The Stacks project

Lemma 42.49.4. In Lemma 42.49.1 assume $Q|_ T$ is isomorphic to a finite locally free $\mathcal{O}_ T$-module of rank $< p$. Denote $C \in A^0(W_\infty \to X)$ the class of Lemma 42.48.1. Then

\[ C \circ c_ p(Q|_{X \times \{ 0\} }) = C \circ (Z \to X)_* \circ c'_ p(Q) = c_ p(Q|_{W_\infty }) \circ C \]

Proof. The first equality holds because $c_ p(Q|_{X \times \{ 0\} }) = (Z \to X)_* \circ c'_ p(Q)$ by Lemma 42.49.1. We may prove the second equality one cycle class at a time (see Lemma 42.35.3). Since the construction of the bivariant classes in the lemma is compatible with base change, we may assume we have some $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ and we have to show that $C \cap (Z \to X)_*(c'_ p(Q) \cap \alpha ) = c_ p(Q|_{W_\infty }) \cap C \cap \alpha $. Observe that

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

as desired. For the first equality we used that $c'_ p(Q) = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C$ where $E \subset W_\infty $ is the inverse image of $Z$ and $c'_ p(Q|_ E)$ is the class constructed in Lemma 42.47.1. The second equality is just the statement that $E \to Z \to X$ is equal to $E \to W_\infty \to X$. For the third equality we choose $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$ whose restriction to $b^{-1}(\mathbf{A}^1_ X)$ is the flat pullback of $\alpha $ so that $C \cap \alpha = i_\infty ^*\beta $ by construction. The fourth equality is Lemma 42.47.4. The fifth equality is the fact that $c_ p(Q)$ is a bivariant class and hence commutes with $i_\infty ^*$. The sixth equality is Lemma 42.48.4. The seventh uses again that $c_ p(Q)$ is a bivariant class. The final holds as $C \cap \alpha = i_\infty ^*\beta $. $\square$

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