The Stacks project

Lemma 42.50.8. In Situation 42.50.1 assume $Z \subset Z' \subset X$ where $Z'$ is a closed subscheme of $X$. Then $P_ p(Z' \to X, E) = (Z \to Z')_* \circ P_ p(Z \to X, E)$, resp. $c_ p(Z' \to X, E) = (Z \to Z')_* \circ c_ p(Z \to X, E)$ (with $\circ $ as in Lemma 42.33.4).

Proof. The construction of $P_ p(Z' \to X, E)$, resp. $c_ p(Z' \to X, E)$ in Lemma 42.50.2 uses the exact same morphism $b : W \to \mathbf{P}^1_ X$ and perfect object $Q$ of $D(\mathcal{O}_ W)$. Then we can use Lemma 42.47.5 to conclude. Some details omitted. $\square$

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