Lemma 42.54.4 (0FV7)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.54: Higher codimension gysin homomorphisms
Lemma 42.54.4 (cite)

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Lemma 42.54.4. Let $(S, \delta )$ be as in Situation 42.7.1. Consider a cartesian diagram

$\xymatrix{ Z' \ar[r] \ar[d]_ g & X' \ar[d]^ f \\ Z \ar[r] & X }$

of schemes locally of finite type over $S$ whose horizontal arrows are closed immersions. Let $\mathcal{N}$ , resp. $\mathcal{N}'$ be a virtual normal sheaf for $Z \subset X$ , resp. $Z' \to X'$ . Assume given a short exact sequence $0 \to \mathcal{N}' \to g^*\mathcal{N} \to \mathcal{E} \to 0$ of finite locally free modules on $Z'$ such that the diagram

$\xymatrix{ g^*\mathcal{N}^\vee \ar[r] \ar[d] & (\mathcal{N}')^\vee \ar[d] \\ g^*\mathcal{C}_{Z/X} \ar[r] & \mathcal{C}_{Z'/X'} }$

commutes. Then we have

$res(c(Z \to X, \mathcal{N})) = c_{top}(\mathcal{E}) \circ c(Z' \to X', \mathcal{N}')$

in $A^*(Z' \to X')^\wedge$ .

Proof. By Lemma 42.54.2 we have $res(c(Z \to X, \mathcal{N})) = c(Z' \to X', g^*\mathcal{N})$ and the equality follows from Lemma 42.54.3. $\square$

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