The Stacks project

Lemma 42.47.2. In Lemma 42.47.1 the bivariant class $P'_ p(E_2)$, resp. $c'_ p(E_2)$ in $A^ p(X_2 \to X)$ does not depend on the choice of $X_1$.

Proof. Suppose that $X_1' \subset X$ is another closed subscheme such that $X = X'_1 \cup X_2$ set theoretically and the restriction $E_2|_{X'_1 \cap X_2}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X'_1 \cap X_2}$-module of rank $< p$ sitting in cohomological degree $0$. Then $X = (X_1 \cap X'_1) \cup X_2$. Hence we can write any element $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ as $i_*\beta + i_{2, *}\alpha _2$ with $\alpha _2 \in \mathop{\mathrm{CH}}\nolimits _ k(X'_2)$ and $\beta \in \mathop{\mathrm{CH}}\nolimits _ k(X_1 \cap X'_1)$. Thus it is clear that $P'_ p(E_2) \cap \alpha = P_ p(E_2) \cap \alpha _2 \in \mathop{\mathrm{CH}}\nolimits _{k - p}(X_2)$, resp. $c'_ p(E_2) \cap \alpha = c_ p(E_2) \cap \alpha _2 \in \mathop{\mathrm{CH}}\nolimits _{k - p}(X_2)$, is independent of whether we use $X_1$ or $X'_1$. Similarly after any base change. $\square$

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