Proposition 43.19.3. Let $X$ be a nonsingular variety. Let $V \subset X$ and $W \subset Y$ be closed subvarieties which intersect properly. Let $Z \subset V \cap W$ be an irreducible component. Then $e(X, V \cdot W, Z) > 0$.
This is one of the main results of [Serre_algebre_locale].
Proof.
By Lemma 43.19.2 we have
Since $\Delta : X \to X \times X$ is a regular immersion (see Lemma 43.13.3), we see that $e(X \times X, \Delta \cdot V \times W, \Delta (Z))$ is a positive integer by Lemma 43.16.3.
$\square$
\[ e(X, V \cdot W, Z) = e(X \times X, \Delta \cdot V \times W, \Delta (Z)) \]
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