The Stacks project

Lemma 42.54.8. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme with virtual normal sheaf $\mathcal{N}$. Let $Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$. Then $c$ and $c(Z \to X, \mathcal{N})$ commute (Remark 42.33.6).

Proof. To check this we may use Lemma 42.35.3. Thus we may assume $X$ is an integral scheme and we have to show $c \cap c(Z \to X, \mathcal{N}) \cap [X] = c(Z \to X, \mathcal{N}) \cap c \cap [X]$ in $\mathop{\mathrm{CH}}\nolimits _*(Z \times _ X Y)$.

If $Z = X$, then $c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{N})$ by Lemma 42.54.5 which commutes with the bivariant class $c$, see Lemma 42.38.9.

Assume that $Z$ is not equal to $X$. By Lemma 42.35.3 it even suffices to prove the result after blowing up $X$ (in a nonzero ideal). Let us blowup $X$ in the ideal sheaf of $Z$. This reduces us to the case where $Z$ is an effective Cartier divisor, see Divisors, Lemma 31.32.4,

If $Z$ is an effective Cartier divisor, then we have

\[ c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{E}) \circ i^* \]

where $i^* \in A^1(Z \to X)$ is the gysin homomorphism associated to $i : Z \to X$ (Lemma 42.33.3) and $\mathcal{E}$ is the dual of the kernel of $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$, see Lemmas 42.54.3 and 42.54.7. Then we conclude because Chern classes are in the center of the bivariant ring (in the strong sense formulated in Lemma 42.38.9) and $c$ commutes with the gysin homomorphism $i^*$ by definition of bivariant classes. $\square$


Comments (0)

There are also:

  • 1 comment(s) on Section 42.54: Higher codimension gysin homomorphisms

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Your email address will not be published.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