Proposition 43.19.3 (0B0V)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 43: Intersection Theory
Section 43.19: Reduction to the diagonal
Proposition 43.19.3 (cite)

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This is one of the main results of [Serre_algebre_locale].

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 $e(X, V \cdot W, Z) > 0$ .

Proof. By Lemma 43.19.2 we have

$e(X, V \cdot W, Z) = e(X \times X, \Delta \cdot V \times W, \Delta (Z))$

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. $\square$

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