The Stacks project

Remark 42.54.11 (Variant for immersions). Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $i : Z \to X$ be an immersion of schemes. In this situation

1. the conormal sheaf $\mathcal{C}_{Z/X}$ of $Z$ in $X$ is defined (Morphisms, Definition 29.31.1),

2. we say a pair consisting of a finite locally free $\mathcal{O}_ Z$-module $\mathcal{N}$ and a surjection $\sigma : \mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ is a virtual normal bundle for the immersion $Z \to X$,

3. choose an open subscheme $U \subset X$ such that $Z \to X$ factors through a closed immersion $Z \to U$ and set $c(Z \to X, \mathcal{N}) = c(Z \to U, \mathcal{N}) \circ (U \to X)^*$.

The bivariant class $c(Z \to X, \mathcal{N})$ does not depend on the choice of the open subscheme $U$. All of the lemmas have immediate counterparts for this slightly more general construction. We omit the details.