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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