Lemma 105.5.1. Consider a commutative diagram

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

of thickenings of algebraic stacks. Assume

$\mathcal{Y}' \to \mathcal{B}'$ is locally of finite type,

$\mathcal{X}' \to \mathcal{B}'$ is flat and locally of finite presentation,

$f$ is flat, and

$\mathcal{X} = \mathcal{B} \times _{\mathcal{B}'} \mathcal{X}'$ and $\mathcal{Y} = \mathcal{B} \times _{\mathcal{B}'} \mathcal{Y}'$.

Then $f'$ is flat and for all $y' \in |\mathcal{Y}'|$ in the image of $|f'|$ the morphism $\mathcal{Y}' \to \mathcal{B}'$ is flat at $y'$.

**Proof.**
Choose an algebraic space $U'$ and a surjective smooth morphism $U' \to \mathcal{B}'$. Choose an algebraic space $V'$ and a surjective smooth morphism $V' \to U' \times _{\mathcal{B}'} \mathcal{Y}'$. Choose an algebraic space $W'$ and a surjective smooth morphism $W' \to V' \times _{\mathcal{Y}'} \mathcal{X}'$. Let $U, V, W$ be the base change of $U', V', W'$ by $\mathcal{B} \to \mathcal{B}'$. Then flatness of $f'$ is equivalent to flatness of $W' \to V'$ and we are given that $W \to V$ is flat. Hence we may apply the lemma in the case of algebraic spaces to the diagram

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

of thickenings of algebraic spaces. See More on Morphisms of Spaces, Lemma 75.18.4. The statement about flatness of $\mathcal{Y}'/\mathcal{B}'$ at points in the image of $|f'|$ follows in the same manner.
$\square$

Lemma 105.5.2. Consider a commutative diagram

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

of thickenings of algebraic stacks. Assume $\mathcal{Y}' \to \mathcal{B}'$ locally of finite type, $\mathcal{X}' \to \mathcal{B}'$ flat and locally of finite presentation, $\mathcal{X} = \mathcal{B} \times _{\mathcal{B}'} \mathcal{X}'$, and $\mathcal{Y} = \mathcal{B} \times _{\mathcal{B}'} \mathcal{Y}'$. Then

$f$ 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 a monomorphism if and only if $f'$ is a monomorphism,

$f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,

$f$ 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 finite if and only if $f'$ is finite, and

add more here.

**Proof.**
In case (1) this follows from Lemma 105.5.1.

In cases (6), (7) this can be proved by the method used in the proof of Lemma 105.5.1. Namely, choose an algebraic space $U'$ and a surjective smooth morphism $U' \to \mathcal{B}'$. Choose an algebraic space $V'$ and a surjective smooth morphism $V' \to U' \times _{\mathcal{B}'} \mathcal{Y}'$. Choose an algebraic space $W'$ and a surjective smooth morphism $W' \to V' \times _{\mathcal{Y}'} \mathcal{X}'$. Let $U, V, W$ be the base change of $U', V', W'$ by $\mathcal{B} \to \mathcal{B}'$. Then the property for $f$, resp. $f'$ is equivalent to the property for of $W' \to V'$, resp. $W \to V$. Hence we may apply the lemma in the case of algebraic spaces to the diagram

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

of thickenings of algebraic spaces. See More on Morphisms of Spaces, Lemma 75.18.5.

In cases (8) and (9) we first see that the assumption for $f$ or $f'$ implies that both $f$ and $f'$ are DM morphisms of algebraic stacks, see Lemma 105.4.1. Then we can choose an algebraic space $U'$ and a surjective smooth morphism $U' \to \mathcal{B}'$. Choose an algebraic space $V'$ and a surjective smooth morphism $V' \to U' \times _{\mathcal{B}'} \mathcal{Y}'$. Choose an algebraic space $W'$ and a surjective étale(!) morphism $W' \to V' \times _{\mathcal{Y}'} \mathcal{X}'$. Let $U, V, W$ be the base change of $U', V', W'$ by $\mathcal{B} \to \mathcal{B}'$. Then $W \to V \times _\mathcal {Y} \mathcal{X}$ is surjective étale as well. Hence the property for $f$, resp. $f'$ is equivalent to the property for of $W' \to V'$, resp. $W \to V$. Hence we may apply the lemma in the case of algebraic spaces to the diagram

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

of thickenings of algebraic spaces. See More on Morphisms of Spaces, Lemma 75.18.5.

In cases (2), (3), (4), (10) we first conclude by Lemma 105.4.1 that $f$ and $f'$ are representable by algebraic spaces. Thus we may choose an algebraic space $U'$ and a surjective smooth morphism $U' \to \mathcal{B}'$, an algebraic space $V'$ and a surjective smooth morphism $V' \to U' \times _{\mathcal{B}'} \mathcal{Y}'$, and then $W' = V' \times _{\mathcal{Y}'} \mathcal{X}'$ will be an algebraic space. Let $U, V, W$ be the base change of $U', V', W'$ by $\mathcal{B} \to \mathcal{B}'$. Then $W = V \times _\mathcal {Y} \mathcal{X}$ as well. Then we have to see that $W' \to V'$ is an isomorphism, resp. an open immersion, resp. a monomorphism, resp. finite, if and only if $W \to V$ has the same property. See Properties of Stacks, Lemma 99.3.3. Thus we conclude by applying the results for algebraic spaces as above.

In the case (5) we first observe that $f$ and $f'$ are locally of finite type by Morphisms of Stacks, Lemma 100.17.8. On the other hand, the morphism $f$ is quasi-DM if and only if $f'$ is by Lemma 105.4.1. The last thing to check to see if $f$ or $f'$ is locally quasi-finite (Morphisms of Stacks, Definition 100.23.2) is a condition on underlying topological spaces which holds for $f$ if and only if it holds for $f'$ by the discussion in the first paragraph of the proof.
$\square$

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