## 76.18 Infinitesimal deformations of algebraic spaces

The following simple lemma is often a convenient tool to check whether an infinitesimal deformation of a map is flat.

Lemma 76.18.1. Let $S$ be a scheme. Let $(f, f') : (X \subset X') \to (Y \subset Y')$ be a morphism of first order thickenings of algebraic spaces over $S$. Assume that $f$ is flat. Then the following are equivalent

1. $f'$ is flat and $X = Y \times _{Y'} X'$, and

2. the canonical map $f^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ is an isomorphism.

Proof. Choose a scheme $V'$ and a surjective étale morphism $V' \to Y'$. Choose a scheme $U'$ and a surjective étale morphism $U' \to X' \times _{Y'} V'$. Set $U = X \times _{X'} U'$ and $V = Y \times _{Y'} V'$. According to our definition of a flat morphism of algebraic spaces we see that the induced map $g : U \to V$ is a flat morphism of schemes and that $f'$ is flat if and only if the corresponding morphism $g' : U' \to V'$ is flat. Also, $X = Y \times _{Y'} X'$ if and only if $U = V \times _{V'} V'$. Finally, the map $f^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ is an isomorphism if and only if $g^*\mathcal{C}_{V/V'} \to \mathcal{C}_{U/U'}$ is an isomorphism. Hence the lemma follows from its analogue for morphisms of schemes, see More on Morphisms, Lemma 37.10.1. $\square$

The following lemma is the “nilpotent” version of the “critère de platitude par fibres”, see Section 76.23.

Lemma 76.18.2. 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

1. $X'$ is flat over $B'$,

2. $f$ is flat,

3. $B \subset B'$ is a finite order thickening, and

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

Then $f'$ is flat and $Y'$ is flat over $B'$ at all points in the image of $f'$.

Proof. 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 $B \to B'$. 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 schemes to the diagram

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

of thickenings of schemes. See More on Morphisms, Lemma 37.10.2. The statement about flatness of $Y'/B'$ at points in the image of $f'$ follows in the same manner. $\square$

Many properties of morphisms of schemes are preserved under flat deformations.

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

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

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

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

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

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

6. $f$ is universally closed if and only if $f'$ is universally closed,

7. $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,

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

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

10. $f$ is universally injective if and only if $f'$ is universally injective,

11. $f$ is affine if and only if $f'$ is affine,

12. $f$ is locally of finite type if and only if $f'$ is locally of finite type,

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

14. $f$ is locally of finite presentation if and only if $f'$ is locally of finite presentation,

15. $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$,

16. $f$ is universally open if and only if $f'$ is universally open,

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

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

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

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

21. $f$ is proper if and only if $f'$ is proper,

22. $f$ is integral if and only if $f'$ is integral,

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

24. $f$ is finite locally free (of rank $d$) if and only if $f'$ is finite locally free (of rank $d$), and

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 following lemma is the “locally nilpotent” version of the “critère de platitude par fibres”, see Section 76.23.

Lemma 76.18.4. 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

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

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

3. $f$ is flat, and

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

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

Proof. 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 $B \to B'$. 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 schemes to the diagram

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

of thickenings of schemes. See More on Morphisms, Lemma 37.10.4. The statement about flatness of $Y'/B'$ at points in the image of $f'$ follows in the same manner. $\square$

Many properties of morphisms of schemes are preserved under flat deformations as in the lemma above.

Lemma 76.18.5. 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 $Y' \to B'$ locally of finite type, $X' \to B'$ flat and locally of finite presentation, $X = B \times _{B'} X'$, and $Y = B \times _{B'} Y'$. Then

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

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

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

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

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

6. $f$ is universally closed if and only if $f'$ is universally closed,

7. $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,

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

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

10. $f$ is universally injective if and only if $f'$ is universally injective,

11. $f$ is affine if and only if $f'$ is affine,

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

13. $f$ is locally of finite type of relative dimension $d$ if and only if $f'$ is locally of finite type of relative dimension $d$,

14. $f$ is universally open if and only if $f'$ is universally open,

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

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

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

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

19. $f$ is proper if and only if $f'$ is proper,

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

21. $f$ is finite locally free (of rank $d$) if and only if $f'$ is finite locally free (of rank $d$), and

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), (15), (16), (17), (18) follow from the corresponding cases of More on Morphisms, Lemma 37.10.5.

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 (14) 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 (18) 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 (19). See Lemma 76.10.3.

The case (20). See Lemma 76.10.3.

The case (21). Assume $f$ finite locally free. By (20) we see that $f'$ is finite. By (2) we see that $f'$ is flat. Also $f'$ is locally finite presentation by Morphisms of Spaces, Lemma 67.28.9. Hence $f'$ is finite locally free by Morphisms of Spaces, Lemma 67.46.6. $\square$

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