Lemma 45.14.4 (0FII)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 45: Weil Cohomology Theories
Section 45.14: Weil cohomology theories, III
Lemma 45.14.4 (cite)

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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). $\square$

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