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The Stacks project

Lemma 42.38.2. 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 42.36.2. 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 42.26.4 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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