Lemma 43.13.3 (0B0Q)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 43: Intersection Theory
Section 43.13: Proper intersections
Lemma 43.13.3 (cite)

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Lemma 43.13.3. Let $X$ be a nonsingular variety. Then the diagonal $\Delta : X \to X \times X$ is a regular immersion of codimension $\dim (X)$.

Proof. In fact, any closed immersion between nonsingular projective varieties is a regular immersion, see Divisors, Lemma 31.22.11. $\square$

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