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The Stacks project

Lemma 76.31.3. Let S be a scheme. Let f : X \to Y be a flat morphism of finite presentation of algebraic spaces over S. Let n_{X/Y} be the function on Y giving the dimension of fibres of f introduced in Lemma 76.31.2. Then n_{X/Y} is lower semi-continuous.

Proof. Let V \to Y be a surjective étale morphism where V is a scheme. If we can show that the composition n_{X/Y} \circ |g| is lower semi-continuous, then the lemma follows as |g| is open. Hence we may assume Y is a scheme. Working locally we may assume V is an affine scheme. Then we can choose an affine scheme U and a surjective étale morphism U \to X. Then n_{X/Y} = n_{U/Y}. Hence we may assume X and Y are both schemes. In this case the lemma follows from More on Morphisms, Lemma 37.30.4. \square


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