In this section we spell out what the results in Section 91.13 mean for deformations of algebraic spaces.

Lemma 91.14.1. Let $S$ be a scheme. Let $i : Z \to Z'$ be a morphism of algebraic spaces over $S$. The following are equivalent

$i$ is a thickening of algebraic spaces as defined in More on Morphisms of Spaces, Section 76.9, and

the associated morphism $i_{small} : (\mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}), \mathcal{O}_ Z) \to (\mathop{\mathit{Sh}}\nolimits (Z'_{\acute{e}tale}), \mathcal{O}_{Z'})$ of ringed topoi (Properties of Spaces, Lemma 66.21.3) is a thickening in the sense of Section 91.9.

**Proof.**
We stress that this is not a triviality.

Assume (1). By More on Morphisms of Spaces, Lemma 76.9.6 the morphism $i$ induces an equivalence of small étale sites and in particular of topoi. Of course $i^\sharp $ is surjective with locally nilpotent kernel by definition of thickenings.

Assume (2). (This direction is less important and more of a curiosity.) For any étale morphism $Y' \to Z'$ we see that $Y = Z \times _{Z'} Y'$ has the same étale topos as $Y'$. In particular, $Y'$ is quasi-compact if and only if $Y$ is quasi-compact because being quasi-compact is a topos theoretic notion (Sites, Lemma 7.17.3). Having said this we see that $Y'$ is quasi-compact and quasi-separated if and only if $Y$ is quasi-compact and quasi-separated (because you can characterize $Y'$ being quasi-separated by saying that for all $Y'_1, Y'_2$ quasi-compact algebraic spaces étale over $Y'$ we have that $Y'_1 \times _{Y'} Y'_2$ is quasi-compact). Take $Y'$ affine. Then the algebraic space $Y$ is quasi-compact and quasi-separated. For any quasi-coherent $\mathcal{O}_ Y$-module $\mathcal{F}$ we have $H^ q(Y, \mathcal{F}) = H^ q(Y', (Y \to Y')_*\mathcal{F})$ because the étale topoi are the same. Then $H^ q(Y', (Y \to Y')_*\mathcal{F}) = 0$ because the pushforward is quasi-coherent (Morphisms of Spaces, Lemma 67.11.2) and $Y$ is affine. It follows that $Y'$ is affine by Cohomology of Spaces, Proposition 69.16.7 (there surely is a proof of this direction of the lemma avoiding this reference). Hence $i$ is an affine morphism. In the affine case it follows easily from the conditions in Section 91.9 that $i$ is a thickening of algebraic spaces.
$\square$

Lemma 91.14.2. Let $S$ be a scheme. Let $Y \subset Y'$ be a first order thickening of algebraic spaces over $S$. Let $f : X \to Y$ be a flat morphism of algebraic spaces over $S$. If there exists a flat morphism $f' : X' \to Y'$ of algebraic spaces over $S$ and an isomorphsm $a : X \to X' \times _{Y'} Y$ over $Y$, then

the set of isomorphism classes of pairs $(f' : X' \to Y', a)$ is principal homogeneous under $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/Y}, f^*\mathcal{C}_{Y/Y'})$, and

the set of automorphisms of $\varphi : X' \to X'$ over $Y'$ which reduce to the identity on $X' \times _{Y'} Y$ is $\mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/Y}, f^*\mathcal{C}_{Y/Y'})$.

**Proof.**
We will apply the material on deformations of ringed topoi to the small étale topoi of the algebraic spaces in the lemma. We may think of $X$ as a closed subspace of $X'$ so that $(f, f') : (X \subset X') \to (Y \subset Y')$ is a morphism of first order thickenings. By Lemma 91.14.1 this translates into a morphism of thickenings of ringed topoi. Then we see from More on Morphisms of Spaces, Lemma 76.18.1 (or from the more general Lemma 91.11.2) that the ideal sheaf of $X$ in $X'$ is equal to $f^*\mathcal{C}_{Y'/Y}$ and this is in fact equivalent to flatness of $X'$ over $Y'$. Hence we have a commutative diagram

\[ \xymatrix{ 0 \ar[r] & f^*\mathcal{C}_{Y/Y'} \ar[r] & \mathcal{O}_{X'} \ar[r] & \mathcal{O}_ X \ar[r] & 0 \\ 0 \ar[r] & f_{small}^{-1}\mathcal{C}_{Y/Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_{Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_ Y \ar[u] \ar[r] & 0 } \]

Please compare with (91.13.0.1). Observe that automorphisms $\varphi $ as in (2) give automorphisms $\varphi ^\sharp : \mathcal{O}_{X'} \to \mathcal{O}_{X'}$ fitting in the diagram above. Conversely, an automorphism $\alpha : \mathcal{O}_{X'} \to \mathcal{O}_{X'}$ fitting into the diagram of sheaves above is equal to $\varphi ^\sharp $ for some automorphism $\varphi $ as in (2) by More on Morphisms of Spaces, Lemma 76.9.2. Finally, by More on Morphisms of Spaces, Lemma 76.9.7 if we find another sheaf of rings $\mathcal{A}$ on $X_{\acute{e}tale}$ fitting into the diagram

\[ \xymatrix{ 0 \ar[r] & f^*\mathcal{C}_{Y/Y'} \ar[r] & \mathcal{A} \ar[r] & \mathcal{O}_ X \ar[r] & 0 \\ 0 \ar[r] & f_{small}^{-1}\mathcal{C}_{Y/Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_{Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_ Y \ar[u] \ar[r] & 0 } \]

then there exists a first order thickening $X \subset X''$ with $\mathcal{O}_{X''} = \mathcal{A}$ and applying More on Morphisms of Spaces, Lemma 76.9.2 once more, we obtain a morphism $(f, f'') : (X \subset X'') \to (Y \subset Y')$ with all the desired properties. Thus part (1) follows from Lemma 91.13.3 and part (2) from part (2) of Lemma 91.13.1. (Note that $\mathop{N\! L}\nolimits _{X/Y}$ as defined for a morphism of algebraic spaces in More on Morphisms of Spaces, Section 76.21 agrees with $\mathop{N\! L}\nolimits _{X/Y}$ as used in Section 91.13.)
$\square$

Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F} \to \mathcal{G}$ be a homomorphism of $\mathcal{O}_ X$-modules (not necessarily quasi-coherent). Consider the functor

\[ F : \left\{ \begin{matrix} \text{extensions of }f^{-1}\mathcal{O}_ B\text{ algebras}
\\ 0 \to \mathcal{F} \to \mathcal{O}' \to \mathcal{O}_ X \to 0
\\ \text{where }\mathcal{F}\text{ is an ideal of square zero}
\end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{extensions of }f^{-1}\mathcal{O}_ B\text{ algebras}
\\ 0 \to \mathcal{G} \to \mathcal{O}' \to \mathcal{O}_ X \to 0
\\ \text{where }\mathcal{G}\text{ is an ideal of square zero}
\end{matrix} \right\} \]

given by pushout.

Lemma 91.14.3. In the situation above assume that $X$ is quasi-compact and quasi-separated and that $DQ_ X(\mathcal{F}) \to DQ_ X(\mathcal{G})$ (Derived Categories of Spaces, Section 75.19) is an isomorphism. Then the functor $F$ is an equivalence of categories.

**Proof.**
Recall that $\mathop{N\! L}\nolimits _{X/B}$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$, see More on Morphisms of Spaces, Lemma 76.21.4. Hence our assumption implies the maps

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathop{N\! L}\nolimits _{X/B}, \mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathop{N\! L}\nolimits _{X/B}, \mathcal{G}) \]

are isomorphisms for all $i$. This implies our functor is fully faithful by Lemma 91.13.1. On the other hand, the functor is essentially surjective by Lemma 91.13.3 because we have the solutions $\mathcal{O}_ X \oplus \mathcal{F}$ and $\mathcal{O}_ X \oplus \mathcal{G}$ in both categories.
$\square$

Let $S$ be a scheme. Let $B \subset B'$ be a first order thickening of algebraic spaces over $S$ with ideal sheaf $\mathcal{J}$ which we view either as a quasi-coherent $\mathcal{O}_ B$-module or as a quasi-coherent sheaf of ideals on $B'$, see More on Morphisms of Spaces, Section 76.9. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F} \to \mathcal{G}$ be a homomorphism of $\mathcal{O}_ X$-modules (not necessarily quasi-coherent). Let $c : f^{-1}\mathcal{J} \to \mathcal{F}$ be a map of $f^{-1}\mathcal{O}_ B$-modules and denote $c' : f^{-1}\mathcal{J} \to \mathcal{G}$ the composition. Consider the functor

\[ FT : \{ \text{solutions to }(08UF) \text{ for }\mathcal{F}\text{ and }c\} \longrightarrow \{ \text{solutions to }(08UF) \text{ for }\mathcal{G}\text{ and }c'\} \]

given by pushout.

Lemma 91.14.4. In the situation above assume that $X$ is quasi-compact and quasi-separated and that $DQ_ X(\mathcal{F}) \to DQ_ X(\mathcal{G})$ (Derived Categories of Spaces, Section 75.19) is an isomorphism. Then the functor $FT$ is an equivalence of categories.

**Proof.**
A solution of (91.13.0.1) for $\mathcal{F}$ in particular gives an extension of $f^{-1}\mathcal{O}_{B'}$-algebras

\[ 0 \to \mathcal{F} \to \mathcal{O}' \to \mathcal{O}_ X \to 0 \]

where $\mathcal{F}$ is an ideal of square zero. Similarly for $\mathcal{G}$. Moreover, given such an extension, we obtain a map $c_{\mathcal{O}'} : f^{-1}\mathcal{J} \to \mathcal{F}$. Thus we are looking at the full subcategory of such extensions of $f^{-1}\mathcal{O}_{B'}$-algebras with $c = c_{\mathcal{O}'}$. Clearly, if $\mathcal{O}'' = F(\mathcal{O}')$ where $F$ is the equivalence of Lemma 91.14.3 (applied to $X \to B'$ this time), then $c_{\mathcal{O}''}$ is the composition of $c_{\mathcal{O}'}$ and the map $\mathcal{F} \to \mathcal{G}$. This proves the lemma.
$\square$

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