The Stacks project

37.28 Dimension of fibres

Lemma 37.28.1. 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 $n$.

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 37.22.1 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 37.22.2. Hence we win. $\square$

Lemma 37.28.2. 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 37.28.3. 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 37.28.2. 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 37.28.2 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 37.28.1 shows that $n_{X/Y}$ is constant in a neighbourhood of $\xi $ in $Z$. This implies what we want. $\square$

Lemma 37.28.4. 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 37.28.2. 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 37.20.7 and 37.20.9. 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 neighbourhoof of $y$ where $n_{X/Y}$ is $\geq n$. $\square$

Lemma 37.28.5. 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 37.28.2. 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 37.28.6. 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 37.28.2. Then $n_{X/Y}$ is locally constant.

Proof. Immediate consequence of Lemmas 37.28.4 and 37.28.5. $\square$


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