## 91.8 Deformations of schemes

In this section we spell out what the results in Section 91.7 mean for deformations of schemes.

Lemma 91.8.1. Let $S \subset S'$ be a first order thickening of schemes. Let $f : X \to S$ be a flat morphism of schemes. If there exists a flat morphism $f' : X' \to S'$ of schemes and an isomorphism $a : X \to X' \times _{S'} S$ over $S$, then

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

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

Proof. First we observe that thickenings of schemes as defined in More on Morphisms, Section 37.2 are the same things as morphisms of schemes which are thickenings in the sense of Section 91.3. We may think of $X$ as a closed subscheme of $X'$ so that $(f, f') : (X \subset X') \to (S \subset S')$ is a morphism of first order thickenings. Then we see from More on Morphisms, Lemma 37.10.1 (or from the more general Lemma 91.5.2) that the ideal sheaf of $X$ in $X'$ is equal to $f^*\mathcal{C}_{S/S'}$. Hence we have a commutative diagram

$\xymatrix{ 0 \ar[r] & f^*\mathcal{C}_{S/S'} \ar[r] & \mathcal{O}_{X'} \ar[r] & \mathcal{O}_ X \ar[r] & 0 \\ 0 \ar[r] & \mathcal{C}_{S/S'} \ar[u] \ar[r] & \mathcal{O}_{S'} \ar[u] \ar[r] & \mathcal{O}_ S \ar[u] \ar[r] & 0 }$

where the vertical arrows are $f$-maps; please compare with (91.7.0.1). Thus part (1) follows from Lemma 91.7.3 and part (2) from part (2) of Lemma 91.7.1. (Note that $\mathop{N\! L}\nolimits _{X/S}$ as defined for a morphism of schemes in More on Morphisms, Section 37.13 agrees with $\mathop{N\! L}\nolimits _{X/S}$ as used in Section 91.7.) $\square$

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