Lemma 42.52.1 (0FBA)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.52: Properties of localized Chern classes
Lemma 42.52.1 (cite)

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Lemma 42.52.1. In Situation 42.50.1 assume $E|_{X \setminus Z}$ is zero. Then

\begin{align*} P_1(Z \to X, E) & = c_1(Z \to X, E), \\ P_2(Z \to X, E) & = c_1(Z \to X, E)^2 - 2c_2(Z \to X, E), \\ P_3(Z \to X, E) & = c_1(Z \to X, E)^3 - 3c_1(Z \to X, E)c_2(Z \to X, E) + 3c_3(Z \to X, E), \end{align*}

and so on where the products are taken in the algebra $A^{(1)}(Z \to X)$ of Remark 42.34.7.

Proof. The statement makes sense because the zero sheaf has rank $< 1$ and hence the classes $c_ p(Z \to X, E)$ are defined for all $p \geq 1$. The result itself follows immediately from the more general Lemma 42.49.6 as the localized Chern classes where defined using the procedure of Lemma 42.49.1 in Section 42.50. $\square$

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