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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)