Lemma 45.14.4. Assume given data (D0), (D1), and (D2') satisfying axioms (A1) – (A4) and (A7). Let $X$ be a smooth projective scheme over $k$. Let $Z \subset X$ be a smooth closed subscheme such that every irreducible component of $Z$ has codimension $r$ in $X$. Assume the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$ is the restriction of an element of $K_0(X)$. If $a \in H^*(X)$ and $a|_ Z = 0$ in $H^*(Z)$, then $\gamma ([Z]) \cup a = 0$.
Proof. Let $b : X' \to X$ be the blowing up. By (A7) it suffices to show that
\[ b^*(\gamma ([Z]) \cup a) = b^*\gamma ([Z]) \cup b^*a = 0 \]
By Lemma 45.14.3 we have
\[ b^*\gamma ([Z]) = \gamma (b^*[Z]) = \gamma ([E] \cdot \theta ) = \gamma ([E]) \cup \gamma (\theta ) \]
Hence because $b^*a$ restricts to zero on $E$ and since $\gamma ([E]) = c^ H_1(\mathcal{O}_{X'}(E))$ we get what we want from (A4). $\square$
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Your email address will not be published.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)