The Stacks project

Lemma 70.6.14. Assumptions and notation as in Situation 70.6.1. Let $d \geq 0$. If

  1. $f$ has relative dimension $\leq d$ (Morphisms of Spaces, Definition 67.33.2), and

  2. $f_0$ is locally of finite type,

then there exists an $i$ such that $f_ i$ has relative dimension $\leq d$.

Proof. Choose an affine scheme $V_0$ and a surjective étale morphism $V_0 \to Y_0$. Choose an affine scheme $U_0$ and a surjective étale morphism $U_0 \to V_0 \times _{Y_0} X_0$. Diagram

\[ \xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 } \]

The vertical arrows are surjective and étale by construction. We can base change this diagram to $B_ i$ or $B$ to get

\[ \vcenter { \xymatrix{ U_ i \ar[d] \ar[r] & V_ i \ar[d] \\ X_ i \ar[r] & Y_ i } } \quad \text{and}\quad \vcenter { \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } } \]

Note that $U_ i, V_ i, U, V$ are affine schemes, the vertical morphisms are surjective étale, and the limit of the morphisms $U_ i \to V_ i$ is $U \to V$. In this situation $X_ i \to Y_ i$ has relative dimension $\leq d$ if and only if $U_ i \to V_ i$ has relative dimension $\leq d$ (as defined in Morphisms, Definition 29.29.1). To see the equivalence, use that the definition for morphisms of algebraic spaces involves Morphisms of Spaces, Definition 67.33.1 which uses étale localization. The same is true for $X \to Y$ and $U \to V$. Since $f_0$ is locally of finite type, so is the morphism $U_0 \to V_0$. Hence the lemma follows from the more general Limits, Lemma 32.18.1. $\square$

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