93.11 Algebraic spaces
The deformation theory of algebraic spaces.
Example 93.11.1 (Algebraic spaces). 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 an algebraic space $X$ 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 $g : X \to Y$ of algebraic spaces over $\Lambda $ 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 algebraic spaces.
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 an algebraic space $X$ 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 93.11.2. Example 93.11.1 satisfies the Rim-Schlessinger condition (RS). In particular, $\mathcal{D}\! \mathit{ef}_ X$ is a deformation category for any algebraic space $X$ 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 Pushouts of Spaces, Lemma 81.6.2. Thus the lemma is a special case of Pushouts of Spaces, Lemma 81.6.7.
$\square$
Lemma 93.11.3. In Example 93.11.1 let $X$ be an algebraic space 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 91.14.2 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 of Spaces, Lemma 76.21.4. 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 More on Morphisms of Spaces, Definition 76.7.2 and Modules on Sites, Definition 18.33.3.
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 91.14.2.
$\square$
Lemma 93.11.4. In Lemma 93.11.3 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 of Spaces, Lemma 76.21.5 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 Spaces, Lemma 75.8.4 the displayed $\mathop{\mathrm{Ext}}\nolimits $-groups are finite $k$-vector spaces and the proof is complete.
$\square$
In Example 93.11.1 if $X$ is a proper algebraic space 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 93.11.2 and 93.11.4 and the general discussion in Section 93.3.
Lemma 93.11.5. In Example 93.11.1 assume $X$ is a proper algebraic space over $k$. 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 93.11.2 and 93.11.4 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.
$\square$
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