The Stacks project

Lemma 42.49.2. In Lemma 42.49.1 let $X' \to X$ be a morphism which is locally of finite type. Denote $Z'$, $b' : W' \to \mathbf{P}^1_{X'}$, and $T' \subset W'_\infty $ the base changes of $Z$, $b : W \to \mathbf{P}^1_ X$, and $T \subset W_\infty $. Set $Q' = (W' \to W)^*Q$. Then the class $P'_ p(Q')$, resp. $c'_ p(Q')$ in $A^ p(Z' \to X')$ constructed as in Lemma 42.49.1 using $b'$, $Q'$, and $T'$ is the restriction (Remark 42.33.5) of the class $P'_ p(Q)$, resp. $c'_ p(Q)$ in $A^ p(Z \to X)$.

Proof. Recall that the construction is as follows

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

Thus the lemma follows from the corresponding base change property for $C$ (Lemma 42.48.2) and the fact that the same base change property holds for the classes constructed in Lemma 42.47.1 (small detail omitted). $\square$

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