Lemma 43.13.4 (0AZP)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 43: Intersection Theory
Section 43.13: Proper intersections
Lemma 43.13.4 (cite)

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Lemma 43.13.4. Let $X$ be a nonsingular variety and let $W,V \subset X$ be closed subvarieties with $\dim (W) = s$ and $\dim (V) = r$ . Then every irreducible component $Z$ of $V \cap W$ has dimension $\geq r + s - \dim (X)$ .

Proof. Since $V \cap W = \Delta ^{-1}(V \times W)$ (scheme theoretically) we conclude by Lemmas 43.13.3 and 43.13.2. $\square$

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