Lemma 42.44.2 (0FAA)—The Stacks project

Lemma 42.44.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$ . Let

$0 \to \mathcal{N}' \to \mathcal{N} \to \mathcal{E} \to 0$

be a short exact sequence of finite locally free $\mathcal{O}_ X$ -modules. Consider the closed embedding

$i : N' = \underline{\mathop{\mathrm{Spec}}}_ X(\text{Sym}((\mathcal{N}')^\vee )) \longrightarrow N = \underline{\mathop{\mathrm{Spec}}}_ X(\text{Sym}(\mathcal{N}^\vee ))$

For $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ we have

$i_*(p')^*\alpha = p^*(c_{top}(\mathcal{E}) \cap \alpha )$

where $p' : N' \to X$ and $p : N \to X$ are the structure morphisms.

Proof. Here $c_{top}(\mathcal{E})$ is the bivariant class defined in Remark 42.38.11. By its very definition, in order to verify the formula, we may assume that $\mathcal{E}$ has constant rank. We may similarly assume $\mathcal{N}'$ and $\mathcal{N}$ have constant ranks, say $r'$ and $r$ , so $\mathcal{E}$ has rank $r - r'$ and $c_{top}(\mathcal{E}) = c_{r - r'}(\mathcal{E})$ . Observe that $p^*\mathcal{E}$ has a canonical section

$s \in \Gamma (N, p^*\mathcal{E}) = \Gamma (X, p_*p^*\mathcal{E}) = \Gamma (X, \mathcal{E} \otimes _{\mathcal{O}_ X} \text{Sym}(\mathcal{N}^\vee ) \supset \Gamma (X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{N}, \mathcal{E}))$

corresponding to the surjection $\mathcal{N} \to \mathcal{E}$ given in the statement of the lemma. The vanishing scheme of this section is exactly $N' \subset N$ . Let $Y \subset X$ be an integral closed subscheme of $\delta$ -dimension $n$ . Then we have

$p^*[Y] = [p^{-1}(Y)]$ since $p^{-1}(Y)$ is integral of $\delta$ -dimension $n + r$ ,
$(p')^*[Y] = [(p')^{-1}(Y)]$ since $(p')^{-1}(Y)$ is integral of $\delta$ -dimension $n + r'$ ,
the restriction of $s$ to $p^{-1}Y$ has vanishing scheme $(p')^{-1}Y$ and the closed immersion $(p')^{-1}Y \to p^{-1}Y$ is a regular immersion (locally cut out by a regular sequence).

We conclude that

$(p')^*[Y] = c_{r - r'}(p^*\mathcal{E}) \cap p^*[Y] \quad \text{in}\quad \mathop{\mathrm{CH}}\nolimits _*(N)$

by Lemma 42.44.1. This proves the lemma. $\square$

The Stacks project

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