The Stacks project

Lemma 71.6.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Let $x \in |D|$. If $\dim _ x(X) < \infty $, then $\dim _ x(D) < \dim _ x(X)$.

Proof. Both the definition of an effective Cartier divisor and of the dimension of an algebraic space at a point (Properties of Spaces, Definition 66.9.1) are étale local. Hence this lemma follows from the case of schemes which is Divisors, Lemma 31.13.5. $\square$

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