The Stacks project

Lemma 42.37.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\pi : P \to X, \mathcal{O}_ P(1))$ be the projective bundle associated to $\mathcal{E}$. For $\alpha \in Z_ k(X)$ the elements $c_ j(\mathcal{E}) \cap \alpha $ are the unique elements $\alpha _ j$ of $\mathop{\mathrm{CH}}\nolimits _{k - j}(X)$ such that $\alpha _0 = \alpha $ and

\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(\alpha _{r - i}) = 0 \]

holds in the Chow group of $P$.

Proof. The uniqueness of $\alpha _0, \ldots , \alpha _ r$ such that $\alpha _0 = \alpha $ and such that the displayed equation holds follows from the projective space bundle formula Lemma 42.35.2. The identity holds by definition for $\alpha = [W]$ where $W$ is an integral closed subscheme of $X$. For a general $k$-cycle $\alpha $ on $X$ write $\alpha = \sum n_ a[W_ a]$ with $n_ a \not= 0$, and $i_ a : W_ a \to X$ pairwise distinct integral closed subschemes. Then the family $\{ W_ a\} $ is locally finite on $X$. Set $P_ a = \pi ^{-1}(W_ a) = \mathbf{P}(\mathcal{E}|_{W_ a})$. Denote $i'_ a : P_ a \to P$ the corresponding closed immersions. Consider the fibre product diagram

\[ \xymatrix{ P' \ar@{=}[r] \ar[d]_{\pi '} & \coprod P_ a \ar[d]_{\coprod \pi _ a} \ar[r]_{\coprod i'_ a} & P \ar[d]^\pi \\ X' \ar@{=}[r] & \coprod W_ a \ar[r]^{\coprod i_ a} & X } \]

The morphism $p : X' \to X$ is proper. Moreover $\pi ' : P' \to X'$ together with the invertible sheaf $\mathcal{O}_{P'}(1) = \coprod \mathcal{O}_{P_ a}(1)$ which is also the pullback of $\mathcal{O}_ P(1)$ is the projective bundle associated to $\mathcal{E}' = p^*\mathcal{E}$. By definition

\[ c_ j(\mathcal{E}) \cap [\alpha ] = \sum i_{a, *}(c_ j(\mathcal{E}|_{W_ a}) \cap [W_ a]). \]

Write $\beta _{a, j} = c_ j(\mathcal{E}|_{W_ a}) \cap [W_ a]$ which is an element of $\mathop{\mathrm{CH}}\nolimits _{k - j}(W_ a)$. We have

\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P_ a}(1))^ i \cap \pi _ a^*(\beta _{a, r - i}) = 0 \]

for each $a$ by definition. Thus clearly we have

\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_{P'}(1))^ i \cap (\pi ')^*(\beta _{r - i}) = 0 \]

with $\beta _ j = \sum n_ a\beta _{a, j} \in \mathop{\mathrm{CH}}\nolimits _{k - j}(X')$. Denote $p' : P' \to P$ the morphism $\coprod i'_ a$. We have $\pi ^*p_*\beta _ j = p'_*(\pi ')^*\beta _ j$ by Lemma 42.15.1. By the projection formula of Lemma 42.25.4 we conclude that

\[ \sum \nolimits _{i = 0}^ r (-1)^ i c_1(\mathcal{O}_ P(1))^ i \cap \pi ^*(p_*\beta _ j) = 0 \]

Since $p_*\beta _ j$ is a representative of $c_ j(\mathcal{E}) \cap \alpha $ we win. $\square$


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