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.)
$\square$
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