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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