## 75.31 Dimension of fibres

This section is the analogue of More on Morphisms, Section 37.30.

Lemma 75.31.1. Let $S$ be a scheme. Let $f : X \to Y$ be a finite type morphism of algebraic spaces over $S$. Let $y \in |Y|$. The following quantities are the same

1. $d = -\infty$ if $y$ is not in the image of $|f|$ and otherwise the minimal integer $d$ such that $f$ has relative dimension $\leq d$ at every $x \in |X|$ mapping to $y$,

2. the dimension of the algebraic space $X_ k = \mathop{\mathrm{Spec}}(k) \times _ Y X$ for any morphism $\mathop{\mathrm{Spec}}(k) \to Y$ in the equivalence class defining $y$.

Proof. To parse this one has to consult Morphisms of Spaces, Definition 66.33.1, Properties of Spaces, Definition 65.9.2, Properties of Spaces, Definition 65.9.1. We will show that the numbers in (1) and (2) are equal for a fixed morphism $\mathop{\mathrm{Spec}}(k) \to Y$. Choose an étale morphism $V \to Y$ where $V$ is an affine scheme and a point $v \in V$ mapping to $y$. Since $V \times _ Y \mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k)$ is surjective étale (by Properties of Spaces, Lemma 65.4.3) we can find a finite separable extension $k'/k$ (by Morphisms, Lemma 29.36.7) and a commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(k') \ar[r] \ar[d] & V \ar[d] \\ \mathop{\mathrm{Spec}}(k) \ar[r] & Y }$

We may replace $X \to Y$ by $V \times _ Y X \to V$ and $X_ k$ by $X_{k'} = \mathop{\mathrm{Spec}}(k') \times _ V (V \times _ Y X)$ because this does not change the dimensions in question by Properties of Spaces, Lemma 65.22.5 and Morphisms of Spaces, Lemma 66.34.3. Thus we may assume that $Y$ is an affine scheme. In this case we may assume that $k = \kappa (y)$ because the dimension of $X_{\kappa (y)}$ and $X_ k$ are the same by the aforementioned Morphisms of Spaces, Lemma 66.34.3 and the fact that for an algebraic space $Z$ over a field $K$ the relative dimension of $Z$ at a point $z \in |Z|$ is the same as $\dim _ z(Z)$ by definition. Assume $Y$ is affine and $k = \kappa (y)$. Then $X$ is quasi-compact we can choose an affine scheme $U$ and an surjective étale morphism $U \to X$. Then $\dim (X_ k) = \dim (U_ k) = \max \dim _ u(U_ k)$ is equal to the number given in (1) by definition. $\square$

Lemma 75.31.2. Let $S$ be a scheme. Let $f : X \to Y$ be a finite type morphism of algebraic spaces over $S$. Let

$n_{X/Y} : |Y| \to \{ -\infty , 0, 1, 2, 3, \ldots \}$

be the function which associates to $y \in |Y|$ the integer discussed in Lemma 75.31.1. 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 immediately from Lemma 75.31.1. $\square$

Lemma 75.31.3. 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 75.31.2. 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. $\square$

Lemma 75.31.4. Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of algebraic spaces over $S$. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma 75.31.2. Then $n_{X/Y}$ is upper semi-continuous.

Proof. Let $Z_ d = \{ x \in |X| : \text{the fibre of }f\text{ at }x\text{ has dimension }> d\}$. Then $Z_ d$ is a closed subset of $|X|$ by Morphisms of Spaces, Lemma 66.34.4. Since $f$ is proper $f(Z_ d)$ is closed in $|Y|$. Since $y \in f(Z_ d) \Leftrightarrow n_{X/Y}(y) > d$ we see that the lemma is true. $\square$

Lemma 75.31.5. Let $S$ be a scheme. Let $f : X \to Y$ be a proper, flat, finitely presented morphism of algebraic spaces over $S$. Let $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$ introduced in Lemma 75.31.2. Then $n_{X/Y}$ is locally constant.

Proof. Immediate consequence of Lemmas 75.31.3 and 75.31.4. $\square$

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