The Stacks project

Lemma 42.49.3. In Lemma 42.49.1 the bivariant class $P'_ p(Q)$, resp. $c'_ p(Q)$ is independent of the choice of the closed subscheme $T$. Moreover, given a proper morphism $g : W' \to W$ which is an isomorphism over $\mathbf{A}^1_ X$, then setting $Q' = g^*Q$ we have $P'_ p(Q) = P'_ p(Q')$, resp. $c'_ p(Q) = c'_ p(Q')$.

Proof. The independence of $T$ follows immediately from Lemma 42.47.2.

Let $g : W' \to W$ be a proper morphism which is an isomorphism over $\mathbf{A}^1_ X$. Observe that taking $T' = g^{-1}(T) \subset W'_\infty $ is a closed subscheme satisfying (A2) hence the operator $P'_ p(Q')$, resp. $c'_ p(Q')$ in $A^ p(Z \to X)$ corresponding to $b' = b \circ g : W' \to \mathbf{P}^1_ X$ and $Q'$ is defined. Denote $E' \subset W'_\infty $ the inverse image of $Z$ in $W'_\infty $. Recall that

\[ c'_ p(Q') = (E' \to Z)_* \circ c'_ p(Q'|_{E'}) \circ C' \]

with $C' \in A^0(W'_\infty \to X)$ and $c'_ p(Q'|_{E'}) \in A^ p(E' \to W'_\infty )$. By Lemma 42.48.3 we have $g_{\infty , *} \circ C' = C$. Observe that $E'$ is also the inverse image of $E$ in $W'_\infty $ by $g_\infty $. Since moreover $Q' = g^*Q$ we find that $c'_ p(Q'|_{E'})$ is simply the restriction of $c'_ p(Q|_ E)$ to schemes lying over $W'_\infty $, see Remark 42.33.5. Thus we obtain

\begin{align*} c'_ p(Q') & = (E' \to Z)_* \circ c'_ p(Q'|_{E'}) \circ C' \\ & = (E \to Z)_* \circ (E' \to E)_* \circ c'_ p(Q|_ E) \circ C' \\ & = (E \to Z)_* \circ c'_ p(Q|_ E) \circ g_{\infty , *} \circ C' \\ & = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C \\ & = c'_ p(Q) \end{align*}

In the third equality we used that $c'_ p(Q|_ E)$ commutes with proper pushforward as it is a bivariant class. The equality $P'_ p(Q) = P'_ p(Q')$ is proved in exactly the same way. $\square$

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