The Stacks project

Lemma 42.51.2. Assume $(S, \delta ), X, Z, b : W \to \mathbf{P}^1_ X, Q, T, p$ satisfy the assumptions of Lemma 42.49.1. Let $F \in D(\mathcal{O}_ X)$ be a perfect object such that

  1. the restriction of $Q$ to $b^{-1}(\mathbf{A}^1_ X)$ is isomorphic to the pullback of $F$,

  2. $F|_{X \setminus Z}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$, and

  3. $Q$ on $W$ and $F$ on $X$ satisfy assumption (3) of Situation 42.50.1.

Then the class $P'_ p(Q)$, resp. $c'_ p(Q)$ in $A^ p(Z \to X)$ constructed in Lemma 42.49.1 is equal to $P_ p(Z \to X, F)$, resp. $c_ p(Z \to X, F)$ from Definition 42.50.3.

Proof. The assumptions are preserved by base change with a morphism $X' \to X$ locally of finite type. Hence it suffices to show that $P_ p(Z \to X, F) \cap \alpha = P'_ p(Q) \cap \alpha $, resp. $c_ p(Z \to X, F) \cap \alpha = c'_ p(Q) \cap \alpha $ for any $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$. Choose $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$ whose restriction to $b^{-1}(\mathbf{A}^1_ X)$ is equal to the flat pullback of $\alpha $ as in the construction of $C$ in Lemma 42.48.1. Denote $W' = b^{-1}(Z)$ and denote $E = W'_\infty \subset W_\infty $ the inverse image of $Z$ by $W_\infty \to X$. The lemma follows from the following sequence of equalities (the case of $P_ p$ is similar)

\begin{align*} c'_ p(Q) \cap \alpha & = (E \to Z)_*(c'_ p(Q|_ E) \cap i_\infty ^*\beta ) \\ & = (E \to Z)_*(c_ p(E \to W_\infty , Q|_{W_\infty }) \cap i_\infty ^*\beta ) \\ & = (W'_\infty \to Z)_*(c_ p(W' \to W, Q) \cap i_\infty ^*\beta ) \\ & = (W'_\infty \to Z)_*((i'_\infty )^*(c_ p(W' \to W, Q) \cap \beta )) \\ & = (W'_\infty \to Z)_*((i'_\infty )^*(c_ p(Z' \to X, F) \cap \beta )) \\ & = (W'_0 \to Z)_*((i'_0)^*(c_ p(Z' \to X, F) \cap \beta )) \\ & = (W'_0 \to Z)_*(c_ p(Z' \to X, F) \cap i_0^*\beta )) \\ & = c_ p(Z \to X, F) \cap \alpha \end{align*}

The first equality is the construction of $c'_ p(Q)$ in Lemma 42.49.1. The second is Lemma 42.50.9. The base change of $W' \to W$ by $W_\infty \to W$ is the morphism $E = W'_\infty \to W_\infty $. Hence the third equality holds by Lemma 42.50.4. The fourth equality, in which $i'_\infty : W'_\infty \to W'$ is the inclusion morphism, follows from the fact that $c_ p(W' \to W, Q)$ is a bivariant class. For the fifth equality, observe that $c_ p(W' \to W, Q)$ and $c_ p(Z' \to X, F)$ restrict to the same bivariant class in $A^ p((b')^{-1} \to b^{-1}(\mathbf{A}^1_ X))$ by assumption (1) of the lemma which says that $Q$ and $F$ restrict to the same object of $D(\mathcal{O}_{b^{-1}(\mathbf{A}^1_ X)})$; use Lemma 42.50.4. Since $(i'_\infty )^*$ annihilates cycles supported on $W'_\infty $ (see Remark 42.29.6) we conclude the fifth equality is true. The sixth equality holds because $W'_\infty $ and $W'_0$ are the pullbacks of the rationally equivalent effective Cartier divisors $D_0, D_\infty $ in $\mathbf{P}^1_ Z$ and hence $i_\infty ^*\beta $ and $i_0^*\beta $ map to the same cycle class on $W'$; namely, both represent the class $c_1(\mathcal{O}_{\mathbf{P}^1_ Z}(1)) \cap c_ p(Z \to X, F_) \cap \beta $ by Lemma 42.29.4. The seventh equality holds because $c_ p(Z \to X, F)$ is a bivariant class. By construction $W'_0 = Z$ and $i_0^*\beta = \alpha $ which explains why the final equality holds. $\square$

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 0FE8. Beware of the difference between the letter 'O' and the digit '0'.