The Stacks project

Lemma 42.38.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $Y$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\alpha $ be a $k$-cycle on $Y$. Then

\[ f^*(c_ j(\mathcal{E}) \cap \alpha ) = c_ j(f^*\mathcal{E}) \cap f^*\alpha \]

Proof. Write $\alpha _ j = c_ j(\mathcal{E}) \cap \alpha $, so $\alpha _0 = \alpha $. By Lemma 42.38.2 we have

\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(\alpha _{r - i}) = 0 \]

in the chow group of the projective bundle $(\pi : P \to Y, \mathcal{O}_ P(1))$ associated to $\mathcal{E}$. Consider the fibre product diagram

\[ \xymatrix{ P_ X = \mathbf{P}(f^*\mathcal{E}) \ar[r]_-{f'} \ar[d]_{\pi _ X} & P \ar[d]^\pi \\ X \ar[r]^ f & Y } \]

Note that $\mathcal{O}_{P_ X}(1)$ is the pullback of $\mathcal{O}_ P(1)$. Apply flat pullback $(f')^*$ (Lemma 42.20.2) to the displayed equation above. By Lemmas 42.26.2 and 42.14.3 we see that

\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P_ X}(1))^ i \cap \pi _ X^*(f^*\alpha _{r - i}) = 0 \]

holds in the chow group of $P_ X$. By the characterization of Lemma 42.38.2 we conclude. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 42.38: Intersecting with Chern classes

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Your email address will not be published.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 02U8. Beware of the difference between the letter 'O' and the digit '0'.