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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