The Stacks project

105.5 Infinitesimal deformations of algebraic stacks

This section is the analogue of More on Morphisms of Spaces, Section 75.18.

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

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

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

  3. $f$ is flat, and

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

  1. $f$ is flat if and only if $f'$ is flat,

  2. $f$ is an isomorphism if and only if $f'$ is an isomorphism,

  3. $f$ is an open immersion if and only if $f'$ is an open immersion,

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

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

  6. $f$ is syntomic if and only if $f'$ is syntomic,

  7. $f$ is smooth if and only if $f'$ is smooth,

  8. $f$ is unramified if and only if $f'$ is unramified,

  9. $f$ is étale if and only if $f'$ is étale,

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

  11. 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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