Lemma 37.34.9. Let $d \geq 0$ be an integer. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation all of whose fibres have dimension $d$. Then there exists a diagram as in Lemma 37.34.1 such that in addition all fibres of $f_0$ have dimension $d$.

Proof. Apply Lemma 37.34.1 to get a cartesian diagram

$\xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^ g \ar[d]^ f \\ S_0 & S \ar[l]_ h }$

of affine schemes with $X_0 \to S_0$ a finite type morphism of schemes of finite type over $\mathbf{Z}$. By Lemma 37.30.3 the set $E \subset S_0$ of points where the fibre of $f_0$ has dimension $d$ is a constructible subset. By Lemma 37.30.2 we have $h(S) \subset E$. Write $S_0 = \mathop{\mathrm{Spec}}(A_0)$ and $S = \mathop{\mathrm{Spec}}(A)$. Write $A = \mathop{\mathrm{colim}}\nolimits _ i A_ i$ as a direct colimit of finite type $A_0$-algebras. By Limits, Lemma 32.4.10 we see that $\mathop{\mathrm{Spec}}(A_ i) \to S_0$ has image contained in $E$ for some $i$. After replacing $S_0$ by $\mathop{\mathrm{Spec}}(A_ i)$ and $X_0$ by $X_0 \times _{S_0} \mathop{\mathrm{Spec}}(A_ i)$ we see that all fibres of $f_0$ have dimension $d$. $\square$

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