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The Stacks project

Lemma 33.45.8. Let k be a field. Let X be proper over k. Let Z \subset X be a closed subscheme of dimension d. Let \mathcal{L}_1, \ldots , \mathcal{L}_ d be invertible \mathcal{O}_ X-modules. Assume there exists an effective Cartier divisor D \subset Z such that \mathcal{L}_1|_ Z \cong \mathcal{O}_ Z(D). Then

(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z) = (\mathcal{L}_2 \cdots \mathcal{L}_ d \cdot D)

Proof. We may replace X by Z and \mathcal{L}_ i by \mathcal{L}_ i|_ Z. Thus we may assume X = Z and \mathcal{L}_1 = \mathcal{O}_ X(D). Then \mathcal{L}_1^{-1} is the ideal sheaf of D and we can consider the short exact sequence

0 \to \mathcal{L}_1^{\otimes -1} \to \mathcal{O}_ X \to \mathcal{O}_ D \to 0

Set P(n_1, \ldots , n_ d) = \chi (X, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}) and Q(n_1, \ldots , n_ d) = \chi (D, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_ D). We conclude from additivity that

P(n_1, \ldots , n_ d) - P(n_1 - 1, n_2, \ldots , n_ d) = Q(n_1, \ldots , n_ d)

Because the total degree of P is at most d, we see that the coefficient of n_1 \ldots n_ d in P is equal to the coefficient of n_2 \ldots n_ d in Q. \square


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