Lemma 76.18.3 (0CG6)—The Stacks project

Lemma 76.18.3. Let $S$ be a scheme. Consider a commutative diagram

$\xymatrix{ (X \subset X') \ar[rr]_{(f, f')} \ar[rd] & & (Y \subset Y') \ar[ld] \\ & (B \subset B') }$

of thickenings of algebraic spaces over $S$ . Assume $B \subset B'$ is a finite order thickening, $X'$ flat over $B'$ , $X = B \times _{B'} X'$ , and $Y = B \times _{B'} Y'$ . Then

$f$ is representable if and only if $f'$ is representable,
$f$ is flat if and only if $f'$ is flat,
$f$ is an isomorphism if and only if $f'$ is an isomorphism,
$f$ is an open immersion if and only if $f'$ is an open immersion,
$f$ is quasi-compact if and only if $f'$ is quasi-compact,
$f$ is universally closed if and only if $f'$ is universally closed,
$f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,
$f$ is a monomorphism if and only if $f'$ is a monomorphism,
$f$ is surjective if and only if $f'$ is surjective,
$f$ is universally injective if and only if $f'$ is universally injective,
$f$ is affine if and only if $f'$ is affine,
$f$ is locally of finite type if and only if $f'$ is locally of finite type,
$f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,
$f$ is locally of finite presentation if and only if $f'$ is locally of finite presentation,
$f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$ ,
$f$ is universally open if and only if $f'$ is universally open,
$f$ is syntomic if and only if $f'$ is syntomic,
$f$ is smooth if and only if $f'$ is smooth,
$f$ is unramified if and only if $f'$ is unramified,
$f$ is étale if and only if $f'$ is étale,
$f$ is proper if and only if $f'$ is proper,
$f$ is integral if and only if $f'$ is integral,
$f$ is finite if and only if $f'$ is finite,
$f$ is finite locally free (of rank $d$ ) if and only if $f'$ is finite locally free (of rank $d$ ), and
add more here.

Proof. Case (1) follows from Lemma 76.10.1.

Choose a scheme $U'$ and a surjective étale morphism $U' \to B'$ . Choose a scheme $V'$ and a surjective étale morphism $V' \to U' \times _{B'} Y'$ . Choose a scheme $W'$ and a surjective étale morphism $W' \to V' \times _{Y'} X'$ . Let $U, V, W$ be the base change of $U', V', W'$ by $B \to B'$ . Consider the diagram

$\xymatrix{ (W \subset W') \ar[rr] \ar[rd] & & (V \subset V') \ar[ld] \\ & (U \subset U') }$

of thickenings of schemes. For any of the properties which are étale local on the source-and-target the result follows immediately from the corresponding result for morphisms of thickenings of schemes applied to the diagram above. Thus cases (2), (12), (13), (14), (15), (17), (18), (19), (20) follow from the corresponding cases of More on Morphisms, Lemma 37.10.3.

Since $X \to X'$ and $Y \to Y'$ are universal homeomorphisms we see that any question about the topology of the maps $X \to Y$ and $X' \to Y'$ has the same answer. Thus we see that cases (5), (6), (9), (10), and (16) hold.

In each of the remaining cases we only prove the implication $f\text{ has }P \Rightarrow f'\text{ has }P$ since the other implication follows from the fact that $P$ is stable under base change, see Spaces, Lemma 65.12.3 and Morphisms of Spaces, Lemmas 67.4.4, 67.10.5, 67.20.5, 67.40.3, 67.45.5, and 67.46.5.

The case (4). Assume $f$ is an open immersion. Then $f'$ is étale by (20) and universally injective by (10) hence $f'$ is an open immersion, see Morphisms of Spaces, Lemma 67.51.2. You can avoid using this lemma at the cost of first using (1) to reduce to the case of schemes.

The case (3). Follows from cases (4) and (9).

The case (7). See Lemma 76.10.1.

The case (8). Assume $f$ is a monomorphism. Consider the diagonal morphism $\Delta _{X'/Y'} : X' \to X' \times _{Y'} X'$ . The base change of $\Delta _{X'/Y'}$ by $B \to B'$ is $\Delta _{X/Y}$ which is an isomorphism by assumption. By (3) we conclude that $\Delta _{X'/Y'}$ is an isomorphism and hence $f'$ is a monomorphism.

The case (11). See Lemma 76.10.1.

The case (21). See Lemma 76.10.2.

The case (22). See Lemma 76.10.1.

The case (23). See Lemma 76.10.2.

The case (24). Assume $f$ finite locally free. By (23) we see that $f'$ is finite. By (2) we see that $f'$ is flat. By (14) $f'$ is locally of finite presentation. Hence $f'$ is finite locally free by Morphisms of Spaces, Lemma 67.46.6. $\square$

The Stacks project

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