Lemma 42.55.4 (0FEH)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.55: Calculating some classes
Lemma 42.55.4 (cite)

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Lemma 42.55.4. In the situation of Lemma 42.55.3 say $\dim _\delta (X) = n$ . Then we have

$c_ t(Z \to X, i_*\mathcal{O}_ Z) \cap [X]_ n = 0$ for $t = 1, \ldots , r - 1$ ,
$c_ r(Z \to X, i_*\mathcal{O}_ Z) \cap [X]_ n = (-1)^{r - 1}(r - 1)![Z]_{n - r}$ ,
$ch_ t(Z \to X, i_*\mathcal{O}_ Z) \cap [X]_ n = 0$ for $t = 0, \ldots , r - 1$ , and
$ch_ r(Z \to X, i_*\mathcal{O}_ Z) \cap [X]_ n = [Z]_{n - r}$ .

Proof. Parts (1) and (2) follow immediately from Lemma 42.55.3 combined with Lemma 42.54.5. Then we deduce parts (3) and (4) using the relationship between $ch_ p = (1/p!)P_ p$ and $c_ p$ given in Lemma 42.52.1. (Namely, $(-1)^{r - 1}(r - 1)!ch_ r = c_ r$ provided $c_1 = c_2 = \ldots = c_{r - 1} = 0$ .) $\square$

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