Lemma 42.54.1 (0FBJ)—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 42.54.1 (cite)

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Lemma 42.54.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let

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

be a cartesian diagram of schemes locally of finite type over $S$ whose horizontal arrows are closed immersions. If $\mathcal{N}$ is a virtual normal sheaf for $Z$ in $X$, then $\mathcal{N}' = g^*\mathcal{N}$ is a virtual normal sheaf for $Z'$ in $X'$.

Proof. This follows from the surjectivity of the map $g^*\mathcal{C}_{Z/X} \to \mathcal{C}_{Z'/X'}$ proved in Morphisms, Lemma 29.31.4. $\square$

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