Lemma 72.18.5 (0EDH)—The Stacks project

Table of contents
Part 4: Algebraic Spaces
Chapter 72: Algebraic Spaces over Fields
Section 72.18: Numerical intersections
Lemma 72.18.5 (cite)

Lemma 72.18.5. In the situation of Definition 72.18.3 the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ is additive: if $\mathcal{L}_ i = \mathcal{L}_ i' \otimes \mathcal{L}_ i''$, then we have

\[ (\mathcal{L}_1 \cdots \mathcal{L}_ i \cdots \mathcal{L}_ d \cdot Z) = (\mathcal{L}_1 \cdots \mathcal{L}_ i' \cdots \mathcal{L}_ d \cdot Z) + (\mathcal{L}_1 \cdots \mathcal{L}_ i'' \cdots \mathcal{L}_ d \cdot Z) \]

Proof. This is true because by Lemma 72.18.1 the function

\[ (n_1, \ldots , n_{i - 1}, n_ i', n_ i'', n_{i + 1}, \ldots , n_ d) \mapsto \chi (Z, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes (\mathcal{L}_ i')^{\otimes n_ i'} \otimes (\mathcal{L}_ i'')^{\otimes n_ i''} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_ Z) \]

is a numerical polynomial of total degree at most $d$ in $d + 1$ variables. $\square$

The Stacks project

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