Lemma 42.49.5 (0FAW)—The Stacks project

Processing math: 100%

Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.49: Preparation for localized Chern classes
Lemma 42.49.5 (cite)

numberstags

Lemma 42.49.5. In Lemma 42.49.1 let $Y \to X$ be a morphism locally of finite type and let $c \in A^*(Y \to X)$ be a bivariant class. Then

$P'_ p(Q) \circ c = c \circ P'_ p(Q) \quad \text{resp.}\quad c'_ p(Q) \circ c = c \circ c'_ p(Q)$

in $A^*(Y \times _ X Z \to X)$ .

Proof. Let $E \subset W_\infty$ be the inverse image of $Z$ . Recall that $P'_ p(Q) = (E \to Z)_* \circ P'_ p(Q|_ E) \circ C$ , resp. $c'_ p(Q) = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C$ where $C$ is as in Lemma 42.48.1 and $P'_ p(Q|_ E)$ , resp. $c'_ p(Q|_ E)$ are as in Lemma 42.47.1. By Lemma 42.48.5 we see that $C$ commutes with $c$ and by Lemma 42.47.6 we see that $P'_ p(Q|_ E)$ , resp. $c'_ p(Q|_ E)$ commutes with $c$ . Since $c$ is a bivariant class it commutes with proper pushforward by $E \to Z$ by definition. This finishes the proof. $\square$

Comments (0)

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).

Comments (0)

Post a comment