Definition 33.45.3. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $i : Z \to X$ be a closed subscheme of dimension $d$. Let $\mathcal{L}_1, \ldots , \mathcal{L}_ d$ be invertible $\mathcal{O}_ X$-modules. We define the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ as the coefficient of $n_1 \ldots n_ d$ in the numerical polynomial
\[ \chi (X, i_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}) = \chi (Z, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_ Z) \]
In the special case that $\mathcal{L}_1 = \ldots = \mathcal{L}_ d = \mathcal{L}$ we write $(\mathcal{L}^ d \cdot Z)$.
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