]$ with $c$ and $c'$. Commuting $c'$ with $f_*$ is allowed as $c'$ is a bivariant class. The final equality is again the construction of the intersection product. $\square$


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Your email address will not be published.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 0FFA. Beware of the difference between the letter 'O' and the digit '0'.