Lemma 37.30.1. Let f : X \to Y be a morphism of schemes. Assume Y irreducible with generic point \eta and f of finite type. If X_\eta has dimension n, then there exists a nonempty open V \subset Y such that for all y \in V the fibre X_ y has dimension n.
37.30 Dimension of fibres
Proof. Let Z = \{ x \in X \mid \dim _ x(X_{f(x)}) > n \} . By Morphisms, Lemma 29.28.4 this is a closed subset of X. By assumption Z_\eta = \emptyset . Hence by Lemma 37.24.1 we may shrink Y and assume that Z = \emptyset . Let Z' = \{ x \in X \mid \dim _ x(X_{f(x)}) > n - 1 \} = \{ x \in X \mid \dim _ x(X_{f(x)}) = n\} . As before this is a closed subset of X. By assumption we have Z'_\eta \not= \emptyset . Hence after shrinking Y we may assume that Z' \to Y is surjective, see Lemma 37.24.2. Hence we win. \square
Lemma 37.30.2. Let f : X \to Y be a morphism of finite type. Let
n_{X/Y} : Y \to \{ 0, 1, 2, 3, \ldots , \infty \}
be the function which associates to y \in Y the dimension of X_ y. If g : Y' \to Y is a morphism then
n_{X'/Y'} = n_{X/Y} \circ g
where X' \to Y' is the base change of f.
Proof. This follows from Morphisms, Lemma 29.28.3. \square
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
Lemma 37.30.4. Let f : X \to Y be a flat morphism of schemes of finite presentation. Let n_{X/Y} be the function on Y giving the dimension of fibres of f introduced in Lemma 37.30.2. Then n_{X/Y} is lower semi-continuous.
Proof. Let W \subset X, W = \coprod _{d \geq 0} U_ d be the open constructed in Lemmas 37.22.7 and 37.22.9. Let y \in Y be a point. If n_{X/Y}(y) = \dim (X_ y) = n, then y is in the image of U_ n \to Y. By Morphisms, Lemma 29.25.10 we see that f(U_ n) is open in Y. Hence there is an open neighbourhood of y where n_{X/Y} is \geq n. \square
Lemma 37.30.5. Let f : X \to Y be a proper 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. Then n_{X/Y} is upper semi-continuous.
Proof. Let Z_ d = \{ x \in X \mid \dim _ x(X_{f(x)}) > d\} . Then Z_ d is a closed subset of X by Morphisms, Lemma 29.28.4. Since f is proper f(Z_ d) is closed. Since y \in f(Z_ d) \Leftrightarrow n_{X/Y}(y) > d we see that the lemma is true. \square
Lemma 37.30.6. Let f : X \to Y be a proper, flat morphism of schemes of finite presentation. Let n_{X/Y} be the function on Y giving the dimension of fibres of f introduced in Lemma 37.30.2. Then n_{X/Y} is locally constant.
Proof. Immediate consequence of Lemmas 37.30.4 and 37.30.5. \square
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