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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