The Stacks project

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

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

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 $U \to V$. Recall that $X_ i \to Y_ i$ is étale if and only if $U_ i \to V_ i$ is étale and similarly $X \to Y$ is étale if and only if $U \to V$ is étale (Morphisms of Spaces, Lemma 67.39.2). Since $f_0$ is locally of finite presentation, so is the morphism $U_0 \to V_0$. Hence the lemma follows from Limits, Lemma 32.8.10. $\square$

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

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

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 $U \to V$. Recall that $X_ i \to Y_ i$ is smooth if and only if $U_ i \to V_ i$ is smooth and similarly $X \to Y$ is smooth if and only if $U \to V$ is smooth (Morphisms of Spaces, Definition 67.37.1). Since $f_0$ is locally of finite presentation, so is the morphism $U_0 \to V_0$. Hence the lemma follows from Limits, Lemma 32.8.9. $\square$

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

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

Note that $U_ i, V_ i, U, V$ are affine schemes, the vertical morphisms are surjective étale, the limit of the morphisms $U_ i \to V_ i$ is $U \to V$, and the morphisms $U_ i \to X_ i \times _{Y_ i} V_ i$ and $U \to X \times _ Y V$ are surjective (as base changes of $U_0 \to X_0 \times _{Y_0} V_0$). In particular, we see that $X_ i \to Y_ i$ is surjective if and only if $U_ i \to V_ i$ is surjective and similarly $X \to Y$ is surjective if and only if $U \to V$ is surjective. Since $f_0$ is locally of finite presentation, so is the morphism $U_0 \to V_0$. Hence the lemma follows from the case of schemes (Limits, Lemma 32.8.15). $\square$

Proof. Recall that a morphism $X \to Y$ is universally injective if and only if the diagonal $X \to X \times _ Y X$ is surjective (Morphisms of Spaces, Definition 67.19.3 and Lemma 67.19.2). Observe that $X_0 \to X_0 \times _{Y_0} X_0$ is of locally of finite presentation (Morphisms of Spaces, Lemma 67.28.10). Hence the lemma follows from Lemma 70.6.4 by considering the morphism $X_0 \to X_0 \times _{Y_0} X_0$. $\square$

Proof. Choose an affine scheme $V_0$ and a surjective étale morphism $V_0 \to Y_0$. Set $V_ i = V_0 \times _{Y_0} Y_ i$ and $V = V_0 \times _{Y_0} Y$. Since $f$ is affine we see that $V \times _ Y X = \mathop{\mathrm{lim}}\nolimits V_ i \times _{Y_ i} X_ i$ is affine. By Lemma 70.5.10 we see that $V_ i \times _{Y_ i} X_ i$ is affine for some $i \geq 0$. For this $i$ the morphism $f_ i$ is affine (Morphisms of Spaces, Lemma 67.20.3). $\square$

Proof. Choose an affine scheme $V_0$ and a surjective étale morphism $V_0 \to Y_0$. Set $V_ i = V_0 \times _{Y_0} Y_ i$ and $V = V_0 \times _{Y_0} Y$. Since $f$ is finite we see that $V \times _ Y X = \mathop{\mathrm{lim}}\nolimits V_ i \times _{Y_ i} X_ i$ is a scheme finite over $V$. By Lemma 70.5.10 we see that $V_ i \times _{Y_ i} X_ i$ is affine for some $i \geq 0$. Increasing $i$ if necessary we find that $V_ i \times _{Y_ i} X_ i \to V_ i$ is finite by Limits, Lemma 32.8.3. For this $i$ the morphism $f_ i$ is finite (Morphisms of Spaces, Lemma 67.45.3). $\square$

Proof. Choose an affine scheme $V_0$ and a surjective étale morphism $V_0 \to Y_0$. Set $V_ i = V_0 \times _{Y_0} Y_ i$ and $V = V_0 \times _{Y_0} Y$. Since $f$ is a closed immersion we see that $V \times _ Y X = \mathop{\mathrm{lim}}\nolimits V_ i \times _{Y_ i} X_ i$ is a closed subscheme of the affine scheme $V$. By Lemma 70.5.10 we see that $V_ i \times _{Y_ i} X_ i$ is affine for some $i \geq 0$. Increasing $i$ if necessary we find that $V_ i \times _{Y_ i} X_ i \to V_ i$ is a closed immersion by Limits, Lemma 32.8.5. For this $i$ the morphism $f_ i$ is a closed immersion (Morphisms of Spaces, Lemma 67.45.3). $\square$

Proof. Apply Lemma 70.6.8 to the diagonal morphism $\Delta _{X_0/Y_0} : X_0 \to X_0 \times _{Y_0} X_0$. (Diagonal morphisms are locally of finite type and the fibre product $X_0 \times _{Y_0} X_0$ is quasi-compact and quasi-separated. Some details omitted.) $\square$

Proof. Being an isomorphism is equivalent to being étale, universally injective, and surjective, see Morphisms of Spaces, Lemma 67.51.2. Thus the lemma follows from Lemmas 70.6.2, 70.6.4, and 70.6.5. $\square$

Proof. Recall that a morphism is a monomorphism if and only if the diagonal is an isomorphism. The morphism $X_0 \to X_0 \times _{Y_0} X_0$ is locally of finite presentation by Morphisms of Spaces, Lemma 67.28.10. Since $X_0 \times _{Y_0} X_0$ is quasi-compact and quasi-separated we conclude from Lemma 70.6.10 that $\Delta _ i : X_ i \to X_ i \times _{Y_ i} X_ i$ is an isomorphism for some $i \geq 0$. For this $i$ the morphism $f_ i$ is a monomorphism. $\square$

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

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

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 $U \to V$. Recall that $\mathcal{F}_ i$ is flat over $Y_ i$ if and only if $\mathcal{F}_ i|_{U_ i}$ is flat over $V_ i$ and similarly $\mathcal{F}$ is flat over $Y$ if and only if $\mathcal{F}|_ U$ is flat over $V$ (Morphisms of Spaces, Definition 67.30.1). Since $f_0$ is locally of finite presentation, so is the morphism $U_0 \to V_0$. Hence the lemma follows from Limits, Lemma 32.10.4. $\square$

Proof. Choose an affine scheme $V_0$ and a surjective étale morphism $V_0 \to Y_0$. Set $V_ i = Y_ i \times _{Y_0} V_0$ and $V = Y \times _{Y_0} V_0$. It suffices to prove that the base change of $f_ i$ to $V_ i$ is proper, see Morphisms of Spaces, Lemma 67.40.2. Thus we may assume $Y_0$ is affine.

By Lemma 70.6.9 we see that $f_ i$ is separated for some $i \geq 0$. Replacing $0$ by $i$ we may assume that $f_0$ is separated. Observe that $f_0$ is quasi-compact. Thus $f_0$ is separated and of finite type. By Cohomology of Spaces, Lemma 69.18.1 we can choose a diagram

where $X_0' \to \mathbf{P}^ n_{Y_0}$ is an immersion, and $\pi : X_0' \to X_0$ is proper and surjective. Introduce $X' = X_0' \times _{Y_0} Y$ and $X_ i' = X_0' \times _{Y_0} Y_ i$. By Morphisms of Spaces, Lemmas 67.40.4 and 67.40.3 we see that $X' \to Y$ is proper. Hence $X' \to \mathbf{P}^ n_ Y$ is a closed immersion (Morphisms of Spaces, Lemma 67.40.6). By Morphisms of Spaces, Lemma 67.40.7 it suffices to prove that $X'_ i \to Y_ i$ is proper for some $i$. By Lemma 70.6.8 we find that $X'_ i \to \mathbf{P}^ n_{Y_ i}$ is a closed immersion for $i$ large enough. Then $X'_ i \to Y_ i$ is proper and we win. $\square$

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

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

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. $\square$