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The Stacks project

Lemma 37.30.3. Let f : X \to Y be a morphism of schemes. Let n_{X/Y} be the function on Y giving the dimension of fibres of f introduced in Lemma 37.30.2. Assume f of finite presentation. Then the level sets

E_ n = \{ y \in Y \mid n_{X/Y}(y) = n\}

of n_{X/Y} are locally constructible in Y.

Proof. Fix n. Let y \in Y. We have to show that there exists an open neighbourhood V of y in Y such that E_ n \cap V is constructible in V. Thus we may assume that Y is affine. Write Y = \mathop{\mathrm{Spec}}(A) and A = \mathop{\mathrm{colim}}\nolimits A_ i as a directed limit of finite type \mathbf{Z}-algebras. By Limits, Lemma 32.10.1 we can find an i and a morphism f_ i : X_ i \to \mathop{\mathrm{Spec}}(A_ i) of finite presentation whose base change to Y recovers f. By Lemma 37.30.2 it suffices to prove the lemma for f_ i. Thus we reduce to the case where Y is the spectrum of a Noetherian ring.

We will use the criterion of Topology, Lemma 5.16.3 to prove that E_ n is constructible in case Y is a Noetherian scheme. To see this let Z \subset Y be an irreducible closed subscheme. We have to show that E_ n \cap Z either contains a nonempty open subset or is not dense in Z. Let \xi \in Z be the generic point. Then Lemma 37.30.1 shows that n_{X/Y} is constant in a neighbourhood of \xi in Z. This implies what we want. \square


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