Lemma 42.55.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{C}$ be a locally free $\mathcal{O}_ X$-module of rank $r$. Consider the morphisms

$X = \underline{\text{Proj}}_ X(\mathcal{O}_ X[T]) \xrightarrow {i} E = \underline{\text{Proj}}_ X(\text{Sym}^*(\mathcal{C})[T]) \xrightarrow {\pi } X$

Then $c_ t(i_*\mathcal{O}_ X) = 0$ for $t = 1, \ldots , r - 1$ and in $A^0(C \to E)$ we have

$p^* \circ \pi _* \circ c_ r(i_*\mathcal{O}_ X) = (-1)^{r - 1}(r - 1)! j^*$

where $j : C \to E$ and $p : C \to X$ are the inclusion and structure morphism of the vector bundle $C = \underline{\mathop{\mathrm{Spec}}}(\text{Sym}^*(\mathcal{C}))$.

Proof. The canonical map $\pi ^*\mathcal{C} \to \mathcal{O}_ E(1)$ vanishes exactly along $i(X)$. Hence the Koszul complex on the map

$\pi ^*\mathcal{C} \otimes \mathcal{O}_ E(-1) \to \mathcal{O}_ E$

is a resolution of $i_*\mathcal{O}_ X$. In particular we see that $i_*\mathcal{O}_ X$ is a perfect object of $D(\mathcal{O}_ E)$ whose Chern classes are defined. The vanishing of $c_ t(i_*\mathcal{O}_ X)$ for $t = 1, \ldots , t - 1$ follows from Lemma 42.55.1. This lemma also gives

$c_ r(i_*\mathcal{O}_ X) = - (r - 1)! c_ r(\pi ^*\mathcal{C} \otimes \mathcal{O}_ E(-1))$

On the other hand, by Lemma 42.43.3 we have

$c_ r(\pi ^*\mathcal{C} \otimes \mathcal{O}_ E(-1)) = (-1)^ r c_ r(\pi ^*\mathcal{C}^\vee \otimes \mathcal{O}_ E(1))$

and $\pi ^*\mathcal{C}^\vee \otimes \mathcal{O}_ E(1)$ has a section $s$ vanishing exactly along $i(X)$.

After replacing $X$ by a scheme locally of finite type over $X$, it suffices to prove that both sides of the equality have the same effect on an element $\alpha \in \mathop{\mathrm{CH}}\nolimits _*(E)$. Since $C \to X$ is a vector bundle, every cycle class on $C$ is of the form $p^*\beta$ for some $\beta \in \mathop{\mathrm{CH}}\nolimits _*(X)$ (Lemma 42.36.3). Hence by Lemma 42.19.3 we can write $\alpha = \pi ^*\beta + \gamma$ where $\gamma$ is supported on $E \setminus C$. Using the equalities above it suffices to show that

$p^*(\pi _*(c_ r(\pi ^*\mathcal{C}^\vee \otimes \mathcal{O}_ E(1)) \cap [W])) = j^*[W]$

when $W \subset E$ is an integral closed subscheme which is either (a) disjoint from $C$ or (b) is of the form $W = \pi ^{-1}Y$ for some integral closed subscheme $Y \subset X$. Using the section $s$ and Lemma 42.44.1 we find in case (a) $c_ r(\pi ^*\mathcal{C}^\vee \otimes \mathcal{O}_ E(1)) \cap [W] = 0$ and in case (b) $c_ r(\pi ^*\mathcal{C}^\vee \otimes \mathcal{O}_ E(1)) \cap [W] = [i(Y)]$. The result follows easily from this; details omitted. $\square$

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