93.9 Schemes
The deformation theory of schemes.
Example 93.9.1 (Schemes). Let $\mathcal{F}$ be the category defined as follows
an object is a pair $(A, X)$ consisting of an object $A$ of $\mathcal{C}_\Lambda $ and a scheme $X$ flat over $A$, and
a morphism $(f, g) : (B, Y) \to (A, X)$ consists of a morphism $f : B \to A$ in $\mathcal{C}_\Lambda $ together with a morphism $g : X \to Y$ such that
\[ \xymatrix{ X \ar[r]_ g \ar[d] & Y \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r]^ f & \mathop{\mathrm{Spec}}(B) } \]
is a cartesian commutative diagram of schemes.
The functor $p : \mathcal{F} \to \mathcal{C}_\Lambda $ sends $(A, X)$ to $A$ and $(f, g)$ to $f$. It is clear that $p$ is cofibred in groupoids. Given a scheme $X$ over $k$, let $x_0 = (k, X)$ be the corresponding object of $\mathcal{F}(k)$. We set
\[ \mathcal{D}\! \mathit{ef}_ X = \mathcal{F}_{x_0} \]
Lemma 93.9.2. Example 93.9.1 satisfies the Rim-Schlessinger condition (RS). In particular, $\mathcal{D}\! \mathit{ef}_ X$ is a deformation category for any scheme $X$ over $k$.
Proof.
Let $A_1 \to A$ and $A_2 \to A$ be morphisms of $\mathcal{C}_\Lambda $. Assume $A_2 \to A$ is surjective. According to Formal Deformation Theory, Lemma 90.16.4 it suffices to show that the functor $\mathcal{F}(A_1 \times _ A A_2) \to \mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$ is an equivalence of categories. Observe that
\[ \xymatrix{ \mathop{\mathrm{Spec}}(A) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(A_2) \ar[d] \\ \mathop{\mathrm{Spec}}(A_1) \ar[r] & \mathop{\mathrm{Spec}}(A_1 \times _ A A_2) } \]
is a pushout diagram as in More on Morphisms, Lemma 37.14.3. Thus the lemma is a special case of More on Morphisms, Lemma 37.14.6.
$\square$
Lemma 93.9.3. In Example 93.9.1 let $X$ be a scheme over $k$. Then
\[ \text{Inf}(\mathcal{D}\! \mathit{ef}_ X) = \text{Ext}^0_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/k}, \mathcal{O}_ X) = \text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X) \]
and
\[ T\mathcal{D}\! \mathit{ef}_ X = \text{Ext}^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X) \]
Proof.
Recall that $\text{Inf}(\mathcal{D}\! \mathit{ef}_ X)$ is the set of automorphisms of the trivial deformation $X' = X \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k[\epsilon ])$ of $X$ to $k[\epsilon ]$ equal to the identity modulo $\epsilon $. By Deformation Theory, Lemma 91.8.1 this is equal to $\text{Ext}^0_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X)$. The equality $\text{Ext}^0_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/k}, \mathcal{O}_ X)$ follows from More on Morphisms, Lemma 37.13.3. The equality $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/k}, \mathcal{O}_ X) = \text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X)$ follows from Morphisms, Lemma 29.32.2.
Recall that $T_{x_0}\mathcal{D}\! \mathit{ef}_ X$ is the set of isomorphism classes of flat deformations $X'$ of $X$ to $k[\epsilon ]$, more precisely, the set of isomorphism classes of $\mathcal{D}\! \mathit{ef}_ X(k[\epsilon ])$. Thus the second statement of the lemma follows from Deformation Theory, Lemma 91.8.1.
$\square$
Lemma 93.9.4. In Lemma 93.9.3 if $X$ is proper over $k$, then $\text{Inf}(\mathcal{D}\! \mathit{ef}_ X)$ and $T\mathcal{D}\! \mathit{ef}_ X$ are finite dimensional.
Proof.
By the lemma we have to show $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X)$ and $\mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X)$ are finite dimensional. By More on Morphisms, Lemma 37.13.4 and the fact that $X$ is Noetherian, we see that $\mathop{N\! L}\nolimits _{X/k}$ has coherent cohomology sheaves zero except in degrees $0$ and $-1$. By Derived Categories of Schemes, Lemma 36.11.7 the displayed $\mathop{\mathrm{Ext}}\nolimits $-groups are finite $k$-vector spaces and the proof is complete.
$\square$
In Example 93.9.1 if $X$ is a proper scheme over $k$, then $\mathcal{D}\! \mathit{ef}_ X$ admits a presentation by a smooth prorepresentable groupoid in functors over $\mathcal{C}_\Lambda $ and a fortiori has a (minimal) versal formal object. This follows from Lemmas 93.9.2 and 93.9.4 and the general discussion in Section 93.3.
Lemma 93.9.5. In Example 93.9.1 assume $X$ is a proper $k$-scheme. Assume $\Lambda $ is a complete local ring with residue field $k$ (the classical case). Then the functor
\[ F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}\! \mathit{ef}_ X(A))/\cong \]
of isomorphism classes of objects has a hull. If $\text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X) = 0$, then $F$ is prorepresentable.
Proof.
The existence of a hull follows immediately from Lemmas 93.9.2 and 93.9.4 and Formal Deformation Theory, Lemma 90.16.6 and Remark 90.15.7.
Assume $\text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X) = 0$. Then $\mathcal{D}\! \mathit{ef}_ X$ and $F$ are equivalent by Formal Deformation Theory, Lemma 90.19.13. Hence $F$ is a deformation functor (because $\mathcal{D}\! \mathit{ef}_ X$ is a deformation category) with finite tangent space and we can apply Formal Deformation Theory, Theorem 90.18.2.
$\square$
Lemma 93.9.6. In Example 93.9.1 let $X$ be a scheme over $k$. Let $U \subset X$ be an open subscheme. There is a natural functor
\[ \mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_ U \]
of deformation categories.
Proof.
Given a deformation of $X$ we can take the corresponding open of it to get a deformation of $U$. We omit the details.
$\square$
Lemma 93.9.7. In Example 93.9.1 let $X = \mathop{\mathrm{Spec}}(P)$ be an affine scheme over $k$. With $\mathcal{D}\! \mathit{ef}_ P$ as in Example 93.8.1 there is a natural equivalence
\[ \mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_ P \]
of deformation categories.
Proof.
The functor sends $(A, Y)$ to $\Gamma (Y, \mathcal{O}_ Y)$. This works because any deformation of $X$ is affine by More on Morphisms, Lemma 37.2.3.
$\square$
Lemma 93.9.8. In Example 93.9.1 let $X$ be a scheme over $k$ Let $p \in X$ be a point. With $\mathcal{D}\! \mathit{ef}_{\mathcal{O}_{X, p}}$ as in Example 93.8.1 there is a natural functor
\[ \mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{\mathcal{O}_{X, p}} \]
of deformation categories.
Proof.
Choose an affine open $U = \mathop{\mathrm{Spec}}(P) \subset X$ containing $p$. Then $\mathcal{O}_{X, p}$ is a localization of $P$. We combine the functors from Lemmas 93.9.6, 93.9.7, and 93.8.7.
$\square$
Situation 93.9.9. Let $\Lambda \to k$ be as in Section 93.3. Let $X$ be a scheme over $k$ which has an affine open covering $X = U_1 \cup U_2$ with $U_{12} = U_1 \cap U_2$ affine too. Write $U_1 = \mathop{\mathrm{Spec}}(P_1)$, $U_2 = \mathop{\mathrm{Spec}}(P_2)$ and $U_{12} = \mathop{\mathrm{Spec}}(P_{12})$. Let $\mathcal{D}\! \mathit{ef}_ X$, $\mathcal{D}\! \mathit{ef}_{U_1}$, $\mathcal{D}\! \mathit{ef}_{U_2}$, and $\mathcal{D}\! \mathit{ef}_{U_{12}}$ be as in Example 93.9.1 and let $\mathcal{D}\! \mathit{ef}_{P_1}$, $\mathcal{D}\! \mathit{ef}_{P_2}$, and $\mathcal{D}\! \mathit{ef}_{P_{12}}$ be as in Example 93.8.1.
Lemma 93.9.10. In Situation 93.9.9 there is an equivalence
\[ \mathcal{D}\! \mathit{ef}_ X = \mathcal{D}\! \mathit{ef}_{P_1} \times _{\mathcal{D}\! \mathit{ef}_{P_{12}}} \mathcal{D}\! \mathit{ef}_{P_2} \]
of deformation categories, see Examples 93.9.1 and 93.8.1.
Proof.
It suffices to show that the functors of Lemma 93.9.6 define an equivalence
\[ \mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{U_1} \times _{\mathcal{D}\! \mathit{ef}_{U_{12}}} \mathcal{D}\! \mathit{ef}_{U_2} \]
because then we can apply Lemma 93.9.7 to translate into rings. To do this we construct a quasi-inverse. Denote $F_ i : \mathcal{D}\! \mathit{ef}_{U_ i} \to \mathcal{D}\! \mathit{ef}_{U_{12}}$ the functor of Lemma 93.9.6. An object of the RHS is given by an $A$ in $\mathcal{C}_\Lambda $, objects $(A, V_1) \to (k, U_1)$ and $(A, V_2) \to (k, U_2)$, and a morphism
\[ g : F_1(A, V_1) \to F_2(A, V_2) \]
Now $F_ i(A, V_ i) = (A, V_{i, 3 - i})$ where $V_{i, 3 - i} \subset V_ i$ is the open subscheme whose base change to $k$ is $U_{12} \subset U_ i$. The morphism $g$ defines an isomorphism $V_{1, 2} \to V_{2, 1}$ of schemes over $A$ compatible with $\text{id} : U_{12} \to U_{12}$ over $k$. Thus $(\{ 1, 2\} , V_ i, V_{i, 3 - i}, g, g^{-1})$ is a glueing data as in Schemes, Section 26.14. Let $Y$ be the glueing, see Schemes, Lemma 26.14.1. Then $Y$ is a scheme over $A$ and the compatibilities mentioned above show that there is a canonical isomorphism $Y \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(k) = X$. Thus $(A, Y) \to (k, X)$ is an object of $\mathcal{D}\! \mathit{ef}_ X$. We omit the verification that this construction is a functor and is quasi-inverse to the given one.
$\square$
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