Deformation Problems

Introduction

The goal of this chapter is to work out examples of the general theory developed in the chapters Formal Deformation Theory, Deformation Theory, The Cotangent Complex.

Section 3 of the paper [Sch] by Schlessinger discusses some examples as well.

Examples of deformation problems

List of things that should go here:

1. Deformations of schemes:

   1. The Rim-Schlessinger condition.

   2. Computing the tangent space.

   3. Computing the infinitesimal deformations.

   4. The deformation category of an affine hypersurface.

2. Deformations of sheaves (for example fix \(X/S\), a finite type point \(s\) of \(S\), and a quasi-coherent sheaf \(\mathcal{F}_s\) over \(X_s\)).

3. Deformations of algebraic spaces (very similar to deformations of schemes; maybe even easier?).

4. Deformations of maps (eg morphisms between schemes; you can fix both or one of the target and/or source).

5. Add more here.

General outline

This section lays out the procedure for discussing the next few examples.

Step I. For each section we fix a Noetherian ring \(\Lambda\) and we fix a finite ring map \(\Lambda \to k\) where \(k\) is a field. As usual we let \(\mathcal{C}_\Lambda = \mathcal{C}_{\Lambda, k}\) be our base category, see Formal Deformation Theory, Definition 06GC.

Step II. In each section we define a category \(\mathcal{F}\) cofibred in groupoids over \(\mathcal{C}_\Lambda\). Occasionally we will consider instead a functor \(F : \mathcal{C}_\Lambda \to \textit{Sets}\).

Step III. We explain to what extent \(\mathcal{F}\) satisfies the Rim-Schlesssinger condition (RS) discussed in Formal Deformation Theory, Section 06J1. Similarly, we may discuss to what extent our \(\mathcal{F}\) satisfies (S1) and (S2) or to what extent \(F\) satisfies the corresponding Schlessinger’s conditions (H1) and (H2). See Formal Deformation Theory, Section 06HV.

Step IV. Let \(x_0\) be an object of \(\mathcal{F}(k)\), in other words an object of \(\mathcal{F}\) over \(k\). In this chapter we will use the notation \[\Deformationcategory_{x_0} = \mathcal{F}_{x_0}\] to denote the predeformation category constructed in Formal Deformation Theory, Remark 06GU. If \(\mathcal{F}\) satisfies (RS), then \(\Deformationcategory_{x_0}\) is a deformation category (Formal Deformation Theory, Lemma 06JC) and satisfies (S1) and (S2) (Formal Deformation Theory, Lemma 06J7). If (S1) and (S2) are satisfied, then an important question is whether the tangent space \[T\Deformationcategory_{x_0} = T_{x_0}\mathcal{F} = T\mathcal{F}_{x_0}\] (see Formal Deformation Theory, Remark 06IK and Definition 06IG) is finite dimensional. Namely, this insures that \(\Deformationcategory_{x_0}\) has a versal formal object (Formal Deformation Theory, Lemma 06IW).

Step V. If \(\mathcal{F}\) passes Step IV, then the next question is whether the \(k\)-vector space \[\text{Inf}(\Deformationcategory_{x_0}) = \text{Inf}_{x_0}(\mathcal{F})\] of infinitesimal automorphisms of \(x_0\) is finite dimensional. Namely, if true, this implies that \(\Deformationcategory_{x_0}\) admits a presentation by a smooth prorepresentable groupoid in functors on \(\mathcal{C}_\Lambda\), see Formal Deformation Theory, Theorem 06L8.

Finite projective modules

This section is just a warmup. Of course finite projective modules should not have any “moduli”.

Example

Let \(\mathcal{F}\) be the category defined as follows

1. an object is a pair \((A, M)\) consisting of an object \(A\) of \(\mathcal{C}_\Lambda\) and a finite projective \(A\)-module \(M\), and

2. a morphism \((f, g) : (B, N) \to (A, M)\) consists of a morphism \(f : B \to A\) in \(\mathcal{C}_\Lambda\) together with a map \(g : N \to M\) which is \(f\)-linear and induces an isomorpism \(N \otimes_{B, f} A \cong M\).

The functor \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) sends \((A, M)\) to \(A\) and \((f, g)\) to \(f\). It is clear that \(p\) is cofibred in groupoids. Given a finite dimensional \(k\)-vector space \(V\), let \(x_0 = (k, V)\) be the corresponding object of \(\mathcal{F}(k)\). We set \[\Deformationcategory_V = \mathcal{F}_{x_0}\]

Since every finite projective module over a local ring is finite free (Algebra, Lemma 00NX) we see that \[\begin{matrix} \text{isomorphism classes} \\ \text{of objects of }\mathcal{F}(A) \end{matrix} = \coprod\nolimits_{n \geq 0} \{*\}\] Although this means that the deformation theory of \(\mathcal{F}\) is essentially trivial, we still work through the steps outlined in Section 0DVM to provide an easy example.

Lemma

Example 0D3I satisfies the Rim-Schlessinger condition (RS). In particular, \(\Deformationcategory_V\) is a deformation category for any finite dimensional vector space \(V\) 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 06J5 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.

Thus we have to show that the category of finite projective modules over \(A_1 \times_A A_2\) is equivalent to the fibre product of the categories of finite projective modules over \(A_1\) and \(A_2\) over the category of finite projective modules over \(A\). This is a special case of More on Algebra, Lemma 0D2J. We recall that the inverse functor sends the triple \((M_1, M_2, \varphi)\) where \(M_1\) is a finite projective \(A_1\)-module, \(M_2\) is a finite projective \(A_2\)-module, and \(\varphi : M_1 \otimes_{A_1} A \to M_2 \otimes_{A_2} A\) is an isomorphism of \(A\)-module, to the finite projective \(A_1 \times_A A_2\)-module \(M_1 \times_\varphi M_2\).

Lemma

In Example 0D3I let \(V\) be a finite dimensional \(k\)-vector space. Then \[T\Deformationcategory_V = (0) \quad\text{and}\quad \text{Inf}(\Deformationcategory_V) = \text{End}_k(V)\] are finite dimensional.

Proof

With \(\mathcal{F}\) as in Example 0D3I set \(x_0 = (k, V) \in \Ob(\mathcal{F}(k))\). Recall that \(T\Deformationcategory_V = T_{x_0}\mathcal{F}\) is the set of isomorphism classes of pairs \((x, \alpha)\) consisting of an object \(x\) of \(\mathcal{F}\) over the dual numbers \(k[\epsilon]\) and a morphism \(\alpha : x \to x_0\) of \(\mathcal{F}\) lying over \(k[\epsilon] \to k\).

Up to isomorphism, there is a unique pair \((M, \alpha)\) consisting of a finite projective module \(M\) over \(k[\epsilon]\) and \(k[\epsilon]\)-linear map \(\alpha : M \to V\) which induces an isomorphism \(M \otimes_{k[\epsilon]} k \to V\). For example, if \(V = k^{\oplus n}\), then we take \(M = k[\epsilon]^{\oplus n}\) with the obvious map \(\alpha\).

Similarly, \(\text{Inf}(\Deformationcategory_V) = \text{Inf}_{x_0}(\mathcal{F})\) is the set of automorphisms of the trivial deformation \(x'_0\) of \(x_0\) over \(k[\epsilon]\). See Formal Deformation Theory, Definition 06JQ for details.

Given \((M, \alpha)\) as in the second paragraph, we see that an element of \(\text{Inf}_{x_0}(\mathcal{F})\) is an automorphism \(\gamma : M \to M\) with \(\gamma \bmod \epsilon = \text{id}\). Then we can write \(\gamma = \text{id}_M + \epsilon \psi\) where \(\psi : M/\epsilon M \to M/\epsilon M\) is \(k\)-linear. Using \(\alpha\) we can think of \(\psi\) as an element of \(\text{End}_k(V)\) and this finishes the proof.

Representations of a group

The deformation theory of representations can be very interesting.

Example

Let \(\Gamma\) be a group. Let \(\mathcal{F}\) be the category defined as follows

1. an object is a triple \((A, M, \rho)\) consisting of an object \(A\) of \(\mathcal{C}_\Lambda\), a finite projective \(A\)-module \(M\), and a homomorphism \(\rho : \Gamma \to \text{GL}_A(M)\), and

2. a morphism \((f, g) : (B, N, \tau) \to (A, M, \rho)\) consists of a morphism \(f : B \to A\) in \(\mathcal{C}_\Lambda\) together with a map \(g : N \to M\) which is \(f\)-linear and \(\Gamma\)-equivariant and induces an isomorpism \(N \otimes_{B, f} A \cong M\).

The functor \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) sends \((A, M, \rho)\) to \(A\) and \((f, g)\) to \(f\). It is clear that \(p\) is cofibred in groupoids. Given a finite dimensional \(k\)-vector space \(V\) and a representation \(\rho_0 : \Gamma \to \text{GL}_k(V)\), let \(x_0 = (k, V, \rho_0)\) be the corresponding object of \(\mathcal{F}(k)\). We set \[\Deformationcategory_{V, \rho_0} = \mathcal{F}_{x_0}\]

Since every finite projective module over a local ring is finite free (Algebra, Lemma 00NX) we see that \[\begin{matrix} \text{isomorphism classes} \\ \text{of objects of }\mathcal{F}(A) \end{matrix} = \coprod\nolimits_{n \geq 0}\quad \begin{matrix} \text{GL}_n(A)\text{-conjugacy classes of}\\ \text{homomorphisms }\rho : \Gamma \to \text{GL}_n(A) \end{matrix}\] This is already more interesting than the discussion in Section 0DVN.

Lemma

Example 0D3J satisfies the Rim-Schlessinger condition (RS). In particular, \(\Deformationcategory_{V, \rho_0}\) is a deformation category for any finite dimensional representation \(\rho_0 : \Gamma \to \text{GL}_k(V)\).

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 06J5 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.

Consider an object \[((A_1, M_1, \rho_1), (A_2, M_2, \rho_2), (\text{id}_A, \varphi))\] of the category \(\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)\). Then, as seen in the proof of Lemma 0DVP, we can consider the finite projective \(A_1 \times_A A_2\)-module \(M_1 \times_\varphi M_2\). Since \(\varphi\) is compatible with the given actions we obtain \[\rho_1 \times \rho_2 : \Gamma \longrightarrow \text{GL}_{A_1 \times_A A_2}(M_1 \times_\varphi M_2)\] Then \((M_1 \times_\varphi M_2, \rho_1 \times \rho_2)\) is an object of \(\mathcal{F}(A_1 \times_A A_2)\). This construction determines a quasi-inverse to our functor.

Lemma

In Example 0D3J let \(\rho_0 : \Gamma \to \text{GL}_k(V)\) be a finite dimensional representation. Then \[T\Deformationcategory_{V, \rho_0} = \Ext^1_{k[\Gamma]}(V, V) = H^1(\Gamma, \text{End}_k(V)) \quad\text{and}\quad \text{Inf}(\Deformationcategory_{V, \rho_0}) = H^0(\Gamma, \text{End}_k(V))\] Thus \(\text{Inf}(\Deformationcategory_{V, \rho_0})\) is always finite dimensional and \(T\Deformationcategory_{V, \rho_0}\) is finite dimensional if \(\Gamma\) is finitely generated.

Proof

We first deal with the infinitesimal automorphisms. Let \(M = V \otimes_k k[\epsilon]\) with induced action \(\rho_0' : \Gamma \to \text{GL}_n(M)\). Then an infinitesimal automorphism, i.e., an element of \(\text{Inf}(\Deformationcategory_{V, \rho_0})\), is given by an automorphism \(\gamma = \text{id} + \epsilon \psi : M \to M\) as in the proof of Lemma 0DVQ, where moreover \(\psi\) has to commute with the action of \(\Gamma\) (given by \(\rho_0\)). Thus we see that \[\text{Inf}(\Deformationcategory_{V, \rho_0}) = H^0(\Gamma, \text{End}_k(V))\] as predicted in the lemma.

Next, let \((k[\epsilon], M, \rho)\) be an object of \(\mathcal{F}\) over \(k[\epsilon]\) and let \(\alpha : M \to V\) be a \(\Gamma\)-equivariant map inducing an isomorphism \(M/\epsilon M \to V\). Since \(M\) is free as a \(k[\epsilon]\)-module we obtain an extension of \(\Gamma\)-modules \[0 \to V \to M \xrightarrow{\alpha} V \to 0\] We omit the detailed construction of the map on the left. Conversely, if we have an extension of \(\Gamma\)-modules as above, then we can use this to make a \(k[\epsilon]\)-module structure on \(M\) and get an object of \(\mathcal{F}(k[\epsilon])\) together with a map \(\alpha\) as above. It follows that \[T\Deformationcategory_{V, \rho_0} = \Ext^1_{k[\Gamma]}(V, V)\] as predicted in the lemma. This is equal to \(H^1(\Gamma, \text{End}_k(V))\) by Étale Cohomology, Lemma 0DVE.

The statement on dimensions follows from Étale Cohomology, Lemma 0DVF.

In Example 0D3J if \(\Gamma\) is finitely generated and \((V, \rho_0)\) is a finite dimensional representation of \(\Gamma\) over \(k\), then \(\Deformationcategory_{V, \rho_0}\) 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 0DVS and 0DVT and the general discussion in Section 0DVM.

Lemma

In Example 0D3J assume \(\Gamma\) finitely generated. Let \(\rho_0 : \Gamma \to \text{GL}_k(V)\) be a finite dimensional representation. Assume \(\Lambda\) is a complete local ring with residue field \(k\) (the classical case). Then the functor \[F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \Ob(\Deformationcategory_{V, \rho_0}(A))/\cong\] of isomorphism classes of objects has a hull. If \(H^0(\Gamma, \text{End}_k(V)) = k\), then \(F\) is prorepresentable.

Proof

The existence of a hull follows from Lemmas 0DVS and 0DVT and Formal Deformation Theory, Lemma 06J7 and Remark 06IZ.

Assume \(H^0(\Gamma, \text{End}_k(V)) = k\). To see that \(F\) is prorepresentable it suffices to show that \(F\) is a deformation functor, see Formal Deformation Theory, Theorem 06JM. In other words, we have to show \(F\) satisfies (RS). For this we can use the criterion of Formal Deformation Theory, Lemma 06J8. The required surjectivity of automorphism groups will follow if we show that \[A \cdot \text{id}_M = \text{End}_{A[\Gamma]}(M)\] for any object \((A, M, \rho)\) of \(\mathcal{F}\) such that \(M \otimes_A k\) is isomorphic to \(V\) as a representation of \(\Gamma\). Since the left hand side is contained in the right hand side, it suffices to show \(\text{length}_A \text{End}_{A[\Gamma]}(M) \leq \text{length}_A A\). Choose pairwise distinct ideals \((0) = I_n \subset \ldots \subset I_1 \subset A\) with \(n = \text{length}(A)\). By correspondingly filtering \(M\), we see that it suffices to prove \(\Hom_{A[\Gamma]}(M, I_tM/I_{t + 1}M)\) has length \(1\). Since \(I_tM/I_{t + 1}M \cong M \otimes_A k\) and since any \(A[\Gamma]\)-module map \(M \to M \otimes_A k\) factors uniquely through the quotient map \(M \to M \otimes_A k\) to give an element of \[\text{End}_{A[\Gamma]}(M \otimes_A k) = \text{End}_{k[\Gamma]}(V) = k\] we conclude.

Continuous representations

A very interesting thing one can do is to take an infinite Galois group and study the deformation theory of its representations, see [Mazur-deforming].

Example

Let \(\Gamma\) be a topological group. Let \(\mathcal{F}\) be the category defined as follows

1. an object is a triple \((A, M, \rho)\) consisting of an object \(A\) of \(\mathcal{C}_\Lambda\), a finite projective \(A\)-module \(M\), and a continuous homomorphism \(\rho : \Gamma \to \text{GL}_A(M)\) where \(\text{GL}_A(M)\) is given the discrete topology¹, and

2. a morphism \((f, g) : (B, N, \tau) \to (A, M, \rho)\) consists of a morphism \(f : B \to A\) in \(\mathcal{C}_\Lambda\) together with a map \(g : N \to M\) which is \(f\)-linear and \(\Gamma\)-equivariant and induces an isomorpism \(N \otimes_{B, f} A \cong M\).

The functor \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) sends \((A, M, \rho)\) to \(A\) and \((f, g)\) to \(f\). It is clear that \(p\) is cofibred in groupoids. Given a finite dimensional \(k\)-vector space \(V\) and a continuous representation \(\rho_0 : \Gamma \to \text{GL}_k(V)\), let \(x_0 = (k, V, \rho_0)\) be the corresponding object of \(\mathcal{F}(k)\). We set \[\Deformationcategory_{V, \rho_0} = \mathcal{F}_{x_0}\]

Since every finite projective module over a local ring is finite free (Algebra, Lemma 00NX) we see that \[\begin{matrix} \text{isomorphism classes} \\ \text{of objects of }\mathcal{F}(A) \end{matrix} = \coprod\nolimits_{n \geq 0}\quad \begin{matrix} \text{GL}_n(A)\text{-conjugacy classes of}\\ \text{continuous homomorphisms }\rho : \Gamma \to \text{GL}_n(A) \end{matrix}\]

Lemma

Example 0D3K satisfies the Rim-Schlessinger condition (RS). In particular, \(\Deformationcategory_{V, \rho_0}\) is a deformation category for any finite dimensional continuous representation \(\rho_0 : \Gamma \to \text{GL}_k(V)\).

Proof

The proof is exactly the same as the proof of Lemma 0DVS.

Lemma

In Example 0D3K let \(\rho_0 : \Gamma \to \text{GL}_k(V)\) be a finite dimensional continuous representation. Then \[T\Deformationcategory_{V, \rho_0} = H^1(\Gamma, \text{End}_k(V)) \quad\text{and}\quad \text{Inf}(\Deformationcategory_{V, \rho_0}) = H^0(\Gamma, \text{End}_k(V))\] Thus \(\text{Inf}(\Deformationcategory_{V, \rho_0})\) is always finite dimensional and \(T\Deformationcategory_{V, \rho_0}\) is finite dimensional if \(\Gamma\) is topologically finitely generated.

Proof

The proof is exactly the same as the proof of Lemma 0DVT.

In Example 0D3K if \(\Gamma\) is topologically finitely generated and \((V, \rho_0)\) is a finite dimensional continuous representation of \(\Gamma\) over \(k\), then \(\Deformationcategory_{V, \rho_0}\) 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 0DVV and 0DVW and the general discussion in Section 0DVM.

Lemma

In Example 0D3K assume \(\Gamma\) is topologically finitely generated. Let \(\rho_0 : \Gamma \to \text{GL}_k(V)\) be a finite dimensional representation. Assume \(\Lambda\) is a complete local ring with residue field \(k\) (the classical case). Then the functor \[F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \Ob(\Deformationcategory_{V, \rho_0}(A))/\cong\] of isomorphism classes of objects has a hull. If \(H^0(\Gamma, \text{End}_k(V)) = k\), then \(F\) is prorepresentable.

Proof

The proof is exactly the same as the proof of Lemma 0ET1.

Graded algebras

We will use the example in this section in the proof that the stack of polarized proper schemes is an algebraic stack. For this reason we will consider commutative graded algebras whose homogeneous parts are finite projective modules (sometimes called “locally finite”).

Example

Let \(\mathcal{F}\) be the category defined as follows

1. an object is a pair \((A, P)\) consisting of an object \(A\) of \(\mathcal{C}_\Lambda\) and a graded \(A\)-algebra \(P\) such that \(P_d\) is a finite projective \(A\)-module for all \(d \geq 0\), and

2. a morphism \((f, g) : (B, Q) \to (A, P)\) consists of a morphism \(f : B \to A\) in \(\mathcal{C}_\Lambda\) together with a map \(g : Q \to P\) which is \(f\)-linear and induces an isomorpism \(Q \otimes_{B, f} A \cong P\).

The functor \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) sends \((A, P)\) to \(A\) and \((f, g)\) to \(f\). It is clear that \(p\) is cofibred in groupoids. Given a graded \(k\)-algebra \(P\) with \(\dim_k(P_d) < \infty\) for all \(d \geq 0\), let \(x_0 = (k, P)\) be the corresponding object of \(\mathcal{F}(k)\). We set \[\Deformationcategory_P = \mathcal{F}_{x_0}\]

Lemma

Example 0D3L satisfies the Rim-Schlessinger condition (RS). In particular, \(\Deformationcategory_P\) is a deformation category for any graded \(k\)-algebra \(P\).

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 06J5 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.

Consider an object \[((A_1, P_1), (A_2, P_2), (\text{id}_A, \varphi))\] of the category \(\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)\). Then we consider \(P_1 \times_\varphi P_2\). Since \(\varphi : P_1 \otimes_{A_1} A \to P_2 \otimes_{A_2} A\) is an isomorphism of graded algebras, we see that the graded pieces of \(P_1 \times_\varphi P_2\) are finite projective \(A_1 \times_A A_2\)-modules, see proof of Lemma 0DVP. Thus \(P_1 \times_\varphi P_2\) is an object of \(\mathcal{F}(A_1 \times_A A_2)\). This construction determines a quasi-inverse to our functor and the proof is complete.

Lemma

In Example 0D3L let \(P\) be a graded \(k\)-algebra. Then \[T\Deformationcategory_P \quad\text{and}\quad \text{Inf}(\Deformationcategory_P) = \text{Der}_k(P, P)\] are finite dimensional if \(P\) is finitely generated over \(k\).

Proof

We first deal with the infinitesimal automorphisms. Let \(Q = P \otimes_k k[\epsilon]\). Then an element of \(\text{Inf}(\Deformationcategory_P)\) is given by an automorphism \(\gamma = \text{id} + \epsilon \delta : Q \to Q\) as above where now \(\delta : P \to P\). The fact that \(\gamma\) is graded implies that \(\delta\) is homogeneous of degree \(0\). The fact that \(\gamma\) is \(k\)-linear implies that \(\delta\) is \(k\)-linear. The fact that \(\gamma\) is multiplicative implies that \(\delta\) is a \(k\)-derivation. Conversely, given a \(k\)-derivation \(\delta : P \to P\) homogeneous of degree \(0\), we obtain an automorphism \(\gamma = \text{id} + \epsilon \delta\) as above. Thus we see that \[\text{Inf}(\Deformationcategory_P) = \text{Der}_k(P, P)\] as predicted in the lemma. Clearly, if \(P\) is generated in degrees \(P_i\), \(0 \leq i \leq N\), then \(\delta\) is determined by the linear maps \(\delta_i : P_i \to P_i\) for \(0 \leq i \leq N\) and we see that \[\dim_k \text{Der}_k(P, P) < \infty\] as desired.

To finish the proof of the lemma we show that there is a finite dimensional deformation space. To do this we choose a presentation \[k[X_1, \ldots, X_n]/(F_1, \ldots, F_m) \longrightarrow P\] of graded \(k\)-algebras where \(\deg(X_i) = d_i\) and \(F_j\) is homogeneous of degree \(e_j\). Let \(Q\) be any graded \(k[\epsilon]\)-algebra finite free in each degree which comes with an isomorphsm \(\alpha : Q/\epsilon Q \to P\) so that \((Q, \alpha)\) defines an element of \(T\Deformationcategory_P\). Choose a homogeneous element \(q_i \in Q\) of degree \(d_i\) mapping to the image of \(X_i\) in \(P\). Then we obtain \[k[\epsilon][X_1, \ldots, X_n] \longrightarrow Q,\quad X_i \longmapsto q_i\] and since \(P = Q/\epsilon Q\) this map is surjective by Nakayama’s lemma. A small diagram chase shows we can choose homogeneous elements \(F_{\epsilon, j} \in k[\epsilon][X_1, \ldots, X_n]\) of degree \(e_j\) mapping to zero in \(Q\) and mapping to \(F_j\) in \(k[X_1, \ldots, X_n]\). Then \[k[\epsilon][X_1, \ldots, X_n]/(F_{\epsilon, 1}, \ldots, F_{\epsilon, m}) \longrightarrow Q\] is a presentation of \(Q\) by flatness of \(Q\) over \(k[\epsilon]\). Write \[F_{\epsilon, j} = F_j + \epsilon G_j\] There is some ambiguity in the vector \((G_1, \ldots, G_m)\). First, using different choices of \(F_{\epsilon, j}\) we can modify \(G_j\) by an arbitrary element of degree \(e_j\) in the kernel of \(k[X_1, \ldots, X_n] \to P\). Hence, instead of \((G_1, \ldots, G_m)\), we remember the element \[(g_1, \ldots, g_m) \in P_{e_1} \oplus \ldots \oplus P_{e_m}\] where \(g_j\) is the image of \(G_j\) in \(P_{e_j}\). Moreover, if we change our choice of \(q_i\) into \(q_i + \epsilon p_i\) with \(p_i\) of degree \(d_i\) then a computation (omitted) shows that \(g_j\) changes into \[g_j^{new} = g_j - \sum\nolimits_{i = 1}^n p_i \partial F_j / \partial X_i\] We conclude that the isomorphism class of \(Q\) is determined by the image of the vector \((G_1, \ldots, G_m)\) in the \(k\)-vector space \[W = \Coker(P_{d_1} \oplus \ldots \oplus P_{d_n} \xrightarrow{(\frac{\partial F_j}{\partial X_i})} P_{e_1} \oplus \ldots \oplus P_{e_m})\] In this way we see that we obtain an injection \[T\Deformationcategory_P \longrightarrow W\] Since \(W\) visibly has finite dimension, we conclude that the lemma is true.

In Example 0D3L if \(P\) is a finitely generated graded \(k\)-algebra, then \(\Deformationcategory_P\) 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 0DVY and 0DVZ and the general discussion in Section 0DVM.

Lemma

In Example 0D3L assume \(P\) is a finitely generated graded \(k\)-algebra. Assume \(\Lambda\) is a complete local ring with residue field \(k\) (the classical case). Then the functor \[F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \Ob(\Deformationcategory_P(A))/\cong\] of isomorphism classes of objects has a hull.

Proof

This follows immediately from Lemmas 0DVY and 0DVZ and Formal Deformation Theory, Lemma 06J7 and Remark 06IZ.

Rings

The deformation theory of rings is the same as the deformation theory of affine schemes. For rings and schemes when we talk about deformations it means we are thinking about flat deformations.

Example

Let \(\mathcal{F}\) be the category defined as follows

1. an object is a pair \((A, P)\) consisting of an object \(A\) of \(\mathcal{C}_\Lambda\) and a flat \(A\)-algebra \(P\), and

2. a morphism \((f, g) : (B, Q) \to (A, P)\) consists of a morphism \(f : B \to A\) in \(\mathcal{C}_\Lambda\) together with a map \(g : Q \to P\) which is \(f\)-linear and induces an isomorpism \(Q \otimes_{B, f} A \cong P\).

The functor \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) sends \((A, P)\) to \(A\) and \((f, g)\) to \(f\). It is clear that \(p\) is cofibred in groupoids. Given a \(k\)-algebra \(P\), let \(x_0 = (k, P)\) be the corresponding object of \(\mathcal{F}(k)\). We set \[\Deformationcategory_P = \mathcal{F}_{x_0}\]

Lemma

Example 0DY1 satisfies the Rim-Schlessinger condition (RS). In particular, \(\Deformationcategory_P\) is a deformation category for any \(k\)-algebra \(P\).

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 06J5 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. This is a special case of More on Algebra, Lemma 08KQ.

Lemma

In Example 0DY1 let \(P\) be a \(k\)-algebra. Then \[T\Deformationcategory_P = \text{Ext}^1_P(\NL_{P/k}, P) \quad\text{and}\quad \text{Inf}(\Deformationcategory_P) = \text{Der}_k(P, P)\]

Proof

Recall that \(\text{Inf}(\Deformationcategory_P)\) is the set of automorphisms of the trivial deformation \(P[\epsilon] = P \otimes_k k[\epsilon]\) of \(P\) to \(k[\epsilon]\) equal to the identity modulo \(\epsilon\). By Deformation Theory, Lemma 08S5 this is equal to \(\Hom_P(\Omega_{P/k}, P)\) which in turn is equal to \(\text{Der}_k(P, P)\) by Algebra, Lemma 00RO.

Recall that \(T\Deformationcategory_P\) is the set of isomorphism classes of flat deformations \(Q\) of \(P\) to \(k[\epsilon]\), more precisely, the set of isomorphism classes of \(\Deformationcategory_P(k[\epsilon])\). Recall that a \(k[\epsilon]\)-algebra \(Q\) with \(Q/\epsilon Q = P\) is flat over \(k[\epsilon]\) if and only if \[0 \to P \xrightarrow{\epsilon} Q \to P \to 0\] is exact. This is proven in More on Morphisms, Lemma 063Y and more generally in Deformation Theory, Lemma 08LI. Thus we may apply Deformation Theory, Lemma 08S7 to see that the set of isomorphism classes of such deformations is equal to \(\text{Ext}^1_P(\NL_{P/k}, P)\).

Lemma

In Example 0DY1 let \(P\) be a smooth \(k\)-algebra. Then \(T\Deformationcategory_P = (0)\).

Proof

By Lemma 0DY3 we have to show \(\text{Ext}^1_P(\NL_{P/k}, P) = (0)\). Since \(k \to P\) is smooth \(\NL_{P/k}\) is quasi-isomorphic to the complex consisting of a finite projective \(P\)-module placed in degree \(0\).

Lemma

In Lemma 0DY3 if \(P\) is a finite type \(k\)-algebra, then

1. \(\text{Inf}(\Deformationcategory_P)\) is finite dimensional if and only if \(\dim(P) = 0\), and

2. \(T\Deformationcategory_P\) is finite dimensional if \(\Spec(P) \to \Spec(k)\) is smooth except at a finite number of points.

Proof

Proof of (1). We view \(\text{Der}_k(P, P)\) as a \(P\)-module. If it has finite dimension over \(k\), then it has finite length as a \(P\)-module, hence it is supported in finitely many closed points of \(\Spec(P)\) (Algebra, Lemma 00J3). Since \(\text{Der}_k(P, P) = \Hom_P(\Omega_{P/k}, P)\) we see that \(\text{Der}_k(P, P)_\mathfrak p = \text{Der}_k(P_\mathfrak p, P_\mathfrak p)\) for any prime \(\mathfrak p \subset P\) (this uses Algebra, Lemmas 00RT, 00RY, and 0583). Let \(\mathfrak p\) be a minimal prime ideal of \(P\) corresponding to an irreducible component of dimension \(d > 0\). Then \(P_\mathfrak p\) is an Artinian local ring essentially of finite type over \(k\) with residue field and \(\Omega_{P_\mathfrak p/k}\) is nonzero for example by Algebra, Lemma 00TT. Any nonzero finite module over an Artinian local ring has both a sub and a quotient module isomorphic to the residue field. Thus we find that \(\text{Der}_k(P_\mathfrak p, P_\mathfrak p) = \Hom_{P_\mathfrak p}(\Omega_{P_\mathfrak p/k}, P_\mathfrak p)\) is nonzero too. Combining all of the above we find that (1) is true.

Proof of (2). For a prime \(\mathfrak p\) of \(P\) we will use that \(\NL_{P_\mathfrak p/k} = (\NL_{P/k})_\mathfrak p\) (Algebra, Lemma 00S7) and we will use that \(\text{Ext}_P^1(\NL_{P/k}, P)_\mathfrak p = \text{Ext}_{P_\mathfrak p}^1(\NL_{P_\mathfrak p/k}, P_\mathfrak p)\) (More on Algebra, Lemma 087R). Given a prime \(\mathfrak p \subset P\) then \(k \to P\) is smooth at \(\mathfrak p\) if and only if \((\NL_{P/k})_\mathfrak p\) is quasi-isomorphic to a finite projective module placed in degree \(0\) (this follows immediately from the definition of a smooth ring map but it also follows from the stronger Algebra, Lemma 07BU).

Assume that \(P\) is smooth over \(k\) at all but finitely many primes. Then these “bad” primes are maximal ideals \(\mathfrak m_1, \ldots, \mathfrak m_n \subset P\) by Algebra, Lemma 0ALW and the fact that the “bad” primes form a closed subset of \(\Spec(P)\). For \(\mathfrak p \not \in \{\mathfrak m_1, \ldots, \mathfrak m_n\}\) we have \(\text{Ext}^1_P(\NL_{P/k}, P)_\mathfrak p = 0\) by the results above. Thus \(\text{Ext}^1_P(\NL_{P/k}, P)\) is a finite \(P\)-module whose support is contained in \(\{\mathfrak m_1, \ldots, \mathfrak m_r\}\). By Algebra, Proposition 02CE for example, we find that the dimension over \(k\) of \(\text{Ext}^1_P(\NL_{P/k}, P)\) is a finite integer combination of \(\dim_k \kappa(\mathfrak m_i)\) and hence finite by the Hilbert Nullstellensatz (Algebra, Theorem 00FV).

In Example 0DY1, let \(P\) be a finite type \(k\)-algebra. Then \(\Deformationcategory_P\) admits a presentation by a smooth prorepresentable groupoid in functors over \(\mathcal{C}_\Lambda\) if and only if \(\dim(P) = 0\). Furthermore, \(\Deformationcategory_P\) has a versal formal object if \(\Spec(P) \to \Spec(k)\) has finitely many singular points. This follows from Lemmas 0DY2 and 0DY4 and the general discussion in Section 0DVM.

Lemma

In Example 0DY1 assume \(P\) is a finite type \(k\)-algebra such that \(\Spec(P) \to \Spec(k)\) is smooth except at a finite number of points. Assume \(\Lambda\) is a complete local ring with residue field \(k\) (the classical case). Then the functor \[F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \Ob(\Deformationcategory_P(A))/\cong\] of isomorphism classes of objects has a hull.

Proof

This follows immediately from Lemmas 0DY2 and 0DY4 and Formal Deformation Theory, Lemma 06J7 and Remark 06IZ.

Lemma

In Example 0DY1 let \(P\) be a \(k\)-algebra. Let \(S \subset P\) be a multiplicative subset. There is a natural functor \[\Deformationcategory_P \longrightarrow \Deformationcategory_{S^{-1}P}\] of deformation categories.

Proof

Given a deformation of \(P\) we can take the localization of it to get a deformation of the localization; this is clear and we encourage the reader to skip the proof. More precisely, let \((A, Q) \to (k, P)\) be a morphism in \(\mathcal{F}\), i.e., an object of \(\Deformationcategory_P\). Let \(S_Q \subset Q\) be the inverse image of \(S\). Then Hence \((A, S_Q^{-1}Q) \to (k, S^{-1}P)\) is the desired object of \(\Deformationcategory_{S^{-1}P}\).

Lemma

In Example 0DY1 let \(P\) be a \(k\)-algebra. Let \(J \subset P\) be an ideal. Denote \((P^h, J^h)\) the henselization of the pair \((P, J)\). There is a natural functor \[\Deformationcategory_P \longrightarrow \Deformationcategory_{P^h}\] of deformation categories.

Proof

Given a deformation of \(P\) we can take the henselization of it to get a deformation of the henselization; this is clear and we encourage the reader to skip the proof. More precisely, let \((A, Q) \to (k, P)\) be a morphism in \(\mathcal{F}\), i.e., an object of \(\Deformationcategory_P\). Denote \(J_Q \subset Q\) the inverse image of \(J\) in \(Q\). Let \((Q^h, J_Q^h)\) be the henselization of the pair \((Q, J_Q)\). Recall that \(Q \to Q^h\) is flat (More on Algebra, Lemma 0AGU) and hence \(Q^h\) is flat over \(A\). By More on Algebra, Lemma 0DYE we see that the map \(Q^h \to P^h\) induces an isomorphism \(Q^h \otimes_A k = Q^h \otimes_Q P = P^h\). Hence \((A, Q^h) \to (k, P^h)\) is the desired object of \(\Deformationcategory_{P^h}\).

Lemma

In Example 0DY1 let \(P\) be a \(k\)-algebra. Assume \(P\) is a local ring and let \(P^{sh}\) be a strict henselization of \(P\). There is a natural functor \[\Deformationcategory_P \longrightarrow \Deformationcategory_{P^{sh}}\] of deformation categories.

Proof

Given a deformation of \(P\) we can take the strict henselization of it to get a deformation of the strict henselization; this is clear and we encourage the reader to skip the proof. More precisely, let \((A, Q) \to (k, P)\) be a morphism in \(\mathcal{F}\), i.e., an object of \(\Deformationcategory_P\). Since the kernel of the surjection \(Q \to P\) is nilpotent, we find that \(Q\) is a local ring with the same residue field as \(P\). Let \(Q^{sh}\) be the strict henselization of \(Q\). Recall that \(Q \to Q^{sh}\) is flat (More on Algebra, Lemma 07QM) and hence \(Q^{sh}\) is flat over \(A\). By Algebra, Lemma 05WS we see that the map \(Q^{sh} \to P^{sh}\) induces an isomorphism \(Q^{sh} \otimes_A k = Q^{sh} \otimes_Q P = P^{sh}\). Hence \((A, Q^{sh}) \to (k, P^{sh})\) is the desired object of \(\Deformationcategory_{P^{sh}}\).

Lemma

In Example 0DY1 let \(P\) be a \(k\)-algebra. Assume \(P\) Noetherian and let \(J \subset P\) be an ideal. Denote \(P^\wedge\) the \(J\)-adic completion. There is a natural functor \[\Deformationcategory_P \longrightarrow \Deformationcategory_{P^\wedge}\] of deformation categories.

Proof

Given a deformation of \(P\) we can take the completion of it to get a deformation of the completion; this is clear and we encourage the reader to skip the proof. More precisely, let \((A, Q) \to (k, P)\) be a morphism in \(\mathcal{F}\), i.e., an object of \(\Deformationcategory_P\). Observe that \(Q\) is a Noetherian ring: the kernel of the surjective ring map \(Q \to P\) is nilpotent and finitely generated and \(P\) is Noetherian; apply Algebra, Lemma 05GH. Denote \(J_Q \subset Q\) the inverse image of \(J\) in \(Q\). Let \(Q^\wedge\) be the \(J_Q\)-adic completion of \(Q\). Recall that \(Q \to Q^\wedge\) is flat (Algebra, Lemma 00MB) and hence \(Q^\wedge\) is flat over \(A\). The induced map \(Q^\wedge \to P^\wedge\) induces an isomorphism \(Q^\wedge \otimes_A k = Q^\wedge \otimes_Q P = P^\wedge\) by Algebra, Lemma 00MA for example. Hence \((A, Q^\wedge) \to (k, P^\wedge)\) is the desired object of \(\Deformationcategory_{P^\wedge}\).

Lemma

In Lemma 0DY3 if \(P = k[[x_1, \ldots, x_n]]/(f)\) for some nonzero \(f \in (x_1, \ldots, x_n)^2\), then

1. \(\text{Inf}(\Deformationcategory_P)\) is finite dimensional if and only if \(n = 1\), and

2. \(T\Deformationcategory_P\) is finite dimensional if \[\sqrt{(f, \partial f/\partial x_1, \ldots, \partial f/\partial x_n)} = (x_1, \ldots, x_n)\]

Proof

Proof of (1). Consider the derivations \(\partial/\partial x_i\) of \(k[[x_1, \ldots, x_n]]\) over \(k\). Write \(f_i = \partial f/\partial x_i\). The derivation \[\theta = \sum h_i \partial/\partial x_i\] of \(k[[x_1, \ldots, x_n]]\) induces a derivation of \(P = k[[x_1, \ldots, x_n]]/(f)\) if and only if \(\sum h_i f_i \in (f)\). Moreover, the induced derivation of \(P\) is zero if and only if \(h_i \in (f)\) for \(i = 1, \ldots, n\). Thus we find \[\Ker((f_1, \ldots, f_n) : P^{\oplus n} \longrightarrow P) \subset \text{Der}_k(P, P)\] The left hand side is a finite dimensional \(k\)-vector space only if \(n = 1\); we omit the proof. We also leave it to the reader to see that the right hand side has finite dimension if \(n = 1\). This proves (1).

Proof of (2). Let \(Q\) be a flat deformation of \(P\) over \(k[\epsilon]\) as in the proof of Lemma 0DY3. Choose lifts \(q_i \in Q\) of the image of \(x_i\) in \(P\). Then \(Q\) is a complete local ring with maximal ideal generated by \(q_1, \ldots, q_n\) and \(\epsilon\) (small argument omitted). Thus we get a surjection \[k[\epsilon][[x_1, \ldots, x_n]] \longrightarrow Q,\quad x_i \longmapsto q_i\] Choose an element of the form \(f + \epsilon g \in k[\epsilon][[x_1, \ldots, x_n]]\) mapping to zero in \(Q\). Observe that \(g\) is well defined modulo \((f)\). Since \(Q\) is flat over \(k[\epsilon]\) we get \[Q = k[\epsilon][[x_1, \ldots, x_n]]/(f + \epsilon g)\] Finally, if we changing the choice of \(q_i\) amounts to changing the coordinates \(x_i\) into \(x_i + \epsilon h_i\) for some \(h_i \in k[[x_1, \ldots, x_n]]\). Then \(f + \epsilon g\) changes into \(f + \epsilon (g + \sum h_i f_i)\) where \(f_i = \partial f/\partial x_i\). Thus we see that the isomorphism class of the deformation \(Q\) is determined by an element of \[k[[x_1, \ldots, x_n]]/ (f, \partial f/\partial x_1, \ldots, \partial f/\partial x_n)\] This has finite dimension over \(k\) if and only if its support is the closed point of \(k[[x_1, \ldots, x_n]]\) if and only if \(\sqrt{(f, \partial f/\partial x_1, \ldots, \partial f/\partial x_n)} = (x_1, \ldots, x_n)\).

Schemes

The deformation theory of schemes.

Example

Let \(\mathcal{F}\) be the category defined as follows

1. an object is a pair \((A, X)\) consisting of an object \(A\) of \(\mathcal{C}_\Lambda\) and a scheme \(X\) flat over \(A\), and

2. 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 \[\xymatrix{ X \ar[r]_g \ar[d] & Y \ar[d] \\ \Spec(A) \ar[r]^f & \Spec(B) }\] 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 \[\Deformationcategory_X = \mathcal{F}_{x_0}\]

Lemma

Example 0DY7 satisfies the Rim-Schlessinger condition (RS). In particular, \(\Deformationcategory_X\) 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 06J5 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 \[\xymatrix{ \Spec(A) \ar[r] \ar[d] & \Spec(A_2) \ar[d] \\ \Spec(A_1) \ar[r] & \Spec(A_1 \times_A A_2) }\] is a pushout diagram as in More on Morphisms, Lemma 07RT. Thus the lemma is a special case of More on Morphisms, Lemma 07RX.

Lemma

In Example 0DY7 let \(X\) be a scheme over \(k\). Then \[\text{Inf}(\Deformationcategory_X) = \text{Ext}^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X) = \Hom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X) = \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X)\] and \[T\Deformationcategory_X = \text{Ext}^1_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X)\]

Proof

Recall that \(\text{Inf}(\Deformationcategory_X)\) is the set of automorphisms of the trivial deformation \(X' = X \times_{\Spec(k)} \Spec(k[\epsilon])\) of \(X\) to \(k[\epsilon]\) equal to the identity modulo \(\epsilon\). By Deformation Theory, Lemma 0D14 this is equal to \(\text{Ext}^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X)\). The equality \(\text{Ext}^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X) = \Hom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X)\) follows from More on Morphisms, Lemma 0D0J. The equality \(\Hom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X) = \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X)\) follows from Morphisms, Lemma 01UR.

Recall that \(T_{x_0}\Deformationcategory_X\) is the set of isomorphism classes of flat deformations \(X'\) of \(X\) to \(k[\epsilon]\), more precisely, the set of isomorphism classes of \(\Deformationcategory_X(k[\epsilon])\). Thus the second statement of the lemma follows from Deformation Theory, Lemma 0D14.

Lemma

In Lemma 0DY9 if \(X\) is proper over \(k\), then \(\text{Inf}(\Deformationcategory_X)\) and \(T\Deformationcategory_X\) are finite dimensional.

Proof

By the lemma we have to show \(\Ext^1_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X)\) and \(\Ext^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X)\) are finite dimensional. By More on Morphisms, Lemma 0D0K and the fact that \(X\) is Noetherian, we see that \(\NL_{X/k}\) has coherent cohomology sheaves zero except in degrees \(0\) and \(-1\). By Derived Categories of Schemes, Lemma 0D0D the displayed \(\Ext\)-groups are finite \(k\)-vector spaces and the proof is complete.

In Example 0DY7 if \(X\) is a proper scheme over \(k\), then \(\Deformationcategory_X\) 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 0DY8 and 0DYA and the general discussion in Section 0DVM.

Lemma

In Example 0DY7 assume \(X\) is a proper \(k\)-scheme. Assume \(\Lambda\) is a complete local ring with residue field \(k\) (the classical case). Then the functor \[F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \Ob(\Deformationcategory_X(A))/\cong\] 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 0DY8 and 0DYA and Formal Deformation Theory, Lemma 06J7 and Remark 06IZ.

Assume \(\text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) = 0\). Then \(\Deformationcategory_X\) and \(F\) are equivalent by Formal Deformation Theory, Lemma 06K0. Hence \(F\) is a deformation functor (because \(\Deformationcategory_X\) is a deformation category) with finite tangent space and we can apply Formal Deformation Theory, Theorem 06JM.

Lemma

Let \(\Lambda\) be a Noetherian local ring with residue field \(k\), and let \(X\) be a scheme over \(k\). For a local object \(A\) of \(\mathcal{A}_\Lambda\) (Formal Deformation Theory, Definition formal-defos-definition-artinian-algebra-functor-cech), with residue field \(\kappa(A)\), let \(F(A)\) be the set of isomorphism classes of pairs \((X_A, \alpha_A)\), where \(X_A\) is a flat \(A\)-scheme and \[\alpha_A : X_A \times_{\Spec(A)} \Spec(\kappa(A)) \longrightarrow X \times_{\Spec(k)} \Spec(\kappa(A))\] is an isomorphism. Isomorphisms of pairs are required to respect \(\alpha_A\). For arbitrary \(A\), define \(F(A)\) as the product of these sets over the local factors of \(A\). If \[\text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) = 0,\] then \(F : \mathcal{A}_\Lambda \to \textit{Sets}\) is strictly prorepresentable. If \(X\) is proper over \(k\), then the corresponding local prorepresenting algebra is Noetherian.

If \(X\) is smooth over \(k\), the tangent space at the marked point is canonically \[H^1(X, \mathcal{T}_{X/k}), \qquad \mathcal{T}_{X/k} = \SheafHom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X).\]

Proof

We first eliminate automorphisms, including after extension of the residue field. Let \(l/k\) be a finite field extension. Finite base change gives \[\text{Der}_l(\mathcal{O}_{X_l}, \mathcal{O}_{X_l}) = \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) \otimes_k l = 0.\] Indeed, after pushing forward along the finite morphism \(X_l \to X\), this is the fact that sheaf homomorphisms into a finite direct sum commute with that direct sum. Lemma 0DY9 identifies this group with the infinitesimal automorphisms of \(X_l\). Formal Deformation Theory, Lemma 06JZ then shows that every automorphism of a marked deformation over a local Artinian ring with residue field \(l\) is trivial.

We verify the two local equalizer tests of Formal Deformation Theory, Theorem formal-defos-theorem-grothendieck-levelt-prorepresentability. Let \(A \to B\) be one of the maps occurring in those tests, with \(A\) local. If a class over \(B\) has equal pullbacks to \(B \otimes_A B\), the equality supplies an isomorphism between the two pullbacks. It is unique by the preceding paragraph. Consequently its three pullbacks to \(B \otimes_A B \otimes_A B\) satisfy the cocycle condition automatically.

Suppose first that \(B\) is a nonzero finite free \(A\)-module. The morphism \(\Spec(B) \to \Spec(A)\) is an fppf covering. Fppf descent data for algebraic spaces are effective by Bootstrap, Lemma 0ADV. Thus the marked flat \(B\)-scheme descends to an algebraic space \(Y\) over \(A\). Flatness descends by Descent on Spaces, Lemma 041W, and the marking descends to an isomorphism \(Y_{\kappa(A)} \cong X_{\kappa(A)}\). Since \(Y_{\kappa(A)} \subset Y\) is a thickening, \(Y\) is a scheme by More on Morphisms of Spaces, Lemma 05ZR. Full faithfulness in fppf descent gives an equivalence between marked flat \(A\)-schemes and the descent data under consideration.

Suppose next that \(A \to B\) is injective and \(B/A\) has length \(1\). Put \(\mathfrak m = \mathfrak m_A\). The proof of Formal Deformation Theory, Lemma formal-defos-lemma-length-one-cech-equalizer gives \[\mathfrak mB = \mathfrak m\] and the required ring equalizer. Hence Descent, Lemma descent-lemma-artinian-small-effective-descent-flat-modules applies. Choose an affine basis of \(X\) together with all of its principal affine refinements. The inverse image of any member of this basis in a marked deformation over \(B\) is affine: it is a nilpotent thickening of the corresponding affine open after base change to \(B/\mathfrak m\). Its coordinate algebra is flat over \(B\). The cited descent lemma descends the underlying module to a flat \(A\)-module. Multiplication, the unit, and all restriction maps descend uniquely by the full faithfulness part of that lemma, and their identities can be checked after base change to \(B\).

For completeness, these descended affine pieces have the required open overlaps. If \(V = D(f) \subset U\) is one of the chosen principal refinements, lift \(f\) to the descended coordinate algebra of \(U\). After base change to \(B\), localization at this lift is the coordinate algebra of the inverse image of \(V\). Full faithfulness identifies its descent with the localization of the descended algebra of \(U\). The affine pieces therefore glue to a flat \(A\)-scheme marked by \(X\) on its special fibre. The same argument applied to morphisms proves full faithfulness. Thus marked flat schemes over \(A\) are again equivalent to the relevant descent data over \(B\).

It follows in both cases that \[F(A) \longrightarrow F(B) \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} F(B \otimes_A B)\] is an equalizer. The functor \(F\) commutes with finite products by definition, so the Grothendieck–Levelt theorem cited above proves strict prorepresentability. Moreover, \(F(K)\) is a singleton for every field \(K \in \mathcal{A}_\Lambda\). Thus the strict representing system has a unique component through its \(k\)-point, and this component may be taken to consist of local Artinian algebras with residue field \(k\).

The tangent space at the marked \(k\)-point is the tangent space of \(\Deformationcategory_X\), because the two functors are equivalent after the automorphisms have been eliminated. Lemma 0DY9 identifies it with \[\text{Ext}^1_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X).\] If \(X\) is smooth, the naive cotangent complex is represented by the finite locally free module \(\Omega_{X/k}\) in degree zero, and this Ext group is \(H^1(X, \mathcal{T}_{X/k})\).

Finally assume \(X\) is proper. Lemma 0DYA says that the tangent space is finite dimensional. In a local strict prorepresenting presentation, this bounds the relative cotangent spaces of all the Artinian quotients. Since \(\Lambda\) is Noetherian, its cotangent space is finite dimensional as well, and hence the absolute cotangent spaces of the quotients are bounded. The inverse limit is therefore Noetherian by the bounded-cotangent-space argument of Formal Deformation Theory, Lemma 06SF.

Lemma

Let \(\Lambda\) be a complete Noetherian local ring with residue field \(k\), and let \(X\) be an abelian variety of dimension \(g\) over \(k\). For a local object \(A\) of \(\mathcal{A}_\Lambda\), let \(F(A)\) be the set of isomorphism classes of abelian schemes \(X_A\) over \(A\) endowed with an isomorphism \[X_A \times_{\Spec(A)} \Spec(\kappa(A)) \longrightarrow X \times_{\Spec(k)} \Spec(\kappa(A))\] of abelian schemes. For arbitrary \(A\), define \(F(A)\) as the product over the local factors. Then \(F\) is strictly prorepresentable. The local prorepresenting algebra at the marked \(k\)-point is isomorphic to \[\Lambda[[t_{ij} \mid 1 \leq i, j \leq g]].\] In particular, the deformation functor of \(X\) as an abelian scheme is formally smooth over \(\Lambda\) and has tangent space of dimension \(g^2\). Moreover, forgetting the group law identifies its restriction to \(\mathcal{C}_\Lambda\) with the functor of isomorphism classes of marked deformations of the underlying scheme \(X\).

Proof

We first discuss prorepresentability. We use the rigidity fact that a homomorphism between abelian schemes over a local Artinian ring which is zero on the closed fibre is zero. Consequently an automorphism of a marked abelian scheme which is the identity on the closed fibre is the identity. This is also the argument which shows that forgetting the group law gives the final bijection in the statement; see [Oort-local-moduli, Proposition 2.2.6]. Indeed, a section lifting the origin exists by smoothness, the rigidity lemma supplies a unique group law with that section as origin, and changing the lift of the origin translates the resulting group scheme.

We may now use the two equalizer tests in Formal Deformation Theory, Theorem formal-defos-theorem-grothendieck-levelt-prorepresentability. The proof is the same descent argument as in Lemma examples-defos-lemma-schemes-formal-moduli-no-infinitesimal-automorphisms. For a nonzero finite free map \(A \to B\), fppf descent gives the underlying marked scheme over \(A\); multiplication, inverse, and the identity section descend by full faithfulness. For an injective map \(A \to B\) with \(B/A\) of length \(1\), the affine-basis argument and effective descent for flat modules used there descend the underlying scheme and the three structure morphisms. The group identities may be checked after base change to \(B\). Properness, finite presentation, and smoothness follow from the corresponding properties of the closed fibre. Rigidity makes the descent isomorphism unique, and hence makes its cocycle condition automatic. Thus the two diagrams \[F(A) \longrightarrow F(B) \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} F(B \otimes_A B)\] are equalizers. The cited theorem proves strict prorepresentability.

It remains to prove formal smoothness. Let \(B \to A\) be a small extension in \(\mathcal{C}_\Lambda\) with kernel \(I\), and let \(X_A\) be an abelian scheme deforming \(X\). The obstruction to lifting its underlying smooth scheme lies in \[\omega(X_A) \in H^2(X, \mathcal{T}_{X/k}) \otimes_k I\] by Cotangent Complex, Lemma 08UZ. Put \(P_A = X_A \times_A X_A\). The two projections and the Kunneth decomposition (Varieties, Lemma 0BED) give injections \[i_1, i_2 : H^2(X, \mathcal{T}_{X/k}) \otimes_k I \longrightarrow H^2(X \times_k X, \mathcal{T}_{X \times_k X/k}) \otimes_k I.\] The product construction of the obstruction gives \[\omega(P_A) = i_1(\omega(X_A)) + i_2(\omega(X_A)).\] Consider the automorphism \(a : P_A \to P_A\) given by \(a(x, y) = (x + y, y)\). Functoriality of the obstruction under this automorphism gives \[\omega(P_A) = (a_0^*)^{-1}d a_0(\omega(P_A)) = 2i_1(\omega(X_A)) + i_2(\omega(X_A)).\] The two displayed equalities imply \(i_1(\omega(X_A)) = 0\). Since \(i_1\) is injective, \(\omega(X_A) = 0\). This argument, including the product formula, is the characteristic-free obstruction calculation in [Oort-local-moduli, pp. 279–280]. Thus the underlying scheme lifts to \(B\), and the forgetful bijection already proved equips the lift with an abelian-scheme structure. Hence \(F|_{\mathcal{C}_\Lambda}\) is smooth.

Finally, Groupoids, Lemma 047I identifies \(\mathcal{T}_{X/k}\) with the trivial vector bundle having fibre the tangent space of \(X\) at the origin. The classical identification \(\dim_k H^1(X, \mathcal{O}_X) = g\) therefore gives \[\dim_k H^1(X, \mathcal{T}_{X/k}) = g^2.\] This is the tangent space by Lemma 0DY9. Formal Deformation Theory, Lemma 0DYL now identifies the local prorepresenting algebra with the displayed power series ring.

Lemma

Let \(k\) be an algebraically closed field, let \(\Lambda\) be a complete Noetherian local ring with residue field \(k\), and let \(X\) be a smooth proper connected curve of genus \(g\) over \(k\). The functor on \(\mathcal{C}_\Lambda\) of isomorphism classes of flat deformations of \(X\), marked on the closed fibre, is prorepresented by \[\begin{cases} \Lambda & \text{if } g = 0,\\ \Lambda[[t]] & \text{if } g = 1,\\ \Lambda[[t_1, \ldots, t_{3g - 3}]] & \text{if } g \geq 2. \end{cases}\] In every case the formal moduli scheme is smooth over \(\Lambda\).

Proof

If \(g = 0\), then \(X \cong \mathbf{P}^1_k\) by Algebraic Curves, Proposition 0C6U. A deformation over an Artinian local \(\Lambda\)-algebra is proper and flat, and hence is \(\mathbf{P}^1\) over that algebra by Deformation Theory, Lemma defos-lemma-projective-space-rigid-complete-local. An automorphism of \(\mathbf{P}^1_k\) lifts because an invertible \(2\) by \(2\) matrix lifts over a surjection of local rings. Thus the marking does not introduce a second isomorphism class, and the functor is the one-point functor represented by \(\Lambda\).

Suppose \(g = 1\). Choose a point of \(X(k)\) and use it as the origin of the resulting abelian variety. The final assertion of Lemma examples-defos-lemma-abelian-variety-formal-moduli-smooth identifies deformations of this abelian variety with deformations of its underlying curve. The same lemma, with \(g = 1\), gives the ring \(\Lambda[[t]]\).

Suppose \(g \geq 2\). Algebraic Curves, Lemma 0E67 gives \(H^0(X, \mathcal{T}_{X/k}) = 0\). Lemma examples-defos-lemma-schemes-formal-moduli-no-infinitesimal-automorphisms therefore gives a Noetherian prorepresenting algebra \(R\). The deformation category is unobstructed by Lemma 0DZQ; a smooth curve is a local complete intersection and has empty singular locus. Formal Deformation Theory, Lemma 0DZK shows that \(R\) is a power series ring over \(\Lambda\) with one variable for each dimension of \(H^1(X, \mathcal{T}_{X/k})\).

For a smooth curve, \(\Omega_{X/k} \cong \omega_X\), and hence \(\mathcal{T}_{X/k} \cong \omega_X^{-1}\). Riemann–Roch (Algebraic Curves, Lemma 0BS6) gives \[\chi(X, \mathcal{T}_{X/k}) = (2 - 2g) - \frac{1}{2}(2g - 2) = 3 - 3g.\] The zeroth cohomology vanishes as above, so \(\dim_k H^1(X, \mathcal{T}_{X/k}) = 3g - 3\), as desired.

Lemma

Let \(X\) be a smooth proper scheme over a field \(k\). Set \[\mathcal{T}_{X/k} = \SheafHom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X).\] If \[H^0(X, \mathcal{T}_{X/k}) = H^2(X, \mathcal{T}_{X/k}) = 0,\] then the functor of isomorphism classes of flat deformations of \(X\) over local Artinian \(k\)-algebras is prorepresented by \[k[[x_1, \ldots, x_r]], \qquad r = \dim_k H^1(X, \mathcal{T}_{X/k}).\] In particular, \(X\) has a universal unobstructed formal deformation over a regular complete local \(k\)-algebra.

Proof

Let \(F\) be the functor of isomorphism classes in \(\Deformationcategory_X\). Since \(X\) is smooth, Lemma 0DY9 identifies its infinitesimal automorphisms with \(H^0(X, \mathcal{T}_{X/k})\) and its tangent space with \(H^1(X, \mathcal{T}_{X/k})\). Thus Lemma 0ET5 gives a prorepresenting object \(R\) for \(F\).

We claim that \(F\) is smooth. By Formal Deformation Theory, Lemma 06HH, it suffices to lift a deformation \(Y \to \Spec(A)\) across a small extension \(B \to A\) with kernel \(I\). The morphism \(Y \to \Spec(A)\) is smooth: it is flat, locally of finite presentation, and its only fibre is \(X\). Cotangent Complex, Lemma 08UZ places the obstruction in \[\Ext^2_{\mathcal{O}_Y} (L_{Y/A}, \mathcal{O}_X \otimes_k I) = \Ext^2_{\mathcal{O}_X} (\Omega_{X/k}, \mathcal{O}_X \otimes_k I) = H^2(X, \mathcal{T}_{X/k}) \otimes_k I = 0.\] Here the coefficient module is supported on the special fibre, and we used smooth base change for differentials. The resulting first order thickening is a scheme by More on Morphisms, Lemma 05YV, and is flat over \(B\) by More on Morphisms, Lemma 063Y. Hence it is the required deformation and the claim follows.

The morphism from the prorepresentable functor \(\underline{R}\) to the final functor is therefore smooth. Formal Deformation Theory, Lemma 06HL shows that \(R \cong k[[x_1, \ldots, x_r]]\). The number of variables is the dimension of the tangent space computed above.

Lemma

In Example 0DY7 let \(X\) be a scheme over \(k\). Let \(U \subset X\) be an open subscheme. There is a natural functor \[\Deformationcategory_X \longrightarrow \Deformationcategory_U\] 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.

Lemma

In Example 0DY7 let \(X = \Spec(P)\) be an affine scheme over \(k\). With \(\Deformationcategory_P\) as in Example 0DY1 there is a natural equivalence \[\Deformationcategory_X \longrightarrow \Deformationcategory_P\] 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 06AD.

Lemma

In Example 0DY7 let \(X\) be a scheme over \(k\) Let \(p \in X\) be a point. With \(\Deformationcategory_{\mathcal{O}_{X, p}}\) as in Example 0DY1 there is a natural functor \[\Deformationcategory_X \longrightarrow \Deformationcategory_{\mathcal{O}_{X, p}}\] of deformation categories.

Proof

Choose an affine open \(U = \Spec(P) \subset X\) containing \(p\). Then \(\mathcal{O}_{X, p}\) is a localization of \(P\). We combine the functors from Lemmas 0DYW, 0DYX, and 0DYS.

Situation

Let \(\Lambda \to k\) be as in Section 0DVM. 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 \(U_1 = \Spec(P_1)\), \(U_2 = \Spec(P_2)\) and \(U_{12} = \Spec(P_{12})\). Let \(\Deformationcategory_X\), \(\Deformationcategory_{U_1}\), \(\Deformationcategory_{U_2}\), and \(\Deformationcategory_{U_{12}}\) be as in Example 0DY7 and let \(\Deformationcategory_{P_1}\), \(\Deformationcategory_{P_2}\), and \(\Deformationcategory_{P_{12}}\) be as in Example 0DY1.

Lemma

In Situation 0DYY there is an equivalence \[\Deformationcategory_X = \Deformationcategory_{P_1} \times_{\Deformationcategory_{P_{12}}} \Deformationcategory_{P_2}\] of deformation categories, see Examples 0DY7 and 0DY1.

Proof

It suffices to show that the functors of Lemma 0DYW define an equivalence \[\Deformationcategory_X \longrightarrow \Deformationcategory_{U_1} \times_{\Deformationcategory_{U_{12}}} \Deformationcategory_{U_2}\] because then we can apply Lemma 0DYX to translate into rings. To do this we construct a quasi-inverse. Denote \(F_i : \Deformationcategory_{U_i} \to \Deformationcategory_{U_{12}}\) the functor of Lemma 0DYW. 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 01JA. Let \(Y\) be the glueing, see Schemes, Lemma 01JB. Then \(Y\) is a scheme over \(A\) and the compatibilities mentioned above show that there is a canonical isomorphism \(Y \times_{\Spec(A)} \Spec(k) = X\). Thus \((A, Y) \to (k, X)\) is an object of \(\Deformationcategory_X\). We omit the verification that this construction is a functor and is quasi-inverse to the given one.

Finite locally free coverings

We formulate the extension problem for coverings in terms of an adic system. The flatness over the coefficient ring in the following situation is essential; see Remark examples-defos-remark-formal-coverings-missing-flatness. The use of the reduced coefficient algebra in the marking is essential as well; see Remark examples-defos-remark-formal-coverings-reduced-marking.

Situation

Let \((\Lambda, \mathfrak m, k)\) be a complete Noetherian local ring and put \(\Lambda_n = \Lambda/\mathfrak m^{n + 1}\). Suppose given

1. Noetherian schemes \(X_n\) flat over \(\Lambda_n\) and cartesian identifications \[X_{n + 1} \times_{\Spec(\Lambda_{n + 1})} \Spec(\Lambda_n) = X_n,\]

2. compatible retrocompact opens \(U_n \subset X_n\) such that, after every base change by a finite \(\Lambda_n\)-algebra, the resulting open is scheme theoretically dense, and

3. a finite locally free morphism \(p_0 : Z_0 \to X_0\) whose restriction over \(U_0\) is finite étale.

For \(A \in \mathcal{A}_\Lambda\), choose \(n\) such that \(\mathfrak m^{n + 1}A = 0\) and set \[X_A = X_n \times_{\Spec(\Lambda_n)} \Spec(A), \qquad U_A = U_n \times_{X_n} X_A.\] These schemes are independent of \(n\) up to canonical isomorphism. Write \(A_{\mathrm{red}} = A/\sqrt{(0)}\). The map \(\Lambda \to A_{\mathrm{red}}\) factors through \(k\). Let \(F(A)\) be the set of isomorphism classes of finite locally free morphisms \(p_A : Z_A \to X_A\) endowed with an isomorphism \[Z_A \times_{\Spec(A)} \Spec(A_{\mathrm{red}}) \longrightarrow Z_0 \times_{\Spec(k)} \Spec(A_{\mathrm{red}})\] over \(X_0 \times_{\Spec(k)} \Spec(A_{\mathrm{red}})\). The isomorphisms used in defining \(F(A)\) are required to respect this marking.

Lemma

In Situation examples-defos-situation-formal-coverings, let \(A\) be an object of \(\mathcal{A}_\Lambda\) and let \(Z_A\) and \(Z'_A\) be two marked coverings. The restriction map \[\Mor_{X_A}(Z_A, Z'_A) \longrightarrow \Mor_{X_{A_{\mathrm{red}}}} (Z_A|_{X_{A_{\mathrm{red}}}}, Z'_A|_{X_{A_{\mathrm{red}}}})\] is injective. In particular, a marked covering has no nontrivial automorphism inducing the identity on its reduction.

Proof

The nilradical of \(A\) is nilpotent. Over \(U_A\), the morphism \(Z'_A \to X_A\) is finite locally free and its reduction is finite étale. It is unramified by More on Morphisms, Lemma 0BPG; since it is already flat and of finite presentation, it is étale. In particular, it is formally unramified. Thus two morphisms as in the statement which agree on the reduction agree after restriction to \(U_A\).

We finish on an affine open \(V \subset X_A\). Write the restrictions of the two coverings as \(\Spec(C)\) and \(\Spec(D)\). The two morphisms correspond to two homomorphisms \(D \to C\). Their difference vanishes after restriction to \(U_A \cap V\). The module \(C\) is finite locally free over \(\mathcal{O}_{X_A}(V)\). Since \(U_A \cap V\) is scheme theoretically dense, Morphisms, Lemma 01RE shows that a section of a finite locally free module which vanishes there is zero. Hence the two algebra homomorphisms, and therefore the two morphisms, are equal.

Lemma

Fix \(A\) in Situation examples-defos-situation-formal-coverings. On open subsets \(V \subset X_A\), isomorphism classes of marked finite locally free coverings form a Zariski sheaf \(G_A\). For \(x \in X_A\), finite presentation identifies its stalk with the corresponding class set over \(\Spec(\mathcal{O}_{X_A,x})\), and completion induces an injection \[(G_A)_x = G_A(\Spec(\mathcal{O}_{X_A,x})) \longrightarrow G_A(\Spec(\widehat{\mathcal{O}}_{X_A,x})).\]

Proof

Finite locally free algebras, their markings, and their isomorphisms glue in the Zariski topology. Lemma examples-defos-lemma-formal-coverings-rigidity makes every isomorphism compatible with a marking unique. Hence local isomorphism classes glue, not merely the underlying groupoid, and \(G_A\) is a sheaf. The usual finite-presentation limit argument spreads a covering and its marking from a local ring to a neighbourhood, giving the assertion about the stalk.

Put \(R = \mathcal{O}_{X_A,x}\). The map \(R \to \widehat R\) is faithfully flat by Algebra, Lemma 00MC. The pullback of \(U_A\) remains scheme theoretically dense by Morphisms, Lemma 081H. Suppose two classes over \(R\) become isomorphic over \(\widehat R\). By Lemma examples-defos-lemma-formal-coverings-rigidity, this isomorphism is unique. Its two pullbacks to \(\widehat R \otimes_R \widehat R\) are consequently equal, so it is an fpqc descent datum. Descent of morphisms gives a unique isomorphism over \(R\), proving injectivity. This is the inclusion in the corrected formula; it is not an equality in general.

Lemma

Let \((Z_n \to X_n)\) be a compatible system of marked finite locally free coverings in Situation examples-defos-situation-formal-coverings.

1. The isomorphism class of the system is the same datum as a compatible element of \(\lim_n G_{\Lambda_n}(X_n)\).

2. Suppose there is a proper scheme \(X\) over \(\Lambda\) with \(X_n = X \times_\Lambda \Lambda_n\). Then the system algebraizes uniquely to a finite locally free covering \(Z \to X\) with the prescribed reductions.

Proof

For (1), an equality between the restriction of the class at level \(n + 1\) and the class at level \(n\) supplies an isomorphism. It is unique by Lemma examples-defos-lemma-formal-coverings-rigidity; hence these isomorphisms automatically satisfy every compatibility condition and define a unique adic system.

For (2), Cohomology of Schemes, Lemma 09ZT algebraizes the compatible finite morphisms and their algebra structures. Let \(\mathcal{B}\) be the resulting coherent \(\mathcal{O}_X\)-algebra. At a point \(x \in X\), every quotient of \(\mathcal{B}_x\) by a power of \(\mathfrak m\mathcal{O}_{X,x}\) is finite locally free over the corresponding quotient of \(\mathcal{O}_{X,x}\). Algebra, Lemma 0523 therefore shows that \(\mathcal{B}_x\) is flat over \(\mathcal{O}_{X,x}\). It is finite, and hence finite free over the local ring \(\mathcal{O}_{X,x}\). Thus \(Z \to X\) is finite locally free. Full faithfulness in Grothendieck existence, or Cohomology of Schemes, Lemma 0A42, gives uniqueness and algebraizes the marking.

Proposition

In Situation examples-defos-situation-formal-coverings, the functor \[F : \mathcal{A}_\Lambda \longrightarrow \textit{Sets}\] is strictly prorepresentable.

Proof

The functor commutes with finite products. We verify the two local Cech tests of Formal Deformation Theory, Theorem formal-defos-theorem-grothendieck-levelt-prorepresentability.

Let \(A \to B\) be a map in \(\mathcal{A}_\Lambda\) with \(A\) local. If \(B\) is a nonzero finite free \(A\)-module, then \(A \to B\) is faithfully flat. Finite locally free algebras satisfy fpqc descent. If a class over \(B\) has equal pullbacks to \(B \otimes_A B\), the required isomorphism between those pullbacks is unique by Lemma examples-defos-lemma-formal-coverings-rigidity; hence it automatically satisfies the cocycle condition and the covering descends. Put \(K=A_{\mathrm{red}}\). Faithful flatness shows that \(K \to B_{\mathrm{red}}\) is injective. It is a finite map from a field to a nonzero reduced Artinian ring, and hence is faithfully flat. The marking and its compatibility descend along this map. This proves the first equalizer test.

Suppose instead that \(A \to B\) is injective and \(B/A\) has length \(1\). Formal Deformation Theory, Lemma formal-defos-lemma-length-one-cech-equalizer gives the ring equalizer and its proof gives \(\mathfrak nB = \mathfrak n\), where \(\mathfrak n\) is the maximal ideal of \(A\). Consider a class over \(B\) whose two pullbacks are equal. As above, rigidity turns the unique isomorphism into a descent datum.

We descend it on an affine \(V = \Spec(R) \subset X_A\). The ring \(R\) is flat over \(A\) by the flatness assumption in Situation examples-defos-situation-formal-coverings. Let \(C\) be the finite locally free algebra of the covering on \(V_B\). It is flat as a \(B\)-module. Descent, Lemma descent-lemma-artinian-small-effective-descent-flat-modules produces an \(A\)-flat module \(M\) with \(M \otimes_A B \cong C\). Full faithfulness for flat modules first descends the \(R\)-action. Put \(K=A/\mathfrak n\) and \(\overline B=B_{\mathrm{red}}\). The injection \(A \to B\) induces an injection \(K \to \overline B\): an element of \(A\) whose image in \(B\) is nilpotent was already nilpotent. Thus \(K \to \overline B\) is faithfully flat. We have \[(M/\mathfrak nM) \otimes_K \overline B = C \otimes_B \overline B.\] The module on the right is finite locally free over \((R/\mathfrak nR) \otimes_K \overline B\). Hence \(M/\mathfrak nM\) is finite locally free over \(R/\mathfrak nR\) by fpqc descent. Since \(\mathfrak n\) is nilpotent, Algebra, Lemma 06A5 shows that \(M\) is flat over \(R\). Lifting finitely many generators modulo \(\mathfrak n\) shows that \(M\) is finite over \(R\), and hence it is finite locally free.

Now \(R\), \(M\), and \(M \otimes_R M\) are flat over \(A\). Full faithfulness in the same descent lemma descends the unit and multiplication of \(C\). The algebra identities hold because they hold after base change to \(B\). The descended algebras agree on affine overlaps by uniqueness and therefore glue. The given marking over \(\overline B\) is compatible with this descent datum. It descends along the faithfully flat map \(K \to \overline B\) to the required marking over \(K\). We have proved the second equalizer test, and the Grothendieck–Levelt criterion proves the proposition.

Remark

If \(U_0 = X_0\), then every covering in the situation is finite étale. Invariance of finite étale coverings under nilpotent thickenings (Fundamental Groups, Lemma 0BQB) makes \(F\) the one-point functor, represented by \(\Lambda\). In the ramified case the strict prorepresenting topological algebra need not be Noetherian.

Remark

The reduced-fibre marking in Situation examples-defos-situation-formal-coverings is necessary when \(\mathcal{A}_\Lambda\) contains all finite-length \(\Lambda\)-algebras. A marking only over \(A/\mathfrak mA\), as in the literal formulation in Exposé 195, is not stable enough for the length-one equalizer test.

For an example, let \(k\) have characteristic different from \(2\) and set \[\Lambda=k[\epsilon]/(\epsilon^2), \qquad A=k[\epsilon,a]/(\epsilon^2,a^2,\epsilon a), \qquad B=k[\epsilon,b]/(\epsilon^2,b^2).\] Make \(A\) a subalgebra of \(B\) by \(a=\epsilon b\). Then \(B/A\) has length \(1\) over the local ring \(A\). Moreover, \[\mathfrak mA=(\epsilon) \quad\text{whereas}\quad \mathfrak mB=(\epsilon,\epsilon b)=(\epsilon,a).\] Take \(X=\Spec(\Lambda[z])\), \(U=D(z)\), and the fixed covering \(Z_0=\Spec(k[z,s]/(s^2-z))\). The finite locally free covering over \(X_A\) defined by \[s^2=z+a\] is finite étale over \(U_A\). After base change to \(B\) it has the required \(B/\mathfrak mB\)-marking and its two pullbacks to \(B\otimes_A B\) agree. It cannot descend as an \(A/\mathfrak mA\)-marked covering: its reduction there has discriminant ideal \((4(z+a))\), whereas the fixed covering has discriminant ideal \((4z)\). These ideals are distinct in \(k[a,z]/(a^2)\). Flat-module descent and rigidity show that no different covering over \(A\) can provide the missing preimage. Thus the literal functor fails the Cech equalizer property even though \(X\) is flat over \(\Lambda\).

Marking over \(A_{\mathrm{red}}\) repairs this defect. For a local \(A\) this is its residue field, and for every injective test map \(A \to B\) the induced map \(A_{\mathrm{red}} \to B_{\mathrm{red}}\) is faithfully flat. Notice that for \(A=\Lambda_n\) the reduced fibre is still \(X_0\), so the adic system and its algebraization retain the prescribed covering \(Z_0\).

Remark

The statement of Exposé 195, Proposition 5.1 does not explicitly assume flatness of the formal scheme over \(\Lambda\). Its 1962 erratum strengthens the nonzerodivisor condition defining \(U\) and replaces an incorrect equality with the completion inclusion of Lemma examples-defos-lemma-formal-coverings-local-completion, but it does not add flatness. Without flatness the prorepresentability assertion is false.

Indeed, let \(k\) have characteristic different from \(2\), put \[B = k[t]/(t^4), \qquad \Lambda = k \oplus kt^2 \oplus kt^3 \subset B,\] and write \(x=t^2\), \(y=t^3\). Thus \(\Lambda \cong k[x,y]/(x,y)^2\), and \(B/\Lambda\) has length \(1\) as a \(\Lambda\)-module. Set \[R = (\Lambda/(x))[z] = k[y,z]/(y^2), \qquad X = \Spec(R), \qquad U = D(z).\] The element \(z\) is a nonzerodivisor both on \(X\) and on its closed fibre. The two rank-two coverings \[\Spec(R[s]/(s^2-z)) \longrightarrow X, \qquad \Spec(R[s]/(s^2-z-y)) \longrightarrow X\] have the same marked closed fibre and are finite étale over \(U\). They are not isomorphic: their discriminant ideals are \((4z)\) and \((4(z+y))\), which are distinct because \(y\) is nonzero modulo \((z)\). After base change along the injective length-one map \(\Lambda \to B\), they become identical, since \((x)B=(t^2,t^3)\) kills \(y\). Thus the map on covering classes is not injective. Every strictly prorepresentable functor sends an injective map of coefficient algebras to an injective map of sets, which gives the contradiction. The flatness hypothesis in Situation examples-defos-situation-formal-coverings prevents exactly this failure of base change.

Lemma

Fix \(A\) in Situation examples-defos-situation-formal-coverings. Suppose that \[X_A \setminus U_A = \{x_1, \ldots, x_r\}\] is a finite set of closed points. Then restriction induces a bijection \[G_A(X_A) \longrightarrow \prod_{i = 1}^r (G_A)_{x_i}.\] Suppose in addition that, for \(R_i = \mathcal{O}_{X_A, x_i}\), there is an element \(f_i \in R_i\) such that \[U_A \times_{X_A} \Spec(R_i) = D(f_i)\] and such that the completion \(\widehat R_i\) is the \(f_i\)-adic completion. Then completion induces a bijection \[G_A(X_A) \longrightarrow \prod_{i = 1}^r G_A(\Spec(\widehat R_i)).\] In particular, the additional hypotheses hold in the one dimensional setup where the complement is locally cut out by the nonzerodivisor used to define \(U_A\).

Proof

The restriction of \(G_A\) to \(U_A\) is the one-point sheaf: finite étale coverings and their morphisms lift uniquely across nilpotent thickenings by Fundamental Groups, Lemma 0BQB. Given germs \(s_i \in (G_A)_{x_i}\), represent them on neighbourhoods \(V_i\) of \(x_i\). Shrinking \(V_i\), we may assume that \(V_i\) contains none of the other \(x_j\). Every intersection \(V_i \cap V_j\) is contained in \(U_A\), so the sections agree there. They also agree with the unique section over \(U_A\). The sheaf property in Lemma examples-defos-lemma-formal-coverings-local-completion glues them to a section over \(X_A\). The same argument proves uniqueness. This gives the first bijection.

It remains to prove that the completion map at \(x_i\) is surjective under the additional hypotheses. Set \(R = R_i\), \(f = f_i\), and \(S = \widehat R_i\). A class over \(S\) restricts over \(S_f\) to a finite étale covering. Its marking and invariance under nilpotent thickenings give a unique isomorphism with the pullback of the fixed covering over \(R_f\). More on Algebra, Theorem 05ES and Remark 05EU glue the two covering algebras and this isomorphism to a finite locally free \(R\)-algebra. The same tensor-compatible equivalence glues the marking. Thus every completed-local class comes from a class over \(R\). Injectivity is Lemma examples-defos-lemma-formal-coverings-local-completion, and the second bijection follows.

For the final assertion, the punctured spectrum of each one dimensional local ring is \(D(f_i)\). Since \(V(f_i)\) is the closed point, the \(f_i\)-adic and maximal-adic topologies are equivalent.

Remark

Normality of the closed fibre does not by itself make the formal moduli space in Proposition examples-defos-proposition-formal-coverings-prorepresentable formally smooth. Here is an explicit obstruction.

Let \(k\) be a field, let \(\Lambda = k[[\epsilon]]\), and put \(\Lambda_n = \Lambda/(\epsilon^{n + 1})\). Take \[R_n = (\Lambda_n[t])_{(\epsilon, t)}, \qquad X_n = \Spec(R_n), \qquad U_n = D(t).\] This is a flat system in Situation examples-defos-situation-formal-coverings: localizations of polynomial algebras are flat, and \(t\) remains a nonzerodivisor after every coefficient base change. Thus \(X_0 = \Spec(k[t]_{(t)})\) is normal. Over \(R_0 = k[t]_{(t)}\), let \(C_0\) be free with basis \(1, x_1, x_2, x_3\) and multiplication \[x_i^2 = t x_i, \qquad x_i x_j = 0 \quad (i \ne j).\] It is a finite locally free covering algebra. Indeed, it is the subalgebra of \(R_0^4\) generated by \(1\) and \(x_i = t e_i\) for three of the standard idempotents, and \(C_0[1/t] = R_0[1/t]^4\) is finite étale.

Set \(A = \Lambda/(\epsilon^2)\) and \(R=(A[t])_{(\epsilon,t)}\). On the free \(R\)-module with the same basis, define a commutative algebra \(C\) by \[\begin{align*} x_1^2 &= t x_1 + \epsilon x_2, & x_2^2 &= t x_2 + \epsilon x_3, & x_3^2 &= t x_3, \\ x_1x_2 &= -\epsilon x_2, & x_1x_3 &= 0, & x_2x_3 &= -\epsilon x_3. \end{align*}\] A direct check on the unordered triples in \(x_1,x_2,x_3\) shows that this multiplication is associative modulo \(\epsilon^2\). Its reduction is \(C_0\), and \(C[1/t]\) is finite étale by invariance under nilpotent thickenings. Consequently \(C\) defines an element of \(F(A)\).

This element does not lift to \(A' = \Lambda/(\epsilon^3)\). Indeed, if a lift existed, lift the displayed basis to it. Its multiplication would have the form \[\widetilde\mu = \mu_0 + \epsilon\nu + \epsilon^2q,\] where \(\mu_0\) is the multiplication on \(C_0\), \(\nu\) is the first-order term displayed above, and \(q\) is an \(R_0\)-bilinear correction. Write \(q_{ij}=q(x_i,x_j)\). The coefficient of \(\epsilon^2\) in associativity for \((x_1x_2)x_2=x_1(x_2^2)\) gives \[-x_3 + \mu_0(q_{12},x_2) - tq_{12} - \mu_0(x_1,q_{22}) = 0.\] Modulo \(t\), if \(a_0\) and \(b_0\) are the scalar coefficients of \(q_{12}\) and \(q_{22}\), respectively, this becomes \[-x_3 + a_0x_2 - b_0x_1 = 0.\] This is impossible because \(1,x_1,x_2,x_3\) remain a basis modulo \(t\). Hence \(F(A') \to F(A)\) is not surjective, even though \(X_0\) is normal, and the asserted formal smoothness fails.

Morphisms of Schemes

The deformation theory of morphisms of schemes. Of course this is just an example of deformations of diagrams of schemes.

Example

Let \(\mathcal{F}\) be the category defined as follows

1. an object is a pair \((A, X \to Y)\) consisting of an object \(A\) of \(\mathcal{C}_\Lambda\) and a morphism \(X \to Y\) of schemes over \(A\) with both \(X\) and \(Y\) flat over \(A\), and

2. a morphism \((f, g, h) : (A', X' \to Y') \to (A, X \to Y)\) consists of a morphism \(f : A' \to A\) in \(\mathcal{C}_\Lambda\) together with morphisms of schemes \(g : X \to X'\) and \(h : Y \to Y'\) such that \[\xymatrix{ X \ar[r]_g \ar[d] & X' \ar[d] \\ Y \ar[r]_h \ar[d] & Y' \ar[d] \\ \Spec(A) \ar[r]^f & \Spec(A') }\] is a commutative diagram of schemes where both squares are cartesian.

The functor \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) sends \((A, X \to Y)\) to \(A\) and \((f, g, h)\) to \(f\). It is clear that \(p\) is cofibred in groupoids. Given a morphism of schemes \(X \to Y\) over \(k\), let \(x_0 = (k, X \to Y)\) be the corresponding object of \(\mathcal{F}(k)\). We set \[\Deformationcategory_{X \to Y} = \mathcal{F}_{x_0}\]

Lemma

Example 0E3T satisfies the Rim-Schlessinger condition (RS). In particular, \(\Deformationcategory_{X \to Y}\) is a deformation category for any morphism of schemes \(X \to Y\) 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 06J5 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 \[\xymatrix{ \Spec(A) \ar[r] \ar[d] & \Spec(A_2) \ar[d] \\ \Spec(A_1) \ar[r] & \Spec(A_1 \times_A A_2) }\] is a pushout diagram as in More on Morphisms, Lemma 07RT. Thus the lemma follows immediately from More on Morphisms, Lemma 07RX as this describes the category of schemes flat over \(A_1 \times_A A_2\) as the fibre product of the category of schemes flat over \(A_1\) with the category of schemes flat over \(A_2\) over the category of schemes flat over \(A\).

Lemma

In Example 0DY7 let \(f : X \to Y\) be a morphism of schemes over \(k\). There is a canonical exact sequence of \(k\)-vector spaces \[\xymatrix{ 0 \ar[r] & \text{Inf}(\Deformationcategory_{X \to Y}) \ar[r] & \text{Inf}(\Deformationcategory_X \times \Deformationcategory_Y) \ar[r] & \text{Der}_k(\mathcal{O}_Y, f_*\mathcal{O}_X) \ar[lld] \\ & T\Deformationcategory_{X \to Y} \ar[r] & T(\Deformationcategory_X \times \Deformationcategory_Y) \ar[r] & \text{Ext}^1_{\mathcal{O}_X}(Lf^*\NL_{Y/k}, \mathcal{O}_X) }\]

Proof

The obvious map of deformation categories \(\Deformationcategory_{X \to Y} \to \Deformationcategory_X \times \Deformationcategory_Y\) gives two of the arrows in the exact sequence of the lemma. Recall that \(\text{Inf}(\Deformationcategory_{X \to Y})\) is the set of automorphisms of the trivial deformation \[f' : X' = X \times_{\Spec(k)} \Spec(k[\epsilon]) \xrightarrow{f \times \text{id}} Y' = Y \times_{\Spec(k)} \Spec(k[\epsilon])\] of \(X \to Y\) to \(k[\epsilon]\) equal to the identity modulo \(\epsilon\). This is clearly the same thing as pairs \((\alpha, \beta) \in \text{Inf}(\Deformationcategory_X \times \Deformationcategory_Y)\) of infinitesimal automorphisms of \(X\) and \(Y\) compatible with \(f'\), i.e., such that \(f' \circ \alpha = \beta \circ f'\). By Deformation Theory, Lemma 08U8 for an arbitrary pair \((\alpha, \beta)\) the difference between the morphism \(f' : X' \to Y'\) and the morphism \(\beta^{-1} \circ f' \circ \alpha : X' \to Y'\) defines an element in \[\text{Der}_k(\mathcal{O}_Y, f_*\mathcal{O}_X) = \Hom_{\mathcal{O}_Y}(\Omega_{Y/k}, f_*\mathcal{O}_X)\] Equality by More on Morphisms, Lemma 0D0J. This defines the last top horizontal arrow and shows exactness in the first two places. For the map \[\text{Der}_k(\mathcal{O}_Y, f_*\mathcal{O}_X) \to T\Deformationcategory_{X \to Y}\] we interpret elements of the source as morphisms \(f_\epsilon : X' \to Y'\) over \(\Spec(k[\epsilon])\) equal to \(f\) modulo \(\epsilon\) using Deformation Theory, Lemma 08U8. We send \(f_\epsilon\) to the isomorphism class of \((f_\epsilon : X' \to Y')\) in \(T\Deformationcategory_{X \to Y}\). Note that \((f_\epsilon : X' \to Y')\) is isomorphic to the trivial deformation \((f' : X' \to Y')\) exactly when \(f_\epsilon = \beta^{-1} \circ f \circ \alpha\) for some pair \((\alpha, \beta)\) which implies exactness in the third spot. Clearly, if some first order deformation \((f_\epsilon : X_\epsilon \to Y_\epsilon)\) maps to zero in \(T(\Deformationcategory_X \times \Deformationcategory_Y)\), then we can choose isomorphisms \(X' \to X_\epsilon\) and \(Y' \to Y_\epsilon\) and we conclude we are in the image of the south-west arrow. Therefore we have exactness at the fourth spot. Finally, given two first order deformations \(X_\epsilon\), \(Y_\epsilon\) of \(X\), \(Y\) there is an obstruction in \[ob(X_\epsilon, Y_\epsilon) \in \text{Ext}^1_{\mathcal{O}_X}(Lf^*\NL_{Y/k}, \mathcal{O}_X)\] which vanishes if and only if \(f : X \to Y\) lifts to \(X_\epsilon \to Y_\epsilon\), see Deformation Theory, Lemma 08U8. This finishes the proof.

Lemma

In Lemma 0E3V if \(X\) and \(Y\) are both proper over \(k\), then \(\text{Inf}(\Deformationcategory_{X \to Y})\) and \(T\Deformationcategory_{X \to Y}\) are finite dimensional.

Proof

Omitted. Hint: argue as in Lemma 0DYA and use the exact sequence of the lemma.

In Example 0E3T if \(X \to Y\) is a morphism of proper schemes over \(k\), then \(\Deformationcategory_{X \to Y}\) 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 0E3U and 0E3W and the general discussion in Section 0DVM.

Lemma

In Example 0E3T assume \(X \to Y\) is a morphism of proper \(k\)-schemes. Assume \(\Lambda\) is a complete local ring with residue field \(k\) (the classical case). Then the functor \[F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \Ob(\Deformationcategory_{X \to Y}(A))/\cong\] of isomorphism classes of objects has a hull. If \(\text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) = \text{Der}_k(\mathcal{O}_Y, \mathcal{O}_Y) = 0\), then \(F\) is prorepresentable.

Proof

The existence of a hull follows immediately from Lemmas 0E3U and 0E3W and Formal Deformation Theory, Lemma 06J7 and Remark 06IZ.

Assume \(\text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) = \text{Der}_k(\mathcal{O}_Y, \mathcal{O}_Y) = 0\). Then the exact sequence of Lemma 0E3V combined with Lemma 0DY9 shows that \(\text{Inf}(\Deformationcategory_{X \to Y}) = 0\). Then \(\Deformationcategory_{X \to Y}\) and \(F\) are equivalent by Formal Deformation Theory, Lemma 06K0. Hence \(F\) is a deformation functor (because \(\Deformationcategory_{X \to Y}\) is a deformation category) with finite tangent space and we can apply Formal Deformation Theory, Theorem 06JM.

Lemma

In Example 0DY7 let \(f : X \to Y\) be a morphism of schemes over \(k\). If \(f_*\mathcal{O}_X = \mathcal{O}_Y\) and \(R^1f_*\mathcal{O}_X = 0\), then the morphism of deformation categories \[\Deformationcategory_{X \to Y} \to \Deformationcategory_X\] is an equivalence.

Proof

We construct a quasi-inverse to the forgetful functor of the lemma. Namely, suppose that \((A, U)\) is an object of \(\Deformationcategory_X\). The given map \(X \to U\) is a finite order thickening and we can use it to identify the underlying topological spaces of \(U\) and \(X\), see More on Morphisms, Section 04EW. Thus we may and do think of \(\mathcal{O}_U\) as a sheaf of \(A\)-algebras on \(X\); moreover the fact that \(U \to \Spec(A)\) is flat, means that \(\mathcal{O}_U\) is flat as a sheaf of \(A\)-modules. In particular, we have a filtration \[0 = \mathfrak m_A^n\mathcal{O}_U \subset \mathfrak m_A^{n - 1}\mathcal{O}_U \subset \ldots \subset \mathfrak m_A^2\mathcal{O}_U \subset \mathfrak m_A\mathcal{O}_U \subset \mathcal{O}_U\] with subquotients equal to \(\mathcal{O}_X \otimes_k \mathfrak m_A^i/\mathfrak m_A^{i + 1}\) by flatness, see More on Morphisms, Lemma 063Y or the more general Deformation Theory, Lemma 08LI. Set \[\mathcal{O}_V = f_*\mathcal{O}_U\] viewed as sheaf of \(A\)-algebras on \(Y\). Since \(R^1f_*\mathcal{O}_X = 0\) we find by the description above that \(R^1f_*(\mathfrak m_A^i\mathcal{O}_U/\mathfrak m_A^{i + 1}\mathcal{O}_U) = 0\) for all \(i\). This implies that the sequences \[0 \to (f_*\mathcal{O}_X) \otimes_k \mathfrak m_A^i/\mathfrak m_A^{i + 1} \to f_*(\mathcal{O}_U/\mathfrak m_A^{i + 1}\mathcal{O}_U) \to f_*(\mathcal{O}_U/\mathfrak m_A^i\mathcal{O}_U) \to 0\] are exact for all \(i\). Reading the references given above backwards (and using induction) we find that \(\mathcal{O}_V\) is a flat sheaf of \(A\)-algebras with \(\mathcal{O}_V/\mathfrak m_A\mathcal{O}_V = \mathcal{O}_Y\). Using More on Morphisms, Lemma 05YV we find that \((Y, \mathcal{O}_V)\) is a scheme, call it \(V\). The equality \(\mathcal{O}_V = f_*\mathcal{O}_U\) defines a morphism of ringed spaces \(U \to V\) which is easily seen to be a morphism of schemes. This finishes the proof by the flatness already established.

Algebraic spaces

The deformation theory of algebraic spaces.

Example

Let \(\mathcal{F}\) be the category defined as follows

1. 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

2. 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 \[\xymatrix{ X \ar[r]_g \ar[d] & Y \ar[d] \\ \Spec(A) \ar[r]^f & \Spec(B) }\] 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 \[\Deformationcategory_X = \mathcal{F}_{x_0}\]

Lemma

Example 0E3Z satisfies the Rim-Schlessinger condition (RS). In particular, \(\Deformationcategory_X\) 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 06J5 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 \[\xymatrix{ \Spec(A) \ar[r] \ar[d] & \Spec(A_2) \ar[d] \\ \Spec(A_1) \ar[r] & \Spec(A_1 \times_A A_2) }\] is a pushout diagram as in Pushouts of Spaces, Lemma 07VX. Thus the lemma is a special case of Pushouts of Spaces, Lemma 07W3.

Lemma

In Example 0E3Z let \(X\) be an algebraic space over \(k\). Then \[\text{Inf}(\Deformationcategory_X) = \text{Ext}^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X) = \Hom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X) = \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X)\] and \[T\Deformationcategory_X = \text{Ext}^1_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X)\]

Proof

Recall that \(\text{Inf}(\Deformationcategory_X)\) is the set of automorphisms of the trivial deformation \(X' = X \times_{\Spec(k)} \Spec(k[\epsilon])\) of \(X\) to \(k[\epsilon]\) equal to the identity modulo \(\epsilon\). By Deformation Theory, Lemma 0D17 this is equal to \(\text{Ext}^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X)\). The equality \(\text{Ext}^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X) = \Hom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X)\) follows from More on Morphisms of Spaces, Lemma 0D0Y. The equality \(\Hom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X) = \text{Der}_k(\mathcal{O}_X, \mathcal{O}_X)\) follows from More on Morphisms of Spaces, Definition 04CT and Modules on Sites, Definition 04BN.

Recall that \(T_{x_0}\Deformationcategory_X\) is the set of isomorphism classes of flat deformations \(X'\) of \(X\) to \(k[\epsilon]\), more precisely, the set of isomorphism classes of \(\Deformationcategory_X(k[\epsilon])\). Thus the second statement of the lemma follows from Deformation Theory, Lemma 0D17.

Lemma

In Lemma 0E41 if \(X\) is proper over \(k\), then \(\text{Inf}(\Deformationcategory_X)\) and \(T\Deformationcategory_X\) are finite dimensional.

Proof

By the lemma we have to show \(\Ext^1_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X)\) and \(\Ext^0_{\mathcal{O}_X}(\NL_{X/k}, \mathcal{O}_X)\) are finite dimensional. By More on Morphisms of Spaces, Lemma 0D0Z and the fact that \(X\) is Noetherian, we see that \(\NL_{X/k}\) has coherent cohomology sheaves zero except in degrees \(0\) and \(-1\). By Derived Categories of Spaces, Lemma 0D0T the displayed \(\Ext\)-groups are finite \(k\)-vector spaces and the proof is complete.

In Example 0E3Z if \(X\) is a proper algebraic space over \(k\), then \(\Deformationcategory_X\) 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 0E40 and 0E42 and the general discussion in Section 0DVM.

Lemma

In Example 0E3Z 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 \[F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \Ob(\Deformationcategory_X(A))/\cong\] 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 0E40 and 0E42 and Formal Deformation Theory, Lemma 06J7 and Remark 06IZ.

Assume \(\text{Der}_k(\mathcal{O}_X, \mathcal{O}_X) = 0\). Then \(\Deformationcategory_X\) and \(F\) are equivalent by Formal Deformation Theory, Lemma 06K0. Hence \(F\) is a deformation functor (because \(\Deformationcategory_X\) is a deformation category) with finite tangent space and we can apply Formal Deformation Theory, Theorem 06JM.

Deformations of completions

In this section we compare the deformation problem posed by an algebra and its completion. We first discuss “liftability”.

Lemma

Let \(A' \to A\) be a surjection of rings with nilpotent kernel. Let \(A' \to P'\) be a flat ring map. Set \(P = P' \otimes_{A'} A\). Let \(M\) be an \(A\)-flat \(P\)-module. Then the following are equivalent

1. there is an \(A'\)-flat \(P'\)-module \(M'\) with \(M' \otimes_{P'} P = M\), and

2. there is an object \(K' \in D^-(P')\) with \(K' \otimes_{P'}^\mathbf{L} P = M\).

Proof

Suppose that \(M'\) is as in (1). Then \[M = M' \otimes_P P' = M' \otimes_{A'} A = M' \otimes_A^\mathbf{L} A' = M' \otimes_{P'}^\mathbf{L} P\] The first two equalities are clear, the third holds because \(M'\) is flat over \(A'\), and the fourth holds by More on Algebra, Lemma 0661. Thus (2) holds. Conversely, suppose \(K'\) is as in (2). We may and do assume \(M\) is nonzero. Let \(t\) be the largest integer such that \(H^t(K')\) is nonzero (exists because \(M\) is nonzero). Then \(H^t(K') \otimes_{P'} P = H^t(K' \otimes_{P'}^\mathbf{L} P)\) is zero if \(t > 0\). Since the kernel of \(P' \to P\) is nilpotent this implies \(H^t(K') = 0\) by Nakayama’s lemma a contradiction. Hence \(t = 0\) (the case \(t < 0\) is absurd as well). Then \(M' = H^0(K')\) is a \(P'\)-module such that \(M = M' \otimes_{P'} P\) and the spectral sequence for Tor gives an injective map \[\text{Tor}_1^{P'}(M', P) \to H^{-1}(M' \otimes_{P'}^\mathbf{L} P) = 0\] By the reference on derived base change above \(0 = \text{Tor}_1^{P'}(M', P) = \text{Tor}_1^{A'}(M', A)\). We conclude that \(M'\) is \(A'\)-flat by Algebra, Lemma 051C.

Lemma

Consider a commutative diagram of Noetherian rings \[\xymatrix{ A' \ar[d] \ar[r] & P' \ar[d] \ar[r] & Q' \ar[d] \\ A \ar[r] & P \ar[r] & Q }\] with cartesian squares, with flat horizontal arrows, and with surjective vertical arrows whose kernels are nilpotent. Let \(J' \subset P'\) be an ideal such that \(P'/J' = Q'/J'Q'\). Let \(M\) be an \(A\)-flat \(P\)-module. Assume for all \(g \in J'\) there exists an \(A'\)-flat \((P')_g\)-module lifting \(M_g\). Then the following are equivalent

1. \(M\) has an \(A'\)-flat lift to a \(P'\)-module, and

2. \(M \otimes_P Q\) has an \(A'\)-flat lift to a \(Q'\)-module.

Proof

Let \(I = \Ker(A' \to A)\). By induction on the integer \(n > 1\) such that \(I^n = 0\) we reduce to the case where \(I\) is an ideal of square zero; details omitted. We translate the condition of liftability of \(M\) into the problem of finding an object of \(D^-(P')\) as in Lemma 0DZ1. The obstruction to doing this is the element \[\omega(M) \in \text{Ext}^2_P(M, M \otimes_P^\mathbf{L} IP) = \text{Ext}^2_P(M, M \otimes_P IP)\] constructed in Deformation Theory, Lemma 0DYR. The equality in the displayed formula holds as \(M \otimes_P^\mathbf{L} IP = M \otimes_P IP\) since \(M\) and \(P\) are \(A\)-flat². The obstruction for lifting \(M \otimes_P Q\) is similarly the element \[\omega(M \otimes_P Q) \in \text{Ext}^2_Q(M \otimes_P Q, (M \otimes_P Q) \otimes_Q IQ)\] which is the image of \(\omega(M)\) by the functoriality of the construction \(\omega(-)\) of Deformation Theory, Lemma 0DYR. By More on Algebra, Lemma 0A6A we have \[\text{Ext}^2_Q(M \otimes_P Q, (M \otimes_P Q) \otimes_Q IQ) = \text{Ext}^2_P(M, M \otimes_P IP) \otimes_P Q\] here we use that \(P\) is Noetherian and \(M\) finite. Our assumption on \(P' \to Q'\) guarantees that for an \(P\)-module \(E\) the map \(E \to E \otimes_P Q\) is bijective on \(J'\)-power torsion, see More on Algebra, Lemma 05EC. Thus we conclude that it suffices to show \(\omega(M)\) is \(J'\)-power torsion. In other words, it suffices to show that \(\omega(M)\) dies in \[\text{Ext}^2_P(M, M \otimes_P IP)_g = \text{Ext}^2_{P_g}(M_g, M_g \otimes_{P_g} IP_g)\] for all \(g \in J'\). Howeover, by the compatibility of formation of \(\omega(M)\) with base change again, we conclude that this is true as \(M_g\) is assumed to have a lift (of course you have to use the whole string of equivalences again).

Lemma

Let \(A' \to A\) be a surjective map of Noetherian rings with nilpotent kernel. Let \(A \to B\) be a finite type flat ring map. Let \(\mathfrak b \subset B\) be an ideal such that \(\Spec(B) \to \Spec(A)\) is syntomic on the complement of \(V(\mathfrak b)\). Then \(B\) has a flat lift to \(A'\) if and only if the \(\mathfrak b\)-adic completion \(B^\wedge\) has a flat lift to \(A'\).

Proof

Choose an \(A\)-algebra surjection \(P = A[x_1, \ldots, x_n] \to B\). Let \(\mathfrak p \subset P\) be the inverse image of \(\mathfrak b\). Set \(P' = A'[x_1, \ldots, x_n]\) and denote \(\mathfrak p' \subset P'\) the inverse image of \(\mathfrak p\). (Of course \(\mathfrak p\) and \(\mathfrak p'\) do not designate prime ideals here.) We will denote \(P^\wedge\) and \((P')^\wedge\) the respective completions.

Suppose \(A' \to B'\) is a flat lift of \(A \to B\), in other words, \(A' \to B'\) is flat and there is an \(A\)-algebra isomorphism \(B = B' \otimes_{A'} A\). Then we can choose an \(A'\)-algebra map \(P' \to B'\) lifting the given surjection \(P \to B\). By Nakayama’s lemma (Algebra, Lemma 00DV) we find that \(B'\) is a quotient of \(P'\). In particular, we find that we can endow \(B'\) with an \(A'\)-flat \(P'\)-module structure lifting \(B\) as an \(A\)-flat \(P\)-module. Conversely, if we can lift \(B\) to a \(P'\)-module \(M'\) flat over \(A'\), then \(M'\) is a cyclic module \(M' \cong P'/J'\) (using Nakayama again) and setting \(B' = P'/J'\) we find a flat lift of \(B\) as an algebra.

Set \(C = B^\wedge\) and \(\mathfrak c = \mathfrak bC\). Suppose that \(A' \to C'\) is a flat lift of \(A \to C\). Then \(C'\) is complete with respect to the inverse image \(\mathfrak c'\) of \(\mathfrak c\) (Algebra, Lemma 0DYC). We choose an \(A'\)-algebra map \(P' \to C'\) lifting the \(A\)-algebra map \(P \to C\). These maps pass through completions to give surjections \(P^\wedge \to C\) and \((P')^\wedge \to C'\) (for the second again using Nakayama’s lemma). In particular, we find that we can endow \(C'\) with an \(A'\)-flat \((P')^\wedge\)-module structure lifting \(C\) as an \(A\)-flat \(P^\wedge\)-module. Conversely, if we can lift \(C\) to a \((P')^\wedge\)-module \(N'\) flat over \(A'\), then \(N'\) is a cyclic module \(N' \cong (P')^\wedge/\tilde J\) (using Nakayama again) and setting \(C' = (P')^\wedge/\tilde J\) we find a flat lift of \(C\) as an algebra.

Observe that \(P' \to (P')^\wedge\) is a flat ring map which induces an isomorphism \(P'/\mathfrak p' = (P')^\wedge/\mathfrak p'(P')^\wedge\). We conclude that our lemma is a consequence of Lemma 0DZ2 provided we can show that \(B_g\) lifts to an \(A'\)-flat \(P'_g\)-module for \(g \in \mathfrak p'\). However, the ring map \(A \to B_g\) is syntomic and hence lifts to an \(A'\)-flat algebra \(B'\) by Smoothing Ring Maps, Proposition 07M8. Since \(A' \to P'_g\) is smooth, we can lift \(P_g \to B_g\) to a surjective map \(P'_g \to B'\) as before and we get what we want.

Notation. Let \(A \to B\) be a ring map. Let \(N\) be a \(B\)-module. We denote \(\text{Exal}_A(B, N)\) the set of isomorphism classes of extensions \[0 \to N \to C \to B \to 0\] of \(A\)-algebras such that \(N\) is an ideal of square zero in \(C\). Given a second such \(0 \to N \to C' \to B \to 0\) an isomorphism is a \(A\)-algebra isomorpism \(C \to C'\) such that the diagram \[\xymatrix{ 0 \ar[r] & N \ar[r] \ar[d]_{\text{id}} & C \ar[r] \ar[d] & B \ar[r] \ar[d]_{\text{id}} & 0 \\ 0 \ar[r] & N \ar[r] & C' \ar[r] & B \ar[r] & 0 }\] commutes. The assignment \(N \mapsto \text{Exal}_A(B, N)\) is a functor which transforms products into products. Hence this is an additive functor and \(\text{Exal}_A(B, N)\) has a natural \(B\)-module structure. In fact, by Deformation Theory, Lemma 08S7 we have \(\text{Exal}_A(B, N) = \text{Ext}^1_B(\NL_{B/A}, N)\).

Lemma

Let \(k\) be a field. Let \(B\) be a finite type \(k\)-algebra. Let \(J \subset B\) be an ideal such that \(\Spec(B) \to \Spec(k)\) is smooth on the complement of \(V(J)\). Let \(N\) be a finite \(B\)-module. Then there is a canonical bijection \[\text{Exal}_k(B, N) \to \text{Exal}_k(B^\wedge, N^\wedge)\] Here \(B^\wedge\) and \(N^\wedge\) are the \(J\)-adic completions.

Proof

The map is given by completion: given \(0 \to N \to C \to B \to 0\) in \(\text{Exal}_k(B, N)\) we send it to the completion \(C^\wedge\) of \(C\) with respect to the inverse image of \(J\). Compare with the proof of Lemma 0DYV.

Since \(k \to B\) is of finite presentation the complex \(\NL_{B/k}\) can be represented by a complex \(N^{-1} \to N^0\) where \(N^i\) is a finite \(B\)-module, see Algebra, Section 00S0 and in particular Algebra, Lemma 00S1. As \(B\) is Noetherian, this means that \(\NL_{B/k}\) is pseudo-coherent. For \(g \in J\) the \(k\)-algebra \(B_g\) is smooth and hence \((\NL_{B/k})_g = \NL_{B_g/k}\) is quasi-isomorphic to a finite projective \(B\)-module sitting in degree \(0\). Thus \(\text{Ext}^i_B(\NL_{B/k}, N)_g = 0\) for \(i \geq 1\) and any \(B\)-module \(N\). By More on Algebra, Lemma 0DYJ we conclude that \[\text{Ext}^1_B(\NL_{B/k}, N) \longrightarrow \lim_n \text{Ext}^1_B(\NL_{B/k}, N/J^n N)\] is an isomorphism for any finite \(B\)-module \(N\).

Injectivity of the map. Suppose that \(0 \to N \to C \to B \to 0\) is in \(\text{Exal}_k(B, N)\) and maps to zero in \(\text{Exal}_k(B^\wedge, N^\wedge)\). Choose a splitting \(C^\wedge = B^\wedge \oplus N^\wedge\). Then the induced map \(C \to C^\wedge \to N^\wedge\) gives maps \(C \to N/J^nN\) for all \(n\). Hence we see that our element is in the kernel of the maps \[\text{Ext}^1_B(\NL_{B/k}, N) \to \text{Ext}^1_B(\NL_{B/k}, N/J^n N)\] for all \(n\). By the previous paragraph we conclude that our element is zero.

Surjectivity of the map. Let \(0 \to N^\wedge \to C' \to B^\wedge \to 0\) be an element of \(\text{Exal}_k(B^\wedge, N^\wedge)\). Pulling back by \(B \to B^\wedge\) we get an element \(0 \to N^\wedge \to C'' \to B \to 0\) in \(\text{Exal}_k(B, N^\wedge)\). we have \[\text{Ext}^1_B(\NL_{B/k}, N^\wedge) = \text{Ext}^1_B(\NL_{B/k}, N) \otimes_B B^\wedge = \text{Ext}^1_B(\NL_{B/k}, N)\] The first equality as \(N^\wedge = N \otimes_B B^\wedge\) (Algebra, Lemma 00MA) and More on Algebra, Lemma 087Q. The second equality because \(\text{Ext}^1_B(\NL_{B/k}, N)\) is \(J\)-power torsion (see above), \(B \to B^\wedge\) is flat and induces an isomorphism \(B/J \to B^\wedge/JB^\wedge\), and More on Algebra, Lemma 05EC. Thus we can find a \(C \in \text{Exal}_k(B, N)\) mapping to \(C''\) in \(\text{Exal}_k(B, N^\wedge)\). Thus \[0 \to N^\wedge \to C' \to B^\wedge \to 0 \quad\text{and}\quad 0 \to N^\wedge \to C^\wedge \to B^\wedge \to 0\] are two elements of \(\text{Exal}_k(B^\wedge, N^\wedge)\) mapping to the same element of \(\text{Exal}_k(B, N^\wedge)\). Taking the difference we get an element \(0 \to N^\wedge \to C' \to B^\wedge \to 0\) of \(\text{Exal}_k(B^\wedge, N^\wedge)\) whose image in \(\text{Exal}_k(B, N^\wedge)\) is zero. This means there exists \[\xymatrix{ 0 \ar[r] & N^\wedge \ar[r] & C' \ar[r] & B^\wedge \ar[r] & 0 \\ & & B \ar[u]^\sigma \ar[ru] }\] Let \(J' \subset C'\) be the inverse image of \(JB^\wedge \subset B^\wedge\). To finish the proof it suffices to note that \(\sigma\) is continuous for the \(J\)-adic topology on \(B\) and the \(J'\)-adic topology on \(C'\) and that \(C'\) is \(J'\)-adically complete by Algebra, Lemma 0DYC (here we also use that \(C'\) is Noetherian; small detail omitted). Namely, this means that \(\sigma\) factors through the completion \(B^\wedge\) and \(C' = 0\) in \(\text{Exal}_k(B^\wedge, N^\wedge)\).

Lemma

In Example 0DY1 let \(P\) be a \(k\)-algebra. Let \(J \subset P\) be an ideal. Denote \(P^\wedge\) the \(J\)-adic completion. If

1. \(k \to P\) is of finite type, and

2. \(\Spec(P) \to \Spec(k)\) is smooth on the complement of \(V(J)\).

then the functor between deformation categories of Lemma 0DYV \[\Deformationcategory_P \longrightarrow \Deformationcategory_{P^\wedge}\] is smooth and induces an isomorphism on tangent spaces.

Proof

We know that \(\Deformationcategory_P\) and \(\Deformationcategory_{P^\wedge}\) are deformation categories by Lemma 0DY2. Thus it suffices to check our functor identifies tangent spaces and a correspondence between liftability, see Formal Deformation Theory, Lemma 0DYP. The property on liftability is proven in Lemma 0DZ3 and the isomorphism on tangent spaces is the special case of Lemma 0DZ4 where \(N = B\).

Deformations of localizations

In this section we compare the deformation problem posed by an algebra and its localization at a multiplicative subset. We first discuss “liftability”.

Lemma

Let \(A' \to A\) be a surjective map of Noetherian rings with nilpotent kernel. Let \(A \to B\) be a finite type flat ring map. Let \(S \subset B\) be a multiplicative subset such that if \(\Spec(B) \to \Spec(A)\) is not syntomic at \(\mathfrak q\), then \(S \cap \mathfrak q = \emptyset\). Then \(B\) has a flat lift to \(A'\) if and only if \(S^{-1}B\) has a flat lift to \(A'\).

Proof

This proof is the same as the proof of Lemma 0DZ3 but easier. We suggest the reader to skip the proof. Choose an \(A\)-algebra surjection \(P = A[x_1, \ldots, x_n] \to B\). Let \(S_P \subset P\) be the inverse image of \(S\). Set \(P' = A'[x_1, \ldots, x_n]\) and denote \(S_{P'} \subset P'\) the inverse image of \(S_P\).

Suppose \(A' \to B'\) is a flat lift of \(A \to B\), in other words, \(A' \to B'\) is flat and there is an \(A\)-algebra isomorphism \(B = B' \otimes_{A'} A\). Then we can choose an \(A'\)-algebra map \(P' \to B'\) lifting the given surjection \(P \to B\). By Nakayama’s lemma (Algebra, Lemma 00DV) we find that \(B'\) is a quotient of \(P'\). In particular, we find that we can endow \(B'\) with an \(A'\)-flat \(P'\)-module structure lifting \(B\) as an \(A\)-flat \(P\)-module. Conversely, if we can lift \(B\) to a \(P'\)-module \(M'\) flat over \(A'\), then \(M'\) is a cyclic module \(M' \cong P'/J'\) (using Nakayama again) and setting \(B' = P'/J'\) we find a flat lift of \(B\) as an algebra.

Set \(C = S^{-1}B\). Suppose that \(A' \to C'\) is a flat lift of \(A \to C\). Elements of \(C'\) which map to invertible elements of \(C\) are invertible. We choose an \(A'\)-algebra map \(P' \to C'\) lifting the \(A\)-algebra map \(P \to C\). By the remark above these maps pass through localizations to give surjections \(S_P^{-1}P \to C\) and \(S_{P'}^{-1}P' \to C'\) (for the second use Nakayama’s lemma). In particular, we find that we can endow \(C'\) with an \(A'\)-flat \(S_{P'}^{-1}P'\)-module structure lifting \(C\) as an \(A\)-flat \(S_P^{-1}P\)-module. Conversely, if we can lift \(C\) to a \(S_{P'}^{-1}P'\)-module \(N'\) flat over \(A'\), then \(N'\) is a cyclic module \(N' \cong S_{P'}^{-1}P'/\tilde J\) (using Nakayama again) and setting \(C' = S_{P'}^{-1}P'/\tilde J\) we find a flat lift of \(C\) as an algebra.

The syntomic locus of a morphism of schemes is open by definition. Let \(J_B \subset B\) be an ideal cutting out the set of points in \(\Spec(B)\) where \(\Spec(B) \to \Spec(A)\) is not syntomic. Denote \(J_P \subset P\) and \(J_{P'} \subset P'\) the corresponding ideals. Observe that \(P' \to S_{P'}^{-1}P'\) is a flat ring map which induces an isomorphism \(P'/J_{P'} = S_{P'}^{-1}P'/J_{P'}S_{P'}^{-1}P'\) by our assumption on \(S\) in the lemma, namely, the assumption in the lemma is exactly that \(B/J_B = S^{-1}(B/J_B)\). We conclude that our lemma is a consequence of Lemma 0DZ2 provided we can show that \(B_g\) lifts to an \(A'\)-flat \(P'_g\)-module for \(g \in J_B\). However, the ring map \(A \to B_g\) is syntomic and hence lifts to an \(A'\)-flat algebra \(B'\) by Smoothing Ring Maps, Proposition 07M8. Since \(A' \to P'_g\) is smooth, we can lift \(P_g \to B_g\) to a surjective map \(P'_g \to B'\) as before and we get what we want.

Lemma

Let \(k\) be a field. Let \(B\) be a finite type \(k\)-algebra. Let \(S \subset B\) be a multiplicative subset ideal such that if \(\Spec(B) \to \Spec(k)\) is not smooth at \(\mathfrak q\) then \(S \cap \mathfrak q = \emptyset\). Let \(N\) be a finite \(B\)-module. Then there is a canonical bijection \[\text{Exal}_k(B, N) \to \text{Exal}_k(S^{-1}B, S^{-1}N)\]

Proof

This proof is the same as the proof of Lemma 0DZ4 but easier. We suggest the reader to skip the proof. The map is given by localization: given \(0 \to N \to C \to B \to 0\) in \(\text{Exal}_k(B, N)\) we send it to the localization \(S_C^{-1}C\) of \(C\) with respect to the inverse image \(S_C \subset C\) of \(S\). Compare with the proof of Lemma 0DYS.

The smooth locus of a morphism of schemes is open by definition. Let \(J \subset B\) be an ideal cutting out the set of points in \(\Spec(B)\) where \(\Spec(B) \to \Spec(A)\) is not smooth. Since \(k \to B\) is of finite presentation the complex \(\NL_{B/k}\) can be represented by a complex \(N^{-1} \to N^0\) where \(N^i\) is a finite \(B\)-module, see Algebra, Section 00S0 and in particular Algebra, Lemma 00S1. As \(B\) is Noetherian, this means that \(\NL_{B/k}\) is pseudo-coherent. For \(g \in J\) the \(k\)-algebra \(B_g\) is smooth and hence \((\NL_{B/k})_g = \NL_{B_g/k}\) is quasi-isomorphic to a finite projective \(B\)-module sitting in degree \(0\). Thus \(\text{Ext}^i_B(\NL_{B/k}, N)_g = 0\) for \(i \geq 1\) and any \(B\)-module \(N\). Finally, we have \[\text{Ext}^1_{S^{-1}B}(\NL_{S^{-1}B/k}, S^{-1}N) = \text{Ext}^1_B(\NL_{B/k}, N) \otimes_B S^{-1}B = \text{Ext}^1_B(\NL_{B/k}, N)\] The first equality by More on Algebra, Lemma 0A6A and Algebra, Lemma 00S7. The second because \(\text{Ext}^1_B(\NL_{B/k}, N)\) is \(J\)-power torsion and elements of \(S\) act invertibly on \(J\)-power torsion modules. This concludes the proof by the description of \(\text{Exal}_A(B, N)\) as \(\text{Ext}^1_B(\NL_{B/A}, N)\) given just above Lemma 0DZ4.

Lemma

In Example 0DY1 let \(P\) be a \(k\)-algebra. Let \(S \subset P\) be a multiplicative subset. If

1. \(k \to P\) is of finite type, and

2. \(\Spec(P) \to \Spec(k)\) is smooth at all points of \(V(g)\) for all \(g \in S\).

then the functor between deformation categories of Lemma 0DYS \[\Deformationcategory_P \longrightarrow \Deformationcategory_{S^{-1}P}\] is smooth and induces an isomorphism on tangent spaces.

Proof

We know that \(\Deformationcategory_P\) and \(\Deformationcategory_{S^{-1}P}\) are deformation categories by Lemma 0DY2. Thus it suffices to check our functor identifies tangent spaces and a correspondence between liftability, see Formal Deformation Theory, Lemma 0DYP. The property on liftability is proven in Lemma 0DZ7 and the isomorphism on tangent spaces is the special case of Lemma 0DZ8 where \(N = B\).

Deformations of henselizations

In this section we compare the deformation problem posed by an algebra and its completion. We first discuss “liftability”.

Lemma

Let \(A' \to A\) be a surjective map of Noetherian rings with nilpotent kernel. Let \(A \to B\) be a finite type flat ring map. Let \(\mathfrak b \subset B\) be an ideal such that \(\Spec(B) \to \Spec(A)\) is syntomic on the complement of \(V(\mathfrak b)\). Let \((B^h, \mathfrak b^h)\) be the henselization of the pair \((B, \mathfrak b)\). Then \(B\) has a flat lift to \(A'\) if and only if \(B^h\) has a flat lift to \(A'\).

Proof

This proof is a cheat. Namely, if \(B\) has a flat lift \(B'\), then taking the henselization \((B')^h\) we obtain a flat lift of \(B^h\) (compare with the proof of Lemma 0DYT). Conversely, suppose that \(C'\) is an \(A'\)-flat lift of \((B')^h\). Then let \(\mathfrak c' \subset C'\) be the inverse image of the ideal \(\mathfrak b^h\). Then the completion \((C')^\wedge\) of \(C'\) with respect to \(\mathfrak c'\) is a lift of \(B^\wedge\) (details omitted). Hence we see that \(B\) has a flat lift by Lemma 0DZ3.

Proof

Choose an \(A\)-algebra surjection \(P = A[x_1, \ldots, x_n] \to B\). Let \(\mathfrak p \subset P\) be the inverse image of \(\mathfrak b\). Set \(P' = A'[x_1, \ldots, x_n]\) and denote \(\mathfrak p' \subset P'\) the inverse image of \(\mathfrak p\). (Of course \(\mathfrak p\) and \(\mathfrak p'\) do not designate prime ideals here.) We will denote \(P^h\) and \((P')^h\) the respective henselizations. We will use that taking henselizations is functorial and that the henselization of a quotient is the corresponding quotient of the henselization, see More on Algebra, Lemmas 09Y6 and 0DYE.

Suppose \(A' \to B'\) is a flat lift of \(A \to B\), in other words, \(A' \to B'\) is flat and there is an \(A\)-algebra isomorphism \(B = B' \otimes_{A'} A\). Then we can choose an \(A'\)-algebra map \(P' \to B'\) lifting the given surjection \(P \to B\). By Nakayama’s lemma (Algebra, Lemma 00DV) we find that \(B'\) is a quotient of \(P'\). In particular, we find that we can endow \(B'\) with an \(A'\)-flat \(P'\)-module structure lifting \(B\) as an \(A\)-flat \(P\)-module. Conversely, if we can lift \(B\) to a \(P'\)-module \(M'\) flat over \(A'\), then \(M'\) is a cyclic module \(M' \cong P'/J'\) (using Nakayama again) and setting \(B' = P'/J'\) we find a flat lift of \(B\) as an algebra.

Set \(C = B^h\) and \(\mathfrak c = \mathfrak bC\). Suppose that \(A' \to C'\) is a flat lift of \(A \to C\). Then \(C'\) is henselian with respect to the inverse image \(\mathfrak c'\) of \(\mathfrak c\) (by More on Algebra, Lemma 0DYD and the fact that the kernel of \(C' \to C\) is nilpotent). We choose an \(A'\)-algebra map \(P' \to C'\) lifting the \(A\)-algebra map \(P \to C\). These maps pass through henselizations to give surjections \(P^h \to C\) and \((P')^h \to C'\) (for the second again using Nakayama’s lemma). In particular, we find that we can endow \(C'\) with an \(A'\)-flat \((P')^h\)-module structure lifting \(C\) as an \(A\)-flat \(P^h\)-module. Conversely, if we can lift \(C\) to a \((P')^h\)-module \(N'\) flat over \(A'\), then \(N'\) is a cyclic module \(N' \cong (P')^h/\tilde J\) (using Nakayama again) and setting \(C' = (P')^h/\tilde J\) we find a flat lift of \(C\) as an algebra.

Observe that \(P' \to (P')^h\) is a flat ring map which induces an isomorphism \(P'/\mathfrak p' = (P')^h/\mathfrak p'(P')^h\) (More on Algebra, Lemma 0AGU). We conclude that our lemma is a consequence of Lemma 0DZ2 provided we can show that \(B_g\) lifts to an \(A'\)-flat \(P'_g\)-module for \(g \in \mathfrak p'\). However, the ring map \(A \to B_g\) is syntomic and hence lifts to an \(A'\)-flat algebra \(B'\) by Smoothing Ring Maps, Proposition 07M8. Since \(A' \to P'_g\) is smooth, we can lift \(P_g \to B_g\) to a surjective map \(P'_g \to B'\) as before and we get what we want.

Lemma

Let \(k\) be a field. Let \(B\) be a finite type \(k\)-algebra. Let \(J \subset B\) be an ideal such that \(\Spec(B) \to \Spec(k)\) is smooth on the complement of \(V(J)\). Let \(N\) be a finite \(B\)-module. Then there is a canonical bijection \[\text{Exal}_k(B, N) \to \text{Exal}_k(B^h, N^h)\] Here \((B^h, J^h)\) is the henselization of \((B, J)\) and \(N^h = N \otimes_B B^h\).

Proof

This proof is the same as the proof of Lemma 0DZ4 but easier. We suggest the reader to skip the proof. The map is given by henselization: given \(0 \to N \to C \to B \to 0\) in \(\text{Exal}_k(B, N)\) we send it to the henselization \(C^h\) of \(C\) with respect to the inverse image \(J_C \subset C\) of \(J\). Compare with the proof of Lemma 0DYT.

Since \(k \to B\) is of finite presentation the complex \(\NL_{B/k}\) can be represented by a complex \(N^{-1} \to N^0\) where \(N^i\) is a finite \(B\)-module, see Algebra, Section 00S0 and in particular Algebra, Lemma 00S1. As \(B\) is Noetherian, this means that \(\NL_{B/k}\) is pseudo-coherent. For \(g \in J\) the \(k\)-algebra \(B_g\) is smooth and hence \((\NL_{B/k})_g = \NL_{B_g/k}\) is quasi-isomorphic to a finite projective \(B\)-module sitting in degree \(0\). Thus \(\text{Ext}^i_B(\NL_{B/k}, N)_g = 0\) for \(i \geq 1\) and any \(B\)-module \(N\). Finally, we have \[\begin{align*} \text{Ext}^1_{B^h}(\NL_{B^h/k}, N^h) & = \text{Ext}^1_{B^h}(\NL_{B/k} \otimes_B B^h, N \otimes_B B^h) \\ & = \text{Ext}^1_B(\NL_{B/k}, N) \otimes_B B^h \\ & = \text{Ext}^1_B(\NL_{B/k}, N) \end{align*}\] The first equality by More on Algebra, Lemma 0D08 (or rather its analogue for henselizations of pairs). The second by More on Algebra, Lemma 0A6A. The third because \(\text{Ext}^1_B(\NL_{B/k}, N)\) is \(J\)-power torsion, the map \(B \to B^h\) is flat and induces an isomorphism \(B/J \to B^h/JB^h\) (More on Algebra, Lemma 0AGU), and More on Algebra, Lemma 05EC. This concludes the proof by the description of \(\text{Exal}_A(B, N)\) as \(\text{Ext}^1_B(\NL_{B/A}, N)\) given just above Lemma 0DZ4.

Lemma

In Example 0DY1 let \(P\) be a \(k\)-algebra. Let \(J \subset P\) be an ideal. Denote \((P^h, J^h)\) the henselization of the pair \((P, J)\). If

1. \(k \to P\) is of finite type, and

2. \(\Spec(P) \to \Spec(k)\) is smooth on the complement of \(V(J)\),

then the functor between deformation categories of Lemma 0DYT \[\Deformationcategory_P \longrightarrow \Deformationcategory_{P^h}\] is smooth and induces an isomorphism on tangent spaces.

Proof

We know that \(\Deformationcategory_P\) and \(\Deformationcategory_{P^h}\) are deformation categories by Lemma 0DY2. Thus it suffices to check our functor identifies tangent spaces and a correspondence between liftability, see Formal Deformation Theory, Lemma 0DYP. The property on liftability is proven in Lemma 0DZB and the isomorphism on tangent spaces is the special case of Lemma 0DZC where \(N = B\).

Application to isolated singularities

We apply the discussion above to study the deformation theory of a finite type algebra with finitely many singular points.

Lemma

In Example 0DY1 let \(P\) be a \(k\)-algebra. Assume that \(k \to P\) is of finite type and that \(\Spec(P) \to \Spec(k)\) is smooth except at the maximal ideals \(\mathfrak m_1, \ldots, \mathfrak m_n\) of \(P\). Let \(P_{\mathfrak m_i}\), \(P_{\mathfrak m_i}^h\), \(P_{\mathfrak m_i}^\wedge\) be the local ring, henselization, completion. Then the maps of deformation categories \[\Deformationcategory_P \to \prod \Deformationcategory_{P_{\mathfrak m_i}} \to \prod \Deformationcategory_{P_{\mathfrak m_i}^h} \to \prod \Deformationcategory_{P_{\mathfrak m_i}^\wedge}\] are smooth and induce isomorphisms on their finite dimensional tangent spaces.

Proof

The tangent space is finite dimensional by Lemma 0DY4. The functors between the categories are constructed in Lemmas 0DYS, 0DYT, and 0DYV (we omit some verifications of the form: the completion of the henselization is the completion).

Set \(J = \mathfrak m_1 \cap \ldots \cap \mathfrak m_n\) and apply Lemma 0DZ5 to get that \(\Deformationcategory_P \to \Deformationcategory_{P^\wedge}\) is smooth and induces an isomorphism on tangent spaces where \(P^\wedge\) is the \(J\)-adic completion of \(P\). However, since \(P^\wedge = \prod P_{\mathfrak m_i}^\wedge\) we see that the map \(\Deformationcategory_P \to \prod \Deformationcategory_{P_{\mathfrak m_i}^\wedge}\) is smooth and induces an isomorphism on tangent spaces.

Let \((P^h, J^h)\) be the henselization of the pair \((P, J)\). Then \(P^h = \prod P_{\mathfrak m_i}^h\) (look at idempotents and use More on Algebra, Lemma 09XI). Hence we can apply Lemma 0DZD to conclude as in the case of completion.

To get the final case it suffices to show that \(\Deformationcategory_{P_{\mathfrak m_i}} \to \Deformationcategory_{P_{\mathfrak m_i}^\wedge}\) is smooth and induce isomorphisms on tangent spaces for each \(i\) separately. To do this, we may replace \(P\) by a principal localization whose only singular point is a maximal ideal \(\mathfrak m\) (corresponding to \(\mathfrak m_i\) in the original \(P\)). Then we can apply Lemma 0DZ9 with multiplicative subset \(S = P \setminus \mathfrak m\) to conclude. Minor details omitted.

Unobstructed deformation problems

Let \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) be a category cofibred in groupoids. Recall that we say \(\mathcal{F}\) is smooth or unobstructed if \(p\) is smooth. This means that given a surjection \(\varphi : A' \to A\) in \(\mathcal{C}_\Lambda\) and \(x \in \Ob(\mathcal{F}(A))\) there exists a morphism \(f : x' \to x\) in \(\mathcal{F}\) with \(p(f) = \varphi\). See Formal Deformation Theory, Section 0DYK. In this section we give some geometrically meaningful examples.

Lemma

In Example 0DY1 let \(P\) be a local complete intersection over \(k\) (Algebra, Definition 00S9). Then \(\Deformationcategory_P\) is unobstructed.

Proof

Let \((A, Q) \to (k, P)\) be an object of \(\Deformationcategory_P\). Then we see that \(A \to Q\) is a syntomic ring map by Algebra, Definition 00SL. Hence for any surjection \(A' \to A\) in \(\mathcal{C}_\Lambda\) we see that there is a morphism \((A', Q') \to (A, Q)\) lifting \(A' \to A\) by Smoothing Ring Maps, Proposition 07M8. This proves the lemma.

Lemma

In Situation 0DYY if \(U_{12} \to \Spec(k)\) is smooth, then the morphism \[\Deformationcategory_X \longrightarrow \Deformationcategory_{U_1} \times \Deformationcategory_{U_2} = \Deformationcategory_{P_1} \times \Deformationcategory_{P_2}\] is smooth. If in addition \(U_1\) is a local complete intersection over \(k\), then \[\Deformationcategory_X \longrightarrow \Deformationcategory_{U_2} = \Deformationcategory_{P_2}\] is smooth.

Proof

The equality signs hold by Lemma 0DYX. Let us think of \(\mathcal{C}_\Lambda\) as a deformation category over \(\mathcal{C}_\Lambda\) as in Formal Deformation Theory, Section 0DYK. Then \[\Deformationcategory_{P_1} \times \Deformationcategory_{P_2} = \Deformationcategory_{P_1} \times_{\mathcal{C}_\Lambda} \Deformationcategory_{P_2},\] see Formal Deformation Theory, Remarks 06GK (0DZJ). Using Lemma 0DYZ the first statement is that the functor \[\Deformationcategory_{P_1} \times_{\Deformationcategory_{P_{12}}} \Deformationcategory_{P_2} \longrightarrow \Deformationcategory_{P_1} \times_{\mathcal{C}_\Lambda} \Deformationcategory_{P_2}\] is smooth. This follows from Formal Deformation Theory, Lemma 0DYN as long as we can show that \(T\Deformationcategory_{P_{12}} = (0)\). This vanishing follows from Lemma 0DZL as \(P_{12}\) is smooth over \(k\). For the second statement it suffices to show that \(\Deformationcategory_{P_1} \to \mathcal{C}_\Lambda\) is smooth, see Formal Deformation Theory, Lemma 06HM. In other words, we have to show \(\Deformationcategory_{P_1}\) is unobstructed, which is Lemma 0DZH.

Lemma

In Example 0DY7 let \(X\) be a scheme over \(k\). Assume

1. \(X\) is separated, finite type over \(k\) and \(\dim(X) \leq 1\),

2. \(X \to \Spec(k)\) is smooth except at the closed points \(p_1, \ldots, p_n \in X\).

Let \(\mathcal{O}_{X, p_1}\), \(\mathcal{O}_{X, p_1}^h\), \(\mathcal{O}_{X, p_1}^\wedge\) be the local ring, henselization, completion. Consider the maps of deformation categories \[\Deformationcategory_X \longrightarrow \prod \Deformationcategory_{\mathcal{O}_{X, p_i}} \longrightarrow \prod \Deformationcategory_{\mathcal{O}_{X, p_i}^h} \longrightarrow \prod \Deformationcategory_{\mathcal{O}_{X, p_i}^\wedge}\] The first arrow is smooth and the second and third arrows are smooth and induce isomorphisms on tangent spaces.

Proof

Choose an affine open \(U_2 \subset X\) containing \(p_1, \ldots, p_n\) and the generic point of every irreducible component of \(X\). This is possible by Varieties, Lemma 0A25 and Properties, Lemma 01ZY. Then \(X \setminus U_2\) is finite and we can choose an affine open \(U_1 \subset X \setminus \{p_1, \ldots, p_n\}\) such that \(X = U_1 \cup U_2\). Set \(U_{12} = U_1 \cap U_2\). Then \(U_1\) and \(U_{12}\) are smooth affine schemes over \(k\). We conclude that \[\Deformationcategory_X \longrightarrow \Deformationcategory_{U_2}\] is smooth by Lemma 0DZN. Applying Lemmas 0DYX and 0DZF we win.

Lemma

In Example 0DY7 let \(X\) be a scheme over \(k\). Assume

1. \(X\) is separated, finite type over \(k\) and \(\dim(X) \leq 1\),

2. \(X\) is a local complete intersection over \(k\), and

3. \(X \to \Spec(k)\) is smooth except at finitely many points.

Then \(\Deformationcategory_X\) is unobstructed.

Proof

Let \(p_1, \ldots, p_n \in X\) be the points where \(X \to \Spec(k)\) isn’t smooth. Choose an affine open \(U_2 \subset X\) containing \(p_1, \ldots, p_n\) and the generic point of every irreducible component of \(X\). This is possible by Varieties, Lemma 0A25 and Properties, Lemma 01ZY. Then \(X \setminus U_2\) is finite and we can choose an affine open \(U_1 \subset X \setminus \{p_1, \ldots, p_n\}\) such that \(X = U_1 \cup U_2\). Set \(U_{12} = U_1 \cap U_2\). Then \(U_1\) and \(U_{12}\) are smooth affine schemes over \(k\). We conclude that \[\Deformationcategory_X \longrightarrow \Deformationcategory_{U_2}\] is smooth by Lemma 0DZN. Applying Lemmas 0DYX and 0DZH we win.

Smoothings

Suppose given a finite type scheme or algebraic space \(X\) over a field \(k\). It is often useful to find a flat morphism of finite type \(Y \to \Spec(k[[t]])\) whose generic fibre is smooth and whose special fibre is isomorphic to \(X\). Such a thing is called a smoothing of \(X\). In this section we will find a smoothing for \(1\)-dimensional separated \(X\) which have isolated local complete intersection singularities.

Lemma

Let \(k\) be a field. Set \(S = \Spec(k[[t]])\) and \(S_n = \Spec(k[t]/(t^n))\). Let \(Y \to S\) be a proper, flat morphism of schemes whose special fibre \(X\) is Cohen-Macaulay and equidimensional of dimension \(d\). Denote \(X_n = Y \times_S S_n\). If for some \(n \geq 1\) the \(d\)th Fitting ideal of \(\Omega_{X_n/S_n}\) contains \(t^{n - 1}\), then the generic fibre of \(Y \to S\) is smooth.

Proof

By More on Morphisms, Lemma 045U we see that \(Y \to S\) is a Cohen-Macaulay morphism. By Morphisms, Lemma 02NM we see that \(Y \to S\) has relative dimension \(d\). By Divisors, Lemma 0C3K the \(d\)th Fitting ideal \(\mathcal{I} \subset \mathcal{O}_Y\) of \(\Omega_{Y/S}\) cuts out the singular locus of the morphism \(Y \to S\). In other words, \(V(\mathcal{I}) \subset Y\) is the closed subset of points where \(Y \to S\) is not smooth. By Divisors, Lemma 0C3I formation of this Fitting ideal commutes with base change. By assumption we see that \(t^{n - 1}\) is a section of \(\mathcal{I} + t^n\mathcal{O}_Y\). Thus for every \(x \in X = V(t) \subset Y\) we conclude that \(t^{n - 1} \in \mathcal{I}_x\) where \(\mathcal{I}_x\) is the stalk at \(x\). This implies that \(V(\mathcal{I}) \subset V(t)\) in an open neighbourhood of \(X\) in \(Y\). Since \(Y \to S\) is proper, this implies \(V(\mathcal{I}) \subset V(t)\) as desired.

Lemma

Let \(k\) be a field. Let \(1 \leq c \leq n\) be integers. Let \(f_1, \ldots, f_c \in k[x_1, \ldots x_n]\) be elements. Let \(a_{ij}\), \(0 \leq i \leq n\), \(1 \leq j \leq c\) be variables. Consider \[g_j = f_j + a_{0j} + a_{1j}x_1 + \ldots + a_{nj}x_n \in k[a_{ij}][x_1, \ldots, x_n]\] Denote \(Y \subset \mathbf{A}^{n + c(n + 1)}_k\) the closed subscheme cut out by \(g_1, \ldots, g_c\). Denote \(\pi : Y \to \mathbf{A}^{c(n + 1)}_k\) the projection onto the affine space with variables \(a_{ij}\). Then there is a nonempty Zariski open of \(\mathbf{A}^{c(n + 1)}_k\) over which \(\pi\) is smooth.

Proof

Recall that the set of points where \(\pi\) is smooth is open. Thus the complement, i.e., the singular locus, is closed. By Chevalley’s theorem (in the form of Morphisms, Lemma 054J) the image of the singular locus is constructible. Hence if the generic point of \(\mathbf{A}^{c(n + 1)}_k\) is not in the image of the singular locus, then the lemma follows (by Topology, Lemma 005K for example). Thus we have to show there is no point \(y \in Y\) where \(\pi\) is not smooth mapping to the generic point of \(\mathbf{A}^{c(n + 1)}_k\). Consider the matrix of partial derivatives \[(\frac{\partial g_j}{\partial x_i}) = (\frac{\partial f_j}{\partial x_i} + a_{ij})\] The image of this matrix in \(\kappa(y)\) must have rank \(< c\) since otherwise \(\pi\) would be smooth at \(y\), see discussion in Smoothing Ring Maps, Section 07C4. Thus we can find \(\lambda_1, \ldots, \lambda_c \in \kappa(y)\) not all zero such that the vector \((\lambda_1, \ldots, \lambda_c)\) is in the kernel of this matrix. After renumbering we may assume \(\lambda_1 \not = 0\). Dividing by \(\lambda_1\) we may assume our vector has the form \((1, \lambda_2, \ldots, \lambda_c)\). Then we obtain \[a_{i1} = - \frac{\partial f_j}{\partial x_1} - \sum\nolimits_{j = 2, \ldots, c} \lambda_j(\frac{\partial f_j}{\partial x_i} + a_{ij})\] in \(\kappa(y)\) for \(i = 1, \ldots, n\). Moreover, since \(y \in Y\) we also have \[a_{0j} = -f_j - a_{1j}x_1 - \ldots - a_{nj}x_n\] in \(\kappa(y)\). This means that the subfield of \(\kappa(y)\) generated by \(a_{ij}\) is contained in the subfield of \(\kappa(y)\) generated by the images of \(x_1, \ldots, x_n, \lambda_2, \ldots, \lambda_c\), and \(a_{ij}\) except for \(a_{i1}\) and \(a_{0j}\). We count and we see that the transcendence degree of this is at most \(c(n + 1) - 1\). Hence \(y\) cannot map to the generic point as desired.

Lemma

Let \(k\) be a field. Let \(A\) be a global complete intersection over \(k\). There exists a flat finite type ring map \(k[[t]] \to B\) with \(B/tB \cong A\) such that \(B[1/t]\) is smooth over \(k((t))\).

Proof

Write \(A = k[x_1, \ldots, x_n]/(f_1, \ldots, f_c)\) as in Algebra, Definition 00S9. We are going to choose \(a_{ij} \in (t) \subset k[[t]]\) and set \[g_j = f_j + a_{0j} + a_{1j}x_1 + \ldots + a_{nj}x_n \in k[[t]][x_1, \ldots, x_n]\] After doing this we take \(B = k[[t]][x_1, \ldots, x_n]/(g_1, \ldots, g_c)\). We claim that \(k[[t]] \to B\) is flat at every prime ideal lying over \((t)\). Namely, the elements \(f_1, \ldots, f_c\) form a regular sequence in the local ring at any prime ideal \(\mathfrak p\) of \(k[x_1, \ldots, x_n]\) containing \(f_1, \ldots, f_c\) (Algebra, Lemma 00SC). Thus \(g_1, \ldots, g_c\) is locally a lift of a regular sequence and we can apply Algebra, Lemma 00MG. Flatness at primes lying over \((0) \subset k[[t]]\) is automatic because \(k((t)) = k[[t]]_{(0)}\) is a field. Thus \(B\) is flat over \(k[[t]]\).

All that remains is to show that for suitable choices of \(a_{ij}\) the generic fibre \(B_{(0)}\) is smooth over \(k((t))\). For this we have to show that we can choose our \(a_{ij}\) so that the induced morphism \[(a_{ij}) : \Spec(k[[t]]) \longrightarrow \mathbf{A}^{c(n + 1)}_k\] maps into the nonempty Zariski open of Lemma 0E7U. This is clear because there is no nonzero polynomial in the \(a_{ij}\) which vanishes on \((t)^{\oplus c(n + 1)}\). (We leave this as an exercise to the reader.)

Lemma

Let \(k\) be a field. Let \(A\) be a finite dimensional \(k\)-algebra which is a local complete intersection over \(k\). Then there is a finite flat \(k[[t]]\)-algebra \(B\) with \(B/tB \cong A\) and \(B[1/t]\) étale over \(k((t))\).

Proof

Since \(A\) is Artinian (Algebra, Lemma 00J6), we can write \(A\) as a product of local Artinian rings (Algebra, Lemma 00JB). Thus it suffices to prove the lemma if \(A\) is local (this uses that being a local complete intersection is preserved under taking principal localizations, see Algebra, Lemma 00SA). In this case \(A\) is a global complete intersection. Consider the algebra \(B\) constructed in Lemma 0E7V. Then \(k[[t]] \to B\) is quasi-finite at the unique prime of \(B\) lying over \((t)\) (Algebra, Definition 00PL). Observe that \(k[[t]]\) is a henselian local ring (Algebra, Lemma 04GM). Thus \(B = B' \times C\) where \(B'\) is finite over \(k[[t]]\) and \(C\) has no prime lying over \((t)\), see Algebra, Lemma 04GG. Then \(B'\) is the ring we are looking for (recall that étale is the same thing as smooth of relative dimension \(0\)).

Lemma

Let \(k\) be a field. Let \(A\) be a \(k\)-algebra. Assume

1. \(A\) is a local ring essentially of finite type over \(k\),

2. \(A\) is a complete intersection over \(k\) (Algebra, Definition 00SD).

Set \(d = \dim(A) + \text{trdeg}_k(\kappa)\) where \(\kappa\) is the residue field of \(A\). Then there exists an integer \(n\) and a flat, essentially of finite type ring map \(k[[t]] \to B\) with \(B/tB \cong A\) such that \(t^n\) is in the \(d\)th Fitting ideal of \(\Omega_{B/k[[t]]}\).

Proof

By Algebra, Lemma 00SF we can write \(A\) as the localization at a prime \(\mathfrak p\) of a global complete intersection \(P\) over \(k\). Observe that \(\dim(P) = d\) by Algebra, Lemma 00P1. By Lemma 0E7V we can find a flat, finite type ring map \(k[[t]] \to Q\) such that \(P \cong Q/tQ\) and such that \(k((t)) \to Q[1/t]\) is smooth. It follows from the construction of \(Q\) in the lemma that \(k[[t]] \to Q\) is a relative global complete intersection of relative dimension \(d\); alternatively, Algebra, Lemma 00SY tells us that \(Q\) or a suitable principal localization of \(Q\) is such a global complete intersection. Hence by Divisors, Lemma 0C3K the \(d\)th Fitting ideal \(I \subset Q\) of \(\Omega_{Q/k[[t]]}\) cuts out the singular locus of \(\Spec(Q) \to \Spec(k[[t]])\). Thus \(t^n \in I\) for some \(n\). Let \(\mathfrak q \subset Q\) be the inverse image of \(\mathfrak p\). Set \(B = Q_\mathfrak q\). The lemma is proved.

Lemma

Let \(X\) be a scheme over a field \(k\). Assume

1. \(X\) is proper over \(k\),

2. \(X\) is a local complete intersection over \(k\),

3. \(X\) has dimension \(\leq 1\), and

4. \(X \to \Spec(k)\) is smooth except at finitely many points.

Then there exists a flat projective morphism \(Y \to \Spec(k[[t]])\) whose generic fibre is smooth and whose special fibre is isomorphic to \(X\).

Proof

Observe that \(X\) is Cohen-Macaulay, see Algebra, Lemma 00SB. Thus \(X = X' \amalg X''\) with \(\dim(X') = 0\) and \(X''\) equidimensional of dimension \(1\), see Morphisms, Lemma 02NM. Since \(X'\) is finite over \(k\) (Varieties, Lemma 06LH) we can find \(Y' \to \Spec(k[[t]])\) with special fibre \(X'\) and generic fibre smooth by Lemma 0E7W. Thus it suffices to prove the lemma for \(X''\). After replacing \(X\) by \(X''\) we have \(X\) is Cohen-Macaulay and equidimensional of dimension \(1\).

We are going to use deformation theory for the situation \(\Lambda = k \to k\). Let \(p_1, \ldots, p_r \in X\) be the closed singular points of \(X\), i.e., the points where \(X \to \Spec(k)\) isn’t smooth. For each \(i\) we pick an integer \(n_i\) and a flat, essentially of finite type ring map \[k[[t]] \longrightarrow B_i\] with \(B_i/tB_i \cong \mathcal{O}_{X, p_i}\) such that \(t^{n_i}\) is in the \(1\)st Fitting ideal of \(\Omega_{B_i/k[[t]]}\). This is possible by Lemma 0E7X. Observe that the system \((B_i/t^nB_i)\) defines a formal object of \(\Deformationcategory_{\mathcal{O}_{X, p_i}}\) over \(k[[t]]\). By Lemma 0DZP the map \[\Deformationcategory_X \longrightarrow \prod\nolimits_{i = 1, \ldots, r} \Deformationcategory_{\mathcal{O}_{X, p_i}}\] is a smooth map between deformation categories. Hence by Formal Deformation Theory, Lemma 06HN there exists a formal object \((X_n)\) in \(\Deformationcategory_X\) mapping to the formal object \(\prod_i (B_i/t^n)\) by the arrow above. By More on Morphisms of Spaces, Lemma 0E7R there exists a projective scheme \(Y\) over \(k[[t]]\) and compatible isomorphisms \(Y \times_{\Spec(k[[t]])} \Spec(k[t]/(t^n)) \cong X_n\). By More on Morphisms, Lemma 0D4G we see that \(Y \to \Spec(k[[t]])\) is flat. Since \(X\) is Cohen-Macaulay and equidimensional of dimension \(1\) we may apply Lemma 0E7T to check \(Y\) has smooth generic fibre³. Choose \(n\) strictly larger than the maximum of the integers \(n_i\) found above. If we can show \(t^{n - 1}\) is in the first Fitting ideal of \(\Omega_{X_n/S_n}\) with \(S_n = \Spec(k[t]/(t^n))\), then the proof is done. To do this it suffices to prove this is true in each of the local rings of \(X_n\) at closed points \(p\). However, if \(p\) corresponds to a smooth point for \(X \to \Spec(k)\), then \(\Omega_{X_n/S_n, p}\) is free of rank \(1\) and the first Fitting ideal is equal to the local ring. If \(p = p_i\) for some \(i\), then \[\Omega_{X_n/S_n, p_i} = \Omega_{(B_i/t^nB_i)/(k[t]/(t^n))} = \Omega_{B_i/k[[t]]}/t^n\Omega_{B_i/k[[t]]}\] Since taking Fitting ideals commutes with base change (with already used this but in this algebraic setting it follows from More on Algebra, Lemma 07ZA), and since \(n - 1 \geq n_i\) we see that \(t^{n - 1}\) is in the Fitting ideal of this module over \(B_i/t^nB_i\) as desired.

Lemma

Let \(k\) be a field and let \(X\) be a scheme over \(k\). Assume

1. \(X\) is separated, finite type over \(k\) and \(\dim(X) \leq 1\),

2. \(X\) is a local complete intersection over \(k\), and

3. \(X \to \Spec(k)\) is smooth except at finitely many points.

Then there exists a flat, separated, finite type morphism \(Y \to \Spec(k[[t]])\) whose generic fibre is smooth and whose special fibre is isomorphic to \(X\).

Proof

If \(X\) is reduced, then we can choose an embedding \(X \subset \overline{X}\) as in Varieties, Lemma 0BXW. Writing \(X = \overline{X} \setminus \{x_1, \ldots, x_n\}\) we see that \(\mathcal{O}_{\overline{X}, x_i}\) is a discrete valuation ring and hence in particular a local complete intersection (Algebra, Definition 00SD). Thus \(\overline{X}\) is a local complete intersection over \(k\) because this holds over the open \(X\) and at the points \(x_i\) by Algebra, Lemma 00SF. Thus we may apply Lemma 0E7Y to find a projective flat morphism \(\overline{Y} \to \Spec(k[[t]])\) whose generic fibre is smooth and whose special fibre is \(\overline{X}\). Then we remove \(x_1, \ldots, x_n\) from \(\overline{Y}\) to obtain \(Y\).

In the general case, write \(X = X' \amalg X''\) where with \(\dim(X') = 0\) and \(X''\) equidimensional of dimension \(1\). Then \(X''\) is reduced and the first paragraph applies to it. On the other hand, \(X'\) can be dealt with as in the proof of Lemma 0E7Y. Some details omitted.