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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