Formal Deformation Theory

Introduction

This chapter develops formal deformation theory in a form applicable later in the Stacks project, closely following Rim [SGA7-I, Exposee VI] and Schlessinger [Sch]. We strongly encourage the reader new to this topic to read the paper by Schlessinger first, as it is sufficiently general for most applications, and Schlessinger’s results are indeed used in most papers that use this kind of formal deformation theory.

Let \(\Lambda\) be a complete Noetherian local ring with residue field \(k\), and let \(\mathcal{C}_\Lambda\) denote the category of Artinian local \(\Lambda\)-algebras with residue field \(k\). Given a functor \(F : \mathcal{C}_\Lambda \to \textit{Sets}\) such that \(F(k)\) is a one element set, Schlessinger’s paper introduced conditions (H1)-(H4) such that:

\(F\) has a “hull” if and only if (H1)-(H3) hold.
\(F\) is prorepresentable if and only if (H1)-(H4) hold.

The purpose of this chapter is to generalize these results in two ways exactly as is done in Rim’s paper:

The functor \(F\) is replaced by a category \(\mathcal{F}\) cofibered in groupoids over \(\mathcal{C}_\Lambda\), see Section 06GB.
We let \(\Lambda\) be a Noetherian ring and \(\Lambda \to k\) a finite ring map to a field. The category \(\mathcal{C}_\Lambda\) is the category of Artinian local \(\Lambda\)-algebras \(A\) endowed with a given identification \(A/\mathfrak m_A = k\).

The analogue of the condition that \(F(k)\) is a one element set is that \(\mathcal{F}(k)\) is the trivial groupoid. If \(\mathcal{F}\) satisfies this condition then we say it is a predeformation category, but in general we do not make this assumption. Rim’s paper [SGA7-I, Exposee VI] is the original source for the results in this document. We also mention the useful paper [Talpo-Vistoli], which discusses deformation theory with groupoids but in less generality than we do here.

An important role is played by the “completion” \(\widehat{\mathcal{C}}_\Lambda\) of the category \(\mathcal{C}_\Lambda\). An object of \(\widehat{\mathcal{C}}_\Lambda\) is a Noetherian complete local \(\Lambda\)-algebra \(R\) whose residue field is identified with \(k\), see Section 06GV. On the one hand \(\mathcal{C}_\Lambda \subset \widehat{\mathcal{C}}_\Lambda\) is a strictly full subcategory and on the other hand \(\widehat{\mathcal{C}}_\Lambda\) is a full subcategory of the category of pro-objects of \(\mathcal{C}_\Lambda\). A functor \(\mathcal{C}_\Lambda \to \textit{Sets}\) is prorepresentable if it is isomorphic to the restriction of a representable functor \(\underline{R} = \Mor_{\widehat{\mathcal{C}}_\Lambda}(R, -)\) to \(\mathcal{C}_\Lambda\) where \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\).

Categories cofibred in groupoids are dual to categories fibred in groupoids; we introduce them in Section 06GA. A smooth morphism of categories cofibred in groupoids over \(\mathcal{C}_\Lambda\) is one that satisfies the infinitesimal lifting criterion for objects, see Section 06HF. This is analogous to the definition of a formally smooth ring map, see Algebra, Definition 00TI and is exactly dual to the notion in Criteria for Representability, Section 06CZ. This is an important notion as we eventually want to prove that certain kinds of categories cofibred in groupoids have a smooth prorepresentable presentation, much like the characterization of algebraic stacks in Algebraic Stacks, Sections 04T3 and 04TJ. A versal formal object of a category \(\mathcal{F}\) cofibred in groupoids over \(\mathcal{C}_\Lambda\) is an object \(\xi \in \widehat{\mathcal{F}}(R)\) of the completion such that the associated morphism \(\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\) is smooth.

In Section 06HV, we define conditions (S1) and (S2) on \(\mathcal{F}\) generalizing Schlessinger’s (H1) and (H2). The analogue of Schlessinger’s (H3)—the condition that \(\mathcal{F}\) has finite dimensional tangent space—is not given a name. A key step in the development of the theory is the existence of versal formal objects for predeformation categories satisfying (S1), (S2) and (H3), see Lemma 06IW. Schlessinger’s notion of a hull for a functor \(F : \mathcal{C}_\Lambda \to \textit{Sets}\) is, in our terminology, a versal formal object \(\xi \in \widehat{F}(R)\) such that the induced map of tangent spaces \(d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to TF\) is an isomorphism. In the literature a hull is often called a “miniversal” object. We do not do so, and here is why. It can happen that a functor has a versal formal object without having a hull. Moreover, we show in Section 06T0 that if a predeformation category has a versal formal object, then it always has a minimal one (as defined in Definition 06T4) which is unique up to isomorphism, see Lemma 06T5. But it can happen that the minimal versal formal object does not induce an isomorphism on tangent spaces! (See Examples 06T7 and 06T9.)

Keeping in mind the differences pointed out above, Theorem 06IX is the direct generalization of (1) above: it recovers Schlessinger’s result in the case that \(\mathcal{F}\) is a functor and it characterizes minimal versal formal objects, in the presence of conditions (S1) and (S2), in terms of the map \(d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to TF\) on tangent spaces.

In Section 06J1, we define Rim’s condition (RS) on \(\mathcal{F}\) generalizing Schlessinger’s (H4). A deformation category is defined as a predeformation category satisfying (RS). The analogue to prorepresentable functors are the categories cofibred in groupoids over \(\mathcal{C}_\Lambda\) which have a presentation by a smooth prorepresentable groupoid in functors on \(\mathcal{C}_\Lambda\), see Definitions 06K3, 06KD, and 06KI. This notion of a presentation takes into account the groupoid structure of the fibers of \(\mathcal{F}\). In Theorem 06L8 we prove that \(\mathcal{F}\) has a presentation by a smooth prorepresentable groupoid in functors if and only if \(\mathcal{F}\) has a finite dimensional tangent space and finite dimensional infinitesimal automorphism space. This is the generalization of (2) above: it reduces to Schlessinger’s result in the case that \(\mathcal{F}\) is a functor. There is a final Section 06TD where we discuss how to use minimal versal formal objects to produce a (unique up to isomorphism) minimal presentation by a smooth prorepresentable groupoid in functors.

We also find the following conceptual explanation for Schlessinger’s conditions. If a predeformation category \(\mathcal{F}\) satisfies (RS), then the associated functor of isomorphism classes \(\overline{\mathcal{F}}: \mathcal{C}_\Lambda \to \textit{Sets}\) satisfies (H1) and (H2) (Lemmas 06J7 and 06I0). Conversely, if a functor \(F : \mathcal{C}_\Lambda \to \textit{Sets}\) arises naturally as the functor of isomorphism classes of a category \(\mathcal{F}\) cofibered in groupoids, then it seems to happen in practice that an argument showing \(F\) satisfies (H1) and (H2) will also show \(\mathcal{F}\) satisfies (RS). Examples are discussed in Deformation Problems, Section 0DVL. Moreover, if \(\mathcal{F}\) satisfies (RS), then condition (H4) for \(\overline{\mathcal{F}}\) has a simple interpretation in terms of extending automorphisms of objects of \(\mathcal{F}\) (Lemma 06J8). These observations suggest that (RS) should be regarded as the fundamental deformation theoretic glueing condition.

Notation and Conventions

A ring is commutative with \(1\). The maximal ideal of a local ring \(A\) is denoted by \(\mathfrak{m}_A\). The set of positive integers is denoted by \(\mathbf{N} = \{1, 2, 3, \ldots\}\). If \(U\) is an object of a category \(\mathcal{C}\), we denote by \(\underline{U}\) the functor \(\Mor_\mathcal{C}(U, -): \mathcal{C} \to \textit{Sets}\), see Remarks 06GK (06GQ). Warning: this may conflict with the notation in other chapters where we sometimes use \(\underline{U}\) to denote \(h_U(-) = \Mor_\mathcal{C}(-, U)\).

Throughout this chapter \(\Lambda\) is a Noetherian ring and \(\Lambda \to k\) is a finite ring map from \(\Lambda\) to a field. The kernel of this map is denoted \(\mathfrak m_\Lambda\) and the image \(k' \subset k\). It turns out that \(\mathfrak m_\Lambda\) is a maximal ideal, \(k' = \Lambda/\mathfrak m_\Lambda\) is a field, and the extension \(k/k'\) is finite. See discussion surrounding (06S2).

The base category

Motivation. An important application of formal deformation theory is to criteria for representability by algebraic spaces. Suppose given a locally Noetherian base \(S\) and a functor \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Let \(k\) be a finite type field over \(S\), i.e., we are given a finite type morphism \(\Spec(k) \to S\). One of Artin’s criteria is that for any element \(x \in F(\Spec(k))\) the predeformation functor associated to the triple \((S, k, x)\) should be prorepresentable. By Morphisms, Lemma 01TA the condition that \(k\) is of finite type over \(S\) means that there exists an affine open \(\Spec(\Lambda) \subset S\) such that \(k\) is a finite \(\Lambda\)-algebra. This motivates why we work throughout this chapter with a base category as follows.

Definition

Let \(\Lambda\) be a Noetherian ring and let \(\Lambda \to k\) be a finite ring map where \(k\) is a field. We define \(\mathcal{C}_\Lambda\) to be the category with

objects are pairs \((A, \varphi)\) where \(A\) is an Artinian local \(\Lambda\)-algebra and where \(\varphi : A/\mathfrak m_A \to k\) is a \(\Lambda\)-algebra isomorphism, and

morphisms \(f : (B, \psi) \to (A, \varphi)\) are local \(\Lambda\)-algebra homomorphisms such that \(\varphi \circ (f \bmod \mathfrak m) = \psi\).

We say we are in the classical case if \(\Lambda\) is a Noetherian complete local ring and \(k\) is its residue field.

Note that if \(\Lambda \to k\) is surjective and if \(A\) is an Artinian local \(\Lambda\)-algebra, then the identification \(\varphi\), if it exists, is unique. Moreover, in this case any \(\Lambda\)-algebra map \(A \to B\) is going to be compatible with the identifications. Hence in this case \(\mathcal{C}_\Lambda\) is just the category of local Artinian \(\Lambda\)-algebras whose residue field “is” \(k\). By abuse of notation we also denote objects of \(\mathcal{C}_\Lambda\) simply \(A\) in the general case. Moreover, we will often write \(A/\mathfrak m = k\), i.e., we will pretend all rings in \(\mathcal{C}_\Lambda\) have residue field \(k\) (since all ring maps in \(\mathcal{C}_\Lambda\) are compatible with the given identifications this should never cause any problems). Throughout the rest of this chapter the base ring \(\Lambda\) and the field \(k\) are fixed. The category \(\mathcal{C}_\Lambda\) will be the base category for the cofibered categories considered below.

Definition

Let \(f: B \to A\) be a ring map in \(\mathcal{C}_\Lambda\). We say \(f\) is a small extension if it is surjective and \(\Ker(f)\) is a nonzero principal ideal which is annihilated by \(\mathfrak{m}_B\).

By the following lemma we can often reduce arguments involving surjective ring maps in \(\mathcal{C}_\Lambda\) to the case of small extensions.

Lemma

Let \(f: B \to A\) be a surjective ring map in \(\mathcal{C}_\Lambda\). Then \(f\) can be factored as a composition of small extensions.

Proof

Let \(I\) be the kernel of \(f\). The maximal ideal \(\mathfrak{m}_B\) is nilpotent since \(B\) is Artinian, say \(\mathfrak{m}_B^n = 0\). Hence we get a factorization \[B = B/I\mathfrak{m}_B^{n-1} \to B/I\mathfrak{m}_B^{n-2} \to \ldots \to B/I \cong A\] of \(f\) into a composition of surjective maps whose kernels are annihilated by the maximal ideal. Thus it suffices to prove the lemma when \(f\) itself is such a map, i.e. when \(I\) is annihilated by \(\mathfrak{m}_B\). In this case \(I\) is a \(k\)-vector space, which has finite dimension, see Algebra, Lemma 00JB. Take a basis \(x_1, \ldots, x_n\) of \(I\) as a \(k\)-vector space to get a factorization \[B \to B/(x_1) \to \ldots \to B/(x_1, \ldots, x_n) \cong A\] of \(f\) into a composition of small extensions.

The next lemma says that we can compute the length of a module over a local \(\Lambda\)-algebra with residue field \(k\) in terms of the length over \(\Lambda\). To explain the notation in the statement, let \(k' \subset k\) be the image of our fixed finite ring map \(\Lambda \to k\). Note that \(k' \subset k\) is a finite extension of rings. Hence \(k'\) is a field and \(k/k'\) is a finite extension of fields, see Algebra, Lemma 00GR. Moreover, as \(\Lambda \to k'\) is surjective we see that its kernel is a maximal ideal \(\mathfrak m_\Lambda\). Thus [06S2]\[\begin{equation} [k : k'] = [k : \Lambda/\mathfrak m_\Lambda] < \infty \end{equation}\] and in the classical case we have \(k = k'\). The notation \(k' = \Lambda/\mathfrak m_\Lambda\) will be fixed throughout this chapter.

Lemma

Let \(A\) be a local \(\Lambda\)-algebra with residue field \(k\). Let \(M\) be an \(A\)-module. Then \([k : k'] \text{length}_A(M) = \text{length}_\Lambda(M)\). In the classical case we have \(\text{length}_A(M) = \text{length}_\Lambda(M)\).

Proof

If \(M\) is a simple \(A\)-module then \(M \cong k\) as an \(A\)-module, see Algebra, Lemma 00J2. In this case \(\text{length}_A(M) = 1\) and \(\text{length}_\Lambda(M) = [k' : k]\), see Algebra, Lemma 00IY. If \(\text{length}_A(M)\) is finite, then the result follows on choosing a filtration of \(M\) by \(A\)-submodules with simple quotients using additivity, see Algebra, Lemma 00IV. If \(\text{length}_A(M)\) is infinite, the result follows from the obvious inequality \(\text{length}_A(M) \leq \text{length}_\Lambda(M)\).

Lemma

Let \(A \to B\) be a ring map in \(\mathcal{C}_\Lambda\). The following are equivalent

\(f\) is surjective,

\(\mathfrak m_A/\mathfrak m_A^2 \to \mathfrak m_B/\mathfrak m_B^2\) is surjective, and

\(\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2) \to \mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)\) is surjective.

Proof

For any ring map \(f : A \to B\) in \(\mathcal{C}_\Lambda\) we have \(f(\mathfrak m_A) \subset \mathfrak m_B\) for example because \(\mathfrak m_A\), \(\mathfrak m_B\) is the set of nilpotent elements of \(A\), \(B\). Suppose \(f\) is surjective. Let \(y \in \mathfrak m_B\). Choose \(x \in A\) with \(f(x) = y\). Since \(f\) induces an isomorphism \(A/\mathfrak m_A \to B/\mathfrak m_B\) we see that \(x \in \mathfrak m_A\). Hence the induced map \(\mathfrak m_A/\mathfrak m_A^2 \to \mathfrak m_B/\mathfrak m_B^2\) is surjective. In this way we see that (1) implies (2).

It is clear that (2) implies (3). The map \(A \to B\) gives rise to a canonical commutative diagram \[\xymatrix{ \mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k \ar[r] \ar[d] & \mathfrak m_A/\mathfrak m_A^2 \ar[r] \ar[d] & \mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2) \ar[r] \ar[d] & 0 \\ \mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k \ar[r] & \mathfrak m_B/\mathfrak m_B^2 \ar[r] & \mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \ar[r] & 0 }\] with exact rows. Hence if (3) holds, then so does (2).

Assume (2). To show that \(A \to B\) is surjective it suffices by Nakayama’s lemma (Algebra, Lemma 00DV) to show that \(A/\mathfrak m_A \to B/\mathfrak m_AB\) is surjective. (Note that \(\mathfrak m_A\) is a nilpotent ideal.) As \(k = A/\mathfrak m_A = B/\mathfrak m_B\) it suffices to show that \(\mathfrak m_AB \to \mathfrak m_B\) is surjective. Applying Nakayama’s lemma once more we see that it suffices to see that \(\mathfrak m_AB/\mathfrak m_A\mathfrak m_B \to \mathfrak m_B/\mathfrak m_B^2\) is surjective which is what we assumed.

If \(A \to B\) is a ring map in \(\mathcal{C}_\Lambda\), then the map \(\mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2) \to \mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)\) is the map on relative cotangent spaces. Here is a formal definition.

Definition

Let \(R \to S\) be a local homomorphism of local rings. The relative cotangent space¹ of \(R\) over \(S\) is the \(S/\mathfrak m_S\)-vector space \(\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)\).

If \(f_1: A_1 \to A\) and \(f_2: A_2 \to A\) are two ring maps, then the fiber product \(A_1 \times_A A_2\) is the subring of \(A_1 \times A_2\) consisting of elements whose two projections to \(A\) are equal. Throughout this chapter we will be considering conditions involving such a fiber product when \(f_1\) and \(f_2\) are in \(\mathcal{C}_\Lambda\). It isn’t always the case that the fibre product is an object of \(\mathcal{C}_\Lambda\).

Example

Let \(p\) be a prime number and let \(n \in \mathbf{N}\). Let \(\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)\) and let \(k = \mathbf{F}_p(x_1, \ldots, x_n)\) with map \(\Lambda \to k\) given by \(t_i \mapsto x_i^p\). Let \(A = k[\epsilon] = k[x]/(x^2)\). Then \(A\) is an object of \(\mathcal{C}_\Lambda\). Suppose that \(D : k \to k\) is a derivation of \(k\) over \(\Lambda\), for example \(D = \partial/\partial x_i\). Then the map \[f_D : k \longrightarrow k[\epsilon], \quad a \mapsto a + D(a)\epsilon\] is a morphism of \(\mathcal{C}_\Lambda\). Set \(A_1 = A_2 = k\) and set \(f_1 = f_{\partial/\partial x_1}\) and \(f_2(a) = a\). Then \(A_1 \times_A A_2 = \{a \in k \mid \partial/\partial x_1(a) = 0\}\) which does not surject onto \(k\). Hence the fibre product isn’t an object of \(\mathcal{C}_\Lambda\).

It turns out that this problem can only occur if the residue field extension \(k/k'\) (06S2) is inseparable and neither \(f_1\) nor \(f_2\) is surjective.

Lemma

Let \(f_1 : A_1 \to A\) and \(f_2 : A_2 \to A\) be ring maps in \(\mathcal{C}_\Lambda\). Then:

If \(f_1\) or \(f_2\) is surjective, then \(A_1 \times_A A_2\) is in \(\mathcal{C}_\Lambda\).

If \(f_2\) is a small extension, then so is \(A_1 \times_A A_2 \to A_1\).

If the field extension \(k/k'\) is separable, then \(A_1 \times_A A_2\) is in \(\mathcal{C}_\Lambda\).

Proof

The ring \(A_1 \times_A A_2\) is a \(\Lambda\)-algebra via the map \(\Lambda \to A_1 \times_A A_2\) induced by the maps \(\Lambda \to A_1\) and \(\Lambda \to A_2\). It is a local ring with unique maximal ideal \[\mathfrak m_{A_1} \times_{\mathfrak m_A} \mathfrak m_{A_2} = \Ker(A_1 \times_A A_2 \longrightarrow k)\] A ring is Artinian if and only if it has finite length as a module over itself, see Algebra, Lemma 00JB. Since \(A_1\) and \(A_2\) are Artinian, Lemma 06GG implies \(\text{length}_\Lambda(A_1)\) and \(\text{length}_\Lambda(A_2)\), and hence \(\text{length}_\Lambda(A_1 \times A_2)\), are all finite. As \(A_1 \times_A A_2 \subset A_1 \times A_2\) is a \(\Lambda\)-submodule, this implies \(\text{length}_{A_1 \times_A A_2}(A_1 \times_A A_2) \leq \text{length}_\Lambda(A_1 \times_A A_2)\) is finite. So \(A_1 \times_A A_2\) is Artinian. Thus the only thing that is keeping \(A_1 \times_A A_2\) from being an object of \(\mathcal{C}_\Lambda\) is the possibility that its residue field maps to a proper subfield of \(k\) via the map \(A_1 \times_A A_2 \to A \to A/\mathfrak m_A = k\) above.

Proof of (1). If \(f_2\) is surjective, then the projection \(A_1 \times_A A_2 \to A_1\) is surjective. Hence the composition \(A_1 \times_A A_2 \to A_1 \to A_1/\mathfrak m_{A_1} = k\) is surjective and we conclude that \(A_1 \times_A A_2\) is an object of \(\mathcal{C}_\Lambda\).

Proof of (2). If \(f_2\) is a small extension then \(A_2 \to A\) and \(A_1 \times_A A_2 \to A_1\) are both surjective with the same kernel. Hence the kernel of \(A_1 \times_A A_2 \to A_1\) is a \(1\)-dimensional \(k\)-vector space and we see that \(A_1 \times_A A_2 \to A_1\) is a small extension.

Proof of (3). Choose \(\overline{x} \in k\) such that \(k = k'(\overline{x})\) (see Fields, Lemma 030N). Let \(P'(T) \in k'[T]\) be the minimal polynomial of \(\overline{x}\) over \(k'\). Since \(k/k'\) is separable we see that \(\text{d}P/\text{d}T(\overline{x}) \not = 0\). Choose a monic \(P \in \Lambda[T]\) which maps to \(P'\) under the surjective map \(\Lambda[T] \to k'[T]\). Because \(A, A_1, A_2\) are henselian, see Algebra, Lemma 06RS, we can find \(x, x_1, x_2 \in A, A_1, A_2\) with \(P(x) = 0, P(x_1) = 0, P(x_2) = 0\) and such that the image of \(x, x_1, x_2\) in \(k\) is \(\overline{x}\). Then \((x_1, x_2) \in A_1 \times_A A_2\) because \(x_1, x_2\) map to \(x \in A\) by uniqueness, see Algebra, Lemma 06RR. Hence the residue field of \(A_1 \times_A A_2\) contains a generator of \(k\) over \(k'\) and we win.

Next we define essential surjections in \(\mathcal{C}_\Lambda\). A necessary and sufficient condition for a surjection in \(\mathcal{C}_\Lambda\) to be essential is given in Lemma 06H0.

Definition

Let \(f: B \to A\) be a ring map in \(\mathcal{C}_\Lambda\). We say \(f\) is an essential surjection if it has the following properties:

\(f\) is surjective.

If \(g: C \to B\) is a ring map in \(\mathcal{C}_\Lambda\) such that \(f \circ g\) is surjective, then \(g\) is surjective.

Using Lemma 06S3, we can characterize essential surjections in \(\mathcal{C}_\Lambda\) as follows.

Lemma

Let \(f: B \to A\) be a ring map in \(\mathcal{C}_\Lambda\). The following are equivalent

\(f\) is an essential surjection,

the map \(B/\mathfrak m_B^2 \to A/\mathfrak m_A^2\) is an essential surjection, and

the map \(B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)\) is an essential surjection.

Proof

Assume (3). Let \(C \to B\) be a ring map in \(\mathcal{C}_\Lambda\) such that \(C \to A\) is surjective. Then \(C \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)\) is surjective too. We conclude that \(C \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)\) is surjective by our assumption. Hence \(C \to B\) is surjective by applying Lemma 06S3 (2 times).

Assume (1). Let \(C \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)\) be a morphism of \(\mathcal{C}_\Lambda\) such that \(C \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)\) is surjective. Set \(C' = C \times_{B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)} B\) which is an object of \(\mathcal{C}_\Lambda\) by Lemma 06GH. Note that \(C' \to A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)\) is still surjective, hence \(C' \to A\) is surjective by Lemma 06S3. Thus \(C' \to B\) is surjective by our assumption. This implies that \(C' \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)\) is surjective, which implies by the construction of \(C'\) that \(C \to B/(\mathfrak m_\Lambda B + \mathfrak m_B^2)\) is surjective.

In the first paragraph we proved (3) \(\Rightarrow\) (1) and in the second paragraph we proved (1) \(\Rightarrow\) (3). The equivalence of (2) and (3) is a special case of the equivalence of (1) and (3), hence we are done.

To analyze essential surjections in \(\mathcal{C}_\Lambda\) a bit more we introduce some notation. Suppose that \(A\) is an object of \(\mathcal{C}_\Lambda\) or more generally any \(\Lambda\)-algebra equipped with a \(\Lambda\)-algebra surjection \(A \to k\). There is a canonical exact sequence [06S6]\[\begin{equation} \mathfrak m_A/\mathfrak m_A^2 \xrightarrow{\text{d}_A} \Omega_{A/\Lambda} \otimes_A k \to \Omega_{k/\Lambda} \to 0 \end{equation}\] see Algebra, Lemma 00RU. Note that \(\Omega_{k/\Lambda} = \Omega_{k/k'}\) with \(k'\) as in (06S2). Let \(H_1(L_{k/\Lambda})\) be the first homology module of the naive cotangent complex of \(k\) over \(\Lambda\), see Algebra, Definition 07BN. Then we can extend (06S6) to the exact sequence [06S7]\[\begin{equation} H_1(L_{k/\Lambda}) \to \mathfrak m_A/\mathfrak m_A^2 \xrightarrow{\text{d}_A} \Omega_{A/\Lambda} \otimes_A k \to \Omega_{k/\Lambda} \to 0, \end{equation}\] see Algebra, Lemma 00S2. If \(B \to A\) is a ring map in \(\mathcal{C}_\Lambda\) or more generally a map of \(\Lambda\)-algebras equipped with \(\Lambda\)-algebra surjections onto \(k\), then we obtain a commutative diagram [06S8]\[\begin{equation} \vcenter{ \xymatrix{ H_1(L_{k/\Lambda}) \ar[r] \ar@{=}[d] & \mathfrak m_B/\mathfrak m_B^2 \ar[r]_{\text{d}_B} \ar[d] & \Omega_{B/\Lambda} \otimes_B k \ar[r] \ar[d] & \Omega_{k/\Lambda} \ar[r] \ar@{=}[d] & 0 \\ H_1(L_{k/\Lambda}) \ar[r] & \mathfrak m_A/\mathfrak m_A^2 \ar[r]^{\text{d}_A} & \Omega_{A/\Lambda} \otimes_A k \ar[r] & \Omega_{k/\Lambda} \ar[r] & 0 } } \end{equation}\] with exact rows.

Lemma

There is a canonical map \[\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \longrightarrow H_1(L_{k/\Lambda}).\] If \(k' \subset k\) is separable (for example if the characteristic of \(k\) is zero), then this map induces an isomorphism \(\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k = H_1(L_{k/\Lambda})\). If \(k = k'\) (for example in the classical case), then \(\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 = H_1(L_{k/\Lambda})\). The composition \[\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \longrightarrow H_1(L_{k/\Lambda}) \longrightarrow \mathfrak m_A/\mathfrak m_A^2\] comes from the canonical map \(\mathfrak m_\Lambda \to \mathfrak m_A\).

Proof

Note that \(H_1(L_{k'/\Lambda}) = \mathfrak m_\Lambda/\mathfrak m_\Lambda^2\) as \(\Lambda \to k'\) is surjective with kernel \(\mathfrak m_\Lambda\). The map arises from functoriality of the naive cotangent complex. If \(k' \subset k\) is separable, then \(k' \to k\) is an étale ring map, see Algebra, Lemma 00U3. Thus its naive cotangent complex has trivial homology groups, see Algebra, Definition 00U1. Then Algebra, Lemma 00S2 applied to the ring maps \(\Lambda \to k' \to k\) implies that \(\mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k = H_1(L_{k/\Lambda})\). We omit the proof of the final statement.

Lemma

Let \(f: B \to A\) be a ring map in \(\mathcal{C}_\Lambda\). Notation as in (06S8).

The equivalent conditions of Lemma 06S3 characterizing when \(f\) is surjective are also equivalent to

\(\Im(\text{d}_B) \to \Im(\text{d}_A)\) is surjective, and

the map \(\Omega_{B/\Lambda} \otimes_B k \to \Omega_{A/\Lambda} \otimes_A k\) is surjective.

The following are equivalent

\(f\) is an essential surjection (see Lemma 06S5),

the map \(\Im(\text{d}_B) \to \Im(\text{d}_A)\) is an isomorphism, and

the map \(\Omega_{B/\Lambda} \otimes_B k \to \Omega_{A/\Lambda} \otimes_A k\) is an isomorphism.

If \(k/k'\) is separable, then \(f\) is an essential surjection if and only if the map \(\mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to \mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)\) is an isomorphism.

If \(f\) is a small extension, then \(f\) is not essential if and only if \(f\) has a section \(s: A \to B\) in \(\mathcal{C}_\Lambda\) with \(f \circ s = \text{id}_A\).

Proof

Proof of (1). It follows from (06S8) that (1)(a) and (1)(b) are equivalent. Also, if \(A \to B\) is surjective, then (1)(a) and (1)(b) hold. Assume (1)(a). Since the kernel of \(\text{d}_A\) is the image of \(H_1(L_{k/\Lambda})\) which also maps to \(\mathfrak m_B/\mathfrak m_B^2\) we conclude that \(\mathfrak m_B/\mathfrak m_B^2 \to \mathfrak m_A/\mathfrak m_A^2\) is surjective. Hence \(B \to A\) is surjective by Lemma 06S3. This finishes the proof of (1).

Proof of (2). The equivalence of (2)(b) and (2)(c) is immediate from (06S8).

Assume (2)(b). Let \(g : C \to B\) be a ring map in \(\mathcal{C}_\Lambda\) such that \(f \circ g\) is surjective. We conclude that \(\mathfrak m_C/\mathfrak m_C^2 \to \mathfrak m_A/\mathfrak m_A^2\) is surjective by Lemma 06S3. Hence \(\Im(\text{d}_C) \to \Im(\text{d}_A)\) is surjective and by the assumption we see that \(\Im(\text{d}_C) \to \Im(\text{d}_B)\) is surjective. It follows that \(C \to B\) is surjective by (1).

Assume (2)(a). Then \(f\) is surjective and we see that \(\Omega_{B/\Lambda} \otimes_B k \to \Omega_{A/\Lambda} \otimes_A k\) is surjective. Let \(K\) be the kernel. Note that \(K = \text{d}_B(\Ker(\mathfrak m_B/\mathfrak m_B^2 \to \mathfrak m_A/\mathfrak m_A^2))\) by (06S8). Choose a splitting \[\Omega_{B/\Lambda} \otimes_B k = \Omega_{A/\Lambda} \otimes_A k \oplus K\] of \(k\)-vector space. The map \(\text{d} : B \to \Omega_{B/\Lambda}\) induces via the projection onto \(K\) a map \(D : B \to K\). Set \(C = \{b \in B \mid D(b) = 0\}\). The Leibniz rule shows that this is a \(\Lambda\)-subalgebra of \(B\). Let \(\overline{x} \in k\). Choose \(x \in B\) mapping to \(\overline{x}\). If \(D(x) \not = 0\), then we can find an element \(y \in \mathfrak m_B\) such that \(D(y) = D(x)\). Hence \(x - y \in C\) is an element which maps to \(\overline{x}\). Thus \(C \to k\) is surjective and \(C\) is an object of \(\mathcal{C}_\Lambda\). Similarly, pick \(\omega \in \Im(\text{d}_A)\). We can find \(x \in \mathfrak m_B\) such that \(\text{d}_B(x)\) maps to \(\omega\) by (1). If \(D(x) \not = 0\), then we can find an element \(y \in \mathfrak m_B\) which maps to zero in \(\mathfrak m_A/\mathfrak m_A^2\) such that \(D(y) = D(x)\). Hence \(z = x - y\) is an element of \(\mathfrak m_C\) whose image \(\text{d}_C(z) \in \Omega_{C/k} \otimes_C k\) maps to \(\omega\). Hence \(\Im(\text{d}_C) \to \Im(\text{d}_A)\) is surjective. We conclude that \(C \to A\) is surjective by (1). Hence \(C \to B\) is surjective by assumption. Hence \(D = 0\), i.e., \(K = 0\), i.e., (2)(c) holds. This finishes the proof of (2).

Proof of (3). If \(k'/k\) is separable, then \(H_1(L_{k/\Lambda}) = \mathfrak m_\Lambda/\mathfrak m_\Lambda^2 \otimes_{k'} k\), see Lemma 06S9. Hence \(\Im(\text{d}_A) = \mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)\) and similarly for \(B\). Thus (3) follows from (2).

Proof of (4). A section \(s\) of \(f\) is not surjective (by definition a small extension has nontrivial kernel), hence \(f\) is not essentially surjective. Conversely, assume \(f\) is a small extension but not an essential surjection. Choose a ring map \(C \to B\) in \(\mathcal{C}_\Lambda\) which is not surjective, such that \(C \to A\) is surjective. Let \(C' \subset B\) be the image of \(C \to B\). Then \(C' \not = B\) but \(C'\) surjects onto \(A\). Since \(f : B \to A\) is a small extension, \(\text{length}_C(B) = \text{length}_C(A) + 1\). Thus \(\text{length}_C(C') \leq \text{length}_C(A)\) since \(C'\) is a proper subring of \(B\). But \(C' \to A\) is surjective, so in fact we must have \(\text{length}_C(C') = \text{length}_C(A)\) and \(C' \to A\) is an isomorphism which gives us our section.

Example

Let \(\Lambda = k[[x]]\) be the power series ring in \(1\) variable over \(k\). Set \(A = k\) and \(B = \Lambda/(x^2)\). Then \(B \to A\) is an essential surjection by Lemma 06H0 because it is a small extension and the map \(B \to A\) does not have a right inverse (in the category \(\mathcal{C}_\Lambda\)). But the map \[k \cong \mathfrak m_B/\mathfrak m_B^2 \longrightarrow \mathfrak m_A/\mathfrak m_A^2 = 0\] is not an isomorphism. Thus in Lemma 06H0 (3) it is necessary to consider the map of relative cotangent spaces \(\mathfrak m_B/(\mathfrak m_\Lambda B + \mathfrak m_B^2) \to \mathfrak m_A/(\mathfrak m_\Lambda A + \mathfrak m_A^2)\).

The completed base category

The following “completion” of the category \(\mathcal{C}_\Lambda\) will serve as the base category of the completion of a category cofibered in groupoids over \(\mathcal{C}_\Lambda\) (Section 06H2).

Definition

Let \(\Lambda\) be a Noetherian ring and let \(\Lambda \to k\) be a finite ring map where \(k\) is a field. We define \(\widehat{\mathcal{C}}_\Lambda\) to be the category with

objects are pairs \((R, \varphi)\) where \(R\) is a Noetherian complete local \(\Lambda\)-algebra and where \(\varphi : R/\mathfrak m_R \to k\) is a \(\Lambda\)-algebra isomorphism, and

morphisms \(f : (S, \psi) \to (R, \varphi)\) are local \(\Lambda\)-algebra homomorphisms such that \(\varphi \circ (f \bmod \mathfrak m) = \psi\).

As in the discussion following Definition 06GC we will usually denote an object of \(\widehat{\mathcal{C}}_\Lambda\) simply \(R\), with the identification \(R/\mathfrak m_R = k\) understood. In this section we discuss some basic properties of objects and morphisms of the category \(\widehat{\mathcal{C}}_\Lambda\) paralleling our discussion of the category \(\mathcal{C}_\Lambda\) in the previous section.

Our first observation is that any object \(A \in \mathcal{C}_\Lambda\) is an object of \(\widehat{\mathcal{C}}_\Lambda\) as an Artinian local ring is always Noetherian and complete with respect to its maximal ideal (which is after all a nilpotent ideal). Moreover, it is clear from the definitions that \(\mathcal{C}_\Lambda \subset \widehat{\mathcal{C}}_\Lambda\) is the strictly full subcategory consisting of all Artinian rings. As it turns out, conversely every object of \(\widehat{\mathcal{C}}_\Lambda\) is a limit of objects of \(\mathcal{C}_\Lambda\).

Suppose that \(R\) is an object of \(\widehat{\mathcal{C}}_\Lambda\). Consider the rings \(R_n = R/\mathfrak m_R^n\) for \(n \in \mathbf{N}\). These are Noetherian local rings with a unique nilpotent prime ideal, hence Artinian, see Algebra, Proposition 00KJ. The ring maps \[\ldots \to R_{n + 1} \to R_n \to \ldots \to R_2 \to R_1 = k\] are all surjective. Completeness of \(R\) by definition means that \(R = \lim R_n\). If \(f : R \to S\) is a ring map in \(\widehat{\mathcal{C}}_\Lambda\) then we obtain a system of ring maps \(f_n : R_n \to S_n\) whose limit is the given map.

Lemma

Let \(f: R \to S\) be a ring map in \(\widehat{\mathcal{C}}_\Lambda\). The following are equivalent

\(f\) is surjective,

the map \(\mathfrak m_R/\mathfrak m_R^2 \to \mathfrak m_S/\mathfrak m_S^2\) is surjective, and

the map \(\mathfrak m_R/(\mathfrak m_\Lambda R + \mathfrak m_R^2) \to \mathfrak m_S/(\mathfrak m_\Lambda S + \mathfrak m_S^2)\) is surjective.

Proof

Note that for \(n \geq 2\) we have the equality of relative cotangent spaces \[\mathfrak m_R/(\mathfrak m_\Lambda R + \mathfrak m_R^2) = \mathfrak m_{R_n}/(\mathfrak m_\Lambda R_n + \mathfrak m_{R_n}^2)\] and similarly for \(S\). Hence by Lemma 06S3 we see that \(R_n \to S_n\) is surjective for all \(n\). Now let \(K_n\) be the kernel of \(R_n \to S_n\). Then the sequences \[0 \to K_n \to R_n \to S_n \to 0\] form an exact sequence of directed inverse systems. The system \((K_n)\) is Mittag-Leffler since each \(K_n\) is Artinian. Hence by Algebra, Lemma 0598 taking limits preserves exactness. So \(\lim R_n \to \lim S_n\) is surjective, i.e., \(f\) is surjective.

Lemma

The category \(\widehat{\mathcal{C}}_\Lambda\) admits pushouts.

Proof

Let \(R \to S_1\) and \(R \to S_2\) be morphisms of \(\widehat{\mathcal{C}}_\Lambda\). Consider the ring \(C = S_1 \otimes_R S_2\). This ring has a finitely generated maximal ideal \(\mathfrak m = \mathfrak m_{S_1} \otimes S_2 + S_1 \otimes \mathfrak m_{S_2}\) with residue field \(k\). Set \(C^\wedge\) equal to the completion of \(C\) with respect to \(\mathfrak m\). Then \(C^\wedge\) is a Noetherian ring complete with respect to the maximal ideal \(\mathfrak m^\wedge = \mathfrak mC^\wedge\) whose residue field is identified with \(k\), see Algebra, Lemma 05GH. Hence \(C^\wedge\) is an object of \(\widehat{\mathcal{C}}_\Lambda\). Then \(S_1 \to C^\wedge\) and \(S_2 \to C^\wedge\) turn \(C^\wedge\) into a pushout over \(R\) in \(\widehat{\mathcal{C}}_\Lambda\) (details omitted).

We will not need the following lemma.

Lemma

The category \(\widehat{\mathcal{C}}_\Lambda\) admits coproducts of pairs of objects.

Proof

Let \(R\) and \(S\) be objects of \(\widehat{\mathcal{C}}_\Lambda\). Consider the ring \(C = R \otimes_\Lambda S\). There is a canonical surjective map \(C \to R \otimes_\Lambda S \to k \otimes_\Lambda k \to k\) where the last map is the multiplication map. The kernel of \(C \to k\) is a maximal ideal \(\mathfrak m\). Note that \(\mathfrak m\) is generated by \(\mathfrak m_R C\), \(\mathfrak m_S C\) and finitely many elements of \(C\) which map to generators of the kernel of \(k \otimes_\Lambda k \to k\). Hence \(\mathfrak m\) is a finitely generated ideal. Set \(C^\wedge\) equal to the completion of \(C\) with respect to \(\mathfrak m\). Then \(C^\wedge\) is a Noetherian ring complete with respect to the maximal ideal \(\mathfrak m^\wedge = \mathfrak mC^\wedge\) with residue field \(k\), see Algebra, Lemma 05GH. Hence \(C^\wedge\) is an object of \(\widehat{\mathcal{C}}_\Lambda\). Then \(R \to C^\wedge\) and \(S \to C^\wedge\) turn \(C^\wedge\) into a coproduct in \(\widehat{\mathcal{C}}_\Lambda\) (details omitted).

An empty coproduct in a category is an initial object of the category. In the classical case \(\widehat{\mathcal{C}}_\Lambda\) has an initial object, namely \(\Lambda\) itself. More generally, if \(k' = k\), then the completion \(\Lambda^\wedge\) of \(\Lambda\) with respect to \(\mathfrak m_\Lambda\) is an initial object. More generally still, if \(k' \subset k\) is separable, then \(\widehat{\mathcal{C}}_\Lambda\) has an initial object too. Namely, choose a monic polynomial \(P \in \Lambda[T]\) such that \(k \cong k'[T]/(P')\) where \(p' \in k'[T]\) is the image of \(P\). Then \(R = \Lambda^\wedge[T]/(P)\) is an initial object, see proof of Lemma 06GH.

If \(R\) is an initial object as above, then we have \(\mathcal{C}_\Lambda = \mathcal{C}_R\) and \(\widehat{\mathcal{C}}_\Lambda = \widehat{\mathcal{C}}_R\) which effectively brings the whole discussion in this chapter back to the classical case. But, if \(k' \subset k\) is inseparable, then an initial object does not exist.

Lemma

Let \(S\) be an object of \(\widehat{\mathcal{C}}_\Lambda\). Then \(\dim_k \text{Der}_\Lambda(S, k) < \infty\).

Proof

Let \(x_1, \ldots, x_n \in \mathfrak m_S\) map to a \(k\)-basis for the relative cotangent space \(\mathfrak m_S/(\mathfrak m_\Lambda S + \mathfrak m_S^2)\). Choose \(y_1, \ldots, y_m \in S\) whose images in \(k\) generate \(k\) over \(k'\). We claim that \(\dim_k \text{Der}_\Lambda(S, k) \leq n + m\). To see this it suffices to prove that if \(D(x_i) = 0\) and \(D(y_j) = 0\), then \(D = 0\). Let \(a \in S\). We can find a polynomial \(P = \sum \lambda_J y^J\) with \(\lambda_J \in \Lambda\) whose image in \(k\) is the same as the image of \(a\) in \(k\). Then we see that \(D(a - P) = D(a) - D(P) = D(a)\) by our assumption that \(D(y_j) = 0\) for all \(j\). Thus we may assume \(a \in \mathfrak m_S\). Write \(a = \sum a_i x_i\) with \(a_i \in S\). By the Leibniz rule \[D(a) = \sum x_iD(a_i) + \sum a_iD(x_i) = \sum x_iD(a_i)\] as we assumed \(D(x_i) = 0\). We have \(\sum x_iD(a_i) = 0\) as multiplication by \(x_i\) is zero on \(k\).

Lemma

Let \(f : R \to S\) be a morphism of \(\widehat{\mathcal{C}}_\Lambda\). If \(\text{Der}_\Lambda(S, k) \to \text{Der}_\Lambda(R, k)\) is injective, then \(f\) is surjective.

Proof

If \(f\) is not surjective, then \(\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)\) is nonzero by Lemma 06GZ. Then also \(Q = S/(f(R) + \mathfrak m_R S + \mathfrak m_S^2)\) is nonzero. Note that \(Q\) is a \(k = R/\mathfrak m_R\)-vector space via \(f\). We turn \(Q\) into an \(S\)-module via \(S \to k\). The quotient map \(D : S \to Q\) is an \(R\)-derivation: if \(a_1, a_2 \in S\), we can write \(a_1 = f(b_1) + a_1'\) and \(a_2 = f(b_2) + a_2'\) for some \(b_1, b_2 \in R\) and \(a_1', a_2' \in \mathfrak m_S\). Then \(b_i\) and \(a_i\) have the same image in \(k\) for \(i = 1, 2\) and \[\begin{align*} a_1a_2 & = (f(b_1) + a_1')(f(b_2) + a_2') \\ & = f(b_1)a_2' + f(b_2)a_1' \\ & = f(b_1)(f(b_2) + a_2') + f(b_2)(f(b_1) + a_1') \\ & = f(b_1)a_2 + f(b_2)a_1 \end{align*}\] in \(Q\) which proves the Leibniz rule. Hence \(D : S \to Q\) is a \(\Lambda\)-derivation which is zero on composing with \(R \to S\). Since \(Q \not = 0\) there also exist derivations \(D : S \to k\) which are zero on composing with \(R \to S\), i.e., \(\text{Der}_\Lambda(S, k) \to \text{Der}_\Lambda(R, k)\) is not injective.

Lemma

Let \(R\) be an object of \(\widehat{\mathcal{C}}_\Lambda\). Let \((J_n)\) be a decreasing sequence of ideals such that \(\mathfrak m_R^n \subset J_n\). Set \(J = \bigcap J_n\). Then the sequence \((J_n/J)\) defines the \(\mathfrak m_{R/J}\)-adic topology on \(R/J\).

Proof

It is clear that \(\mathfrak m_{R/J}^n \subset J_n/J\). Thus it suffices to show that for every \(n\) there exists an \(N\) such that \(J_N/J \subset \mathfrak m_{R/J}^n\). This is equivalent to \(J_N \subset \mathfrak m_R^n + J\). For each \(n\) the ring \(R/\mathfrak m_R^n\) is Artinian, hence there exists a \(N_n\) such that \[J_{N_n} + \mathfrak m_R^n = J_{N_n + 1} + \mathfrak m_R^n = \ldots\] Set \(E_n = (J_{N_n} + \mathfrak m_R^n)/\mathfrak m_R^n\). Set \(E = \lim E_n \subset \lim R/\mathfrak m_R^n = R\). Note that \(E \subset J\) as for any \(f \in E\) and any \(m\) we have \(f \in J_m + \mathfrak m_R^n\) for all \(n \gg 0\), so \(f \in J_m\) by Krull’s intersection theorem, see Algebra, Lemma 00IP. Since the transition maps \(E_n \to E_{n - 1}\) are all surjective, we see that \(J\) surjects onto \(E_n\). Hence for \(N = N_n\) works.

Lemma

Let \(\ldots \to A_3 \to A_2 \to A_1\) be a sequence of surjective ring maps in \(\mathcal{C}_\Lambda\). If \(\dim_k (\mathfrak m_{A_n}/\mathfrak m_{A_n}^2)\) is bounded, then \(S = \lim A_n\) is an object in \(\widehat{\mathcal{C}}_\Lambda\) and the ideals \(I_n = \Ker(S \to A_n)\) define the \(\mathfrak m_S\)-adic topology on \(S\).

Proof

We will use freely that the maps \(S \to A_n\) are surjective for all \(n\). Note that the maps \(\mathfrak m_{A_{n + 1}}/\mathfrak m_{A_{n + 1}}^2 \to \mathfrak m_{A_n}/\mathfrak m_{A_n}^2\) are surjective, see Lemma 06GZ. Hence for \(n\) sufficiently large the dimension \(\dim_k (\mathfrak m_{A_n}/\mathfrak m_{A_n}^2)\) stabilizes to an integer, say \(r\). Thus we can find \(x_1, \ldots, x_r \in \mathfrak m_S\) whose images in \(A_n\) generate \(\mathfrak m_{A_n}\). Moreover, pick \(y_1, \ldots, y_t \in S\) whose images in \(k\) generate \(k\) over \(\Lambda\). Then we get a ring map \(P = \Lambda[z_1, \ldots, z_{r + t}] \to S\), \(z_i \mapsto x_i\) and \(z_{r + j} \mapsto y_j\) such that the composition \(P \to S \to A_n\) is surjective for all \(n\). Let \(\mathfrak m \subset P\) be the kernel of \(P \to k\). Let \(R = P^\wedge\) be the \(\mathfrak m\)-adic completion of \(P\); this is an object of \(\widehat{\mathcal{C}}_\Lambda\). Since we still have the compatible system of (surjective) maps \(R \to A_n\) we get a map \(R \to S\). Set \(J_n = \Ker(R \to A_n)\). Set \(J = \bigcap J_n\). By Lemma 06SE we see that \(R/J = \lim R/J_n = \lim A_n = S\) and that the ideals \(J_n/J = I_n\) define the \(\mathfrak m\)-adic topology. (Note that for each \(n\) we have \(\mathfrak m_R^{N_n} \subset J_n\) for some \(N_n\) and not necessarily \(N_n = n\), so a renumbering of the ideals \(J_n\) may be necessary before applying the lemma.)

Lemma

Let \(R', R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\). Suppose that \(R = R' \oplus I\) for some ideal \(I\) of \(R\). Let \(x_1, \ldots, x_r \in I\) map to a basis of \(I/\mathfrak m_R I\). Set \(S = R'[[X_1, \ldots, X_r]]\) and consider the \(R'\)-algebra map \(S \to R\) mapping \(X_i\) to \(x_i\). Assume that for every \(n \gg 0\) the map \(S/\mathfrak m_S^n \to R/\mathfrak m_R^n\) has a left inverse in \(\mathcal{C}_\Lambda\). Then \(S \to R\) is an isomorphism.

Proof

As \(R = R' \oplus I\) we have \[\mathfrak m_R/\mathfrak m_R^2 = \mathfrak m_{R'}/\mathfrak m_{R'}^2 \oplus I/\mathfrak m_RI\] and similarly \[\mathfrak m_S/\mathfrak m_S^2 = \mathfrak m_{R'}/\mathfrak m_{R'}^2 \oplus \bigoplus kX_i\] Hence for \(n > 1\) the map \(S/\mathfrak m_S^n \to R/\mathfrak m_R^n\) induces an isomorphism on cotangent spaces. Thus a left inverse \(h_n : R/\mathfrak m_R^n \to S/\mathfrak m_S^n\) is surjective by Lemma 06GZ. Since \(h_n\) is injective as a left inverse it is an isomorphism. Thus the canonical surjections \(S/\mathfrak m_S^n \to R/\mathfrak m_R^n\) are all isomorphisms and we win.

Categories cofibered in groupoids

In developing the theory we work with categories cofibered in groupoids. We assume as known the definition and basic properties of categories fibered in groupoids, see Categories, Section 003S.

Definition

Let \(\mathcal{C}\) be a category. A category cofibered in groupoids over \(\mathcal{C}\) is a category \(\mathcal{F}\) equipped with a functor \(p: \mathcal{F} \to \mathcal{C}\) such that \(\mathcal{F}^{opp}\) is a category fibered in groupoids over \(\mathcal{C}^{opp}\) via \(p^{opp}: \mathcal{F}^{opp} \to \mathcal{C}^{opp}\).

Explicitly, \(p: \mathcal{F} \to \mathcal{C}\) is cofibered in groupoids if the following two conditions hold:

For every morphism \(f: U \to V\) in \(\mathcal{C}\) and every object \(x\) lying over \(U\), there is a morphism \(x \to y\) of \(\mathcal{F}\) lying over \(f\).
For every pair of morphisms \(a: x \to y\) and \(b: x \to z\) of \(\mathcal{F}\) and any morphism \(f: p(y) \to p(z)\) such that \(p(b) = f \circ p(a)\), there exists a unique morphism \(c: y \to z\) of \(\mathcal F\) lying over \(f\) such that \(b = c \circ a\).

Remarks

Everything about categories fibered in groupoids translates directly to the cofibered setting. The following remarks are meant to fix notation. Let \(\mathcal{C}\) be a category.

We often omit the functor \(p: \mathcal{F} \to \mathcal{C}\) from the notation.

The fiber category over an object \(U\) in \(\mathcal{C}\) is denoted by \(\mathcal{F}(U)\). Its objects are those of \(\mathcal{F}\) lying over \(U\) and its morphisms are those of \(\mathcal{F}\) lying over \(\text{id}_U\). If \(x, y\) are objects of \(\mathcal{F}(U)\), we sometimes write \(\Mor_U(x, y)\) for \(\Mor_{\mathcal{F}(U)}(x, y)\).

The fibre categories \(\mathcal{F}(U)\) are groupoids, see Categories, Lemma 003V. Hence the morphisms in \(\mathcal{F}(U)\) are all isomorphisms. We sometimes write \(\text{Aut}_U(x)\) for \(\Mor_{\mathcal{F}(U)}(x, x)\).

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}\), let \(f: U \to V\) be a morphism in \(\mathcal{C}\), and let \(x \in \Ob(\mathcal{F}(U))\). A pushforward of \(x\) along \(f\) is a morphism \(x \to y\) of \(\mathcal{F}\) lying over \(f\). A pushforward is unique up to unique isomorphism (see the discussion following Categories, Definition 02XK). We sometimes write \(x \to f_*x\) for “the” pushforward of \(x\) along \(f\).

A choice of pushforwards for \(\mathcal{F}\) is the choice of a pushforward of \(x\) along \(f\) for every pair \((x, f)\) as above. We can make such a choice of pushforwards for \(\mathcal{F}\) by the axiom of choice.

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}\). Given a choice of pushforwards for \(\mathcal{F}\), there is an associated pseudo-functor \(\mathcal{C} \to \textit{Groupoids}\). We will never use this construction so we give no details.

A morphism of categories cofibered in groupoids over \(\mathcal{C}\) is a functor commuting with the projections to \(\mathcal{C}\). If \(\mathcal{F}\) and \(\mathcal{F}'\) are categories cofibered in groupoids over \(\mathcal{C}\), we denote the morphisms from \(\mathcal{F}\) to \(\mathcal{F}'\) by \(\Mor_\mathcal{C}(\mathcal{F}, \mathcal{F}')\).

Categories cofibered in groupoids form a \((2, 1)\)-category \(\text{Cof}(\mathcal{C})\). Its 1-morphisms are the morphisms described in (06GL). If \(p : \mathcal{F} \to C\) and \(p': \mathcal{F}' \to \mathcal{C}\) are categories cofibered in groupoids and \(\varphi, \psi : \mathcal{F} \to \mathcal{F}'\) are \(1\)-morphisms, then a 2-morphism \(t : \varphi \to \psi\) is a morphism of functors such that \(p'(t_x) = \text{id}_{p(x)}\) for all \(x \in \Ob(\mathcal{F})\).

Let \(F : \mathcal{C} \to \textit{Groupoids}\) be a functor. There is a category cofibered in groupoids \(\mathcal{F} \to \mathcal{C}\) associated to \(F\) as follows. An object of \(\mathcal{F}\) is a pair \((U, x)\) where \(U \in \Ob(\mathcal{C})\) and \(x \in \Ob(F(U))\). A morphism \((U, x) \to (V, y)\) is a pair \((f, a)\) where \(f \in \Mor_\mathcal{C}(U, V)\) and \(a \in \Mor_{F(V)}(F(f)(x), y)\). The functor \(\mathcal{F} \to \mathcal{C}\) sends \((U, x)\) to \(U\). See Categories, Section 0048.

Let \(\mathcal{F}\) be cofibered in groupoids over \(\mathcal{C}\). For \(U \in \Ob(\mathcal{C})\) set \(\overline{\mathcal{F}}(U)\) equal to the set of isomorphisms classes of the category \(\mathcal{F}(U)\). If \(f : U \to V\) is a morphism of \(\mathcal{C}\), then we obtain a map of sets \(\overline{\mathcal{F}}(U) \to \overline{\mathcal{F}}(V)\) by mapping the isomorphism class of \(x\) to the isomorphism class of a pushforward \(f_*x\) of \(x\) see (06SH). Then \(\overline{\mathcal{F}} : \mathcal{C} \to \textit{Sets}\) is a functor. Similarly, if \(\varphi : \mathcal{F} \to \mathcal{G}\) is a morphism of cofibered categories, we denote by \(\overline{\varphi}: \overline{\mathcal{F}} \to \overline{\mathcal{G}}\) the associated morphism of functors.

Let \(F: \mathcal{C} \to \textit{Sets}\) be a functor. We can think of a set as a discrete category, i.e., as a groupoid with only identity morphisms. Then the construction (06GN) associates to \(F\) a category cofibered in sets. This defines a fully faithful embedding of the category of functors \(\mathcal{C} \to \textit{Sets}\) to the category of categories cofibered in groupoids over \(\mathcal{C}\). We identify the category of functors with its image under this embedding. Hence if \(F : \mathcal{C} \to \textit{Sets}\) is a functor, we denote the associated category cofibered in sets also by \(F\); and if \(\varphi : F \to G\) is a morphism of functors, we denote still by \(\varphi\) the corresponding morphism of categories cofibered in sets, and vice-versa. See Categories, Section 0042.

Let \(U\) be an object of \(\mathcal{C}\). We write \(\underline{U}\) for the functor \(\Mor_\mathcal{C}(U, -): \mathcal{C} \to \textit{Sets}\). This defines a fully faithful embedding of \(\mathcal C^{opp}\) into the category of functors \(\mathcal{C} \to \textit{Sets}\). Hence, if \(f : U \to V\) is a morphism, we are justified in denoting still by \(f\) the induced morphism \(\underline{V} \to \underline{U}\), and vice-versa.

Fiber products of categories cofibered in groupoids: If \(\mathcal{F} \to \mathcal{H}\) and \(\mathcal{G} \to \mathcal{H}\) are morphisms of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\), then a construction of their 2-fiber product is given by the construction for their 2-fiber product as categories over \(\mathcal{C}_\Lambda\), as described in Categories, Lemma 0040.

Products of categories cofibered in groupoids: If \(\mathcal{F}\) and \(\mathcal{G}\) are categories cofibered in groupoids over \(\mathcal{C}_\Lambda\) then their product is defined to be the \(2\)-fiber product \(\mathcal{F} \times_{\mathcal{C}_\Lambda} \mathcal{G}\) as described in Categories, Lemma 0040.

Restricting the base category: Let \(p : \mathcal{F} \to \mathcal{C}\) be a category cofibered in groupoids, and let \(\mathcal{C}'\) be a full subcategory of \(\mathcal{C}\). The restriction \(\mathcal{F}|_{\mathcal{C}'}\) is the full subcategory of \(\mathcal{F}\) whose objects lie over objects of \(\mathcal{C}'\). It is a category cofibered in groupoids via the functor \(p|_{\mathcal{C}'}: \mathcal{F}|_{\mathcal{C}'} \to \mathcal{C}'\).

Prorepresentable functors and predeformation categories

Our basic goal is to understand categories cofibered in groupoids over \(\mathcal{C}_\Lambda\) and \(\widehat{\mathcal{C}}_\Lambda\). Since \(\mathcal{C}_\Lambda\) is a full subcategory of \(\widehat{\mathcal{C}}_\Lambda\) we can restrict categories cofibred in groupoids over \(\widehat{\mathcal{C}}_\Lambda\) to \(\mathcal{C}_\Lambda\), see Remarks 06GK (06GR). In particular we can do this with functors, in particular with representable functors. The functors on \(\mathcal{C}_\Lambda\) one obtains in this way are called prorepresentable functors.

Definition

Let \(F : \mathcal{C}_\Lambda \to \textit{Sets}\) be a functor. We say \(F\) is prorepresentable if there exists an isomorphism \(F \cong \underline{R}|_{\mathcal{C}_\Lambda}\) of functors for some \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\).

Note that if \(F : \mathcal{C}_\Lambda \to \textit{Sets}\) is prorepresentable by \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\), then \[F(k) = \Mor_{\widehat{\mathcal{C}}_\Lambda}(R, k) = \{*\}\] is a singleton. The categories cofibered in groupoids over \(\mathcal{C}_\Lambda\) that are arise in deformation theory will often satisfy an analogous condition.

Definition

A predeformation category \(\mathcal{F}\) is a category cofibered in groupoids over \(\mathcal{C}_\Lambda\) such that \(\mathcal{F}(k)\) is equivalent to a category with a single object and a single morphism, i.e., \(\mathcal{F}(k)\) contains at least one object and there is a unique morphism between any two objects. A morphism of predeformation categories is a morphism of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\).

A feature of a predeformation category is the following. Let \(x_0 \in \Ob(\mathcal{F}(k))\). Then every object of \(\mathcal{F}\) comes equipped with a unique morphism to \(x_0\). Namely, if \(x\) is an object of \(\mathcal{F}\) over \(A\), then we can choose a pushforward \(x \to q_*x\) where \(q : A \to k\) is the quotient map. There is a unique isomorphism \(q_*x \to x_0\) and the composition \(x \to q_*x \to x_0\) is the desired morphism.

Remark

We say that a functor \(F: \mathcal{C}_\Lambda \to \textit{Sets}\) is a predeformation functor if the associated cofibered set is a predeformation category, i.e. if \(F(k)\) is a one element set. Thus if \(\mathcal{F}\) is a predeformation category, then \(\overline{\mathcal{F}}\) is a predeformation functor.

Remark

Let \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) be a category cofibered in groupoids, and let \(x \in \Ob(\mathcal{F}(k))\). We denote by \(\mathcal{F}_x\) the category of objects over \(x\). An object of \(\mathcal{F}_x\) is an arrow \(y \to x\). A morphism \((y \to x) \to (z \to x)\) in \(\mathcal{F}_x\) is a commutative diagram \[\xymatrix{ y \ar[rr] \ar[dr] & & z \ar[dl] \\ & x & }\] There is a forgetful functor \(\mathcal{F}_x \to \mathcal{F}\). We define the functor \(p_x : \mathcal{F}_x \to \mathcal{C}_\Lambda\) as the composition \(\mathcal{F}_x \to \mathcal{F} \xrightarrow{p} \mathcal{C}_\Lambda\). Then \(p_x : \mathcal{F}_x \to \mathcal{C}_\Lambda\) is a predeformation category (proof omitted). In this way we can pass from an arbitrary category cofibered in groupoids over \(\mathcal{C}_\Lambda\) to a predeformation category at any \(x \in \Ob(\mathcal{F}(k))\).

Formal objects and completion categories

In this section we discuss how to go between categories cofibred in groupoids over \(\mathcal{C}_\Lambda\) to categories cofibred in groupoids over \(\widehat{\mathcal{C}}_\Lambda\) and vice versa.

Definition

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). The category \(\widehat{\mathcal{F}}\) of formal objects of \(\mathcal{F}\) is the category with the following objects and morphisms.

A formal object \(\xi = (R, \xi_n, f_n)\) of \(\mathcal{F}\) consists of an object \(R\) of \(\widehat{\mathcal{C}}_\Lambda\), and a collection indexed by \(n \in \mathbf{N}\) of objects \(\xi_n\) of \(\mathcal{F}(R/\mathfrak m_R^n)\) and morphisms \(f_n : \xi_{n + 1} \to \xi_n\) lying over the projection \(R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n\).

Let \(\xi = (R, \xi_n, f_n)\) and \(\eta = (S, \eta_n, g_n)\) be formal objects of \(\mathcal{F}\). A morphism \(a : \xi \to \eta\) of formal objects consists of a map \(a_0 : R \to S\) in \(\widehat{\mathcal{C}}_\Lambda\) and a collection \(a_n : \xi_n \to \eta_n\) of morphisms of \(\mathcal{F}\) lying over \(R/\mathfrak m_R^n \to S/\mathfrak m_S^n\), such that for every \(n\) the diagram \[\xymatrix{ \xi_{n + 1} \ar[r]_{f_n} \ar[d]_{a_{n + 1}} & \xi_n \ar[d]^{a_n} \\ \eta_{n + 1} \ar[r]^{g_n} & \eta_n }\] commutes.

The category of formal objects comes with a functor \(\widehat{p}: \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda\) which sends an object \((R, \xi_n, f_n)\) to \(R\) and a morphism \((R, \xi_n, f_n) \to (S, \eta_n, g_n)\) to the map \(R \to S\).

Lemma

Let \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) be a category cofibered in groupoids. Then \(\widehat{p} : \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda\) is a category cofibered in groupoids.

Proof

Let \(R \to S\) be a ring map in \(\widehat{\mathcal{C}}_\Lambda\). Let \((R, \xi_n, f_n)\) be an object of \(\widehat{\mathcal{F}}\). For each \(n\) choose a pushforward \(\xi_n \to \eta_n\) of \(\xi_n\) along \(R/\mathfrak m_R^n \to S/\mathfrak m_S^n\). For each \(n\) there exists a unique morphism \(g_n : \eta_{n + 1} \to \eta_n\) in \(\mathcal{F}\) lying over \(S/\mathfrak m_S^{n + 1} \to S/\mathfrak m_S^n\) such that \[\xymatrix{ \xi_{n + 1} \ar[d] \ar[r]_{f_n} & \xi_n \ar[d] \\ \eta_{n + 1} \ar[r]^{g_n} & \eta_n }\] commutes (by the first axiom of a category cofibred in groupoids). Hence we obtain a morphism \((R, \xi_n, f_n) \to (S, \eta_n, g_n)\) lying over \(R \to S\), i.e., the first axiom of a category cofibred in groupoids holds for \(\widehat{\mathcal{F}}\). To see the second axiom suppose that we have morphisms \(a : (R, \xi_n, f_n) \to (S, \eta_n, g_n)\) and \(b : (R, \xi_n, f_n) \to (T, \theta_n, h_n)\) in \(\widehat{\mathcal{F}}\) and a morphism \(c_0 : S \to T\) in \(\widehat{\mathcal{C}}_\Lambda\) such that \(c_0 \circ a_0 = b_0\). By the second axiom of a category cofibred in groupoids for \(\mathcal{F}\) we obtain unique maps \(c_n : \eta_n \to \theta_n\) lying over \(S/\mathfrak m_S^n \to T/\mathfrak m_T^n\) such that \(c_n \circ a_n = b_n\). Setting \(c = (c_n)_{n \geq 0}\) gives the desired morphism \(c : (S, \eta_n, g_n) \to (T, \theta_n, h_n)\) in \(\widehat{\mathcal{F}}\) (we omit the verification that \(h_n \circ c_{n + 1} = c_n \circ g_n\)).

Definition

Let \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) be a category cofibered in groupoids. The category cofibered in groupoids \(\widehat{p} : \widehat{\mathcal F} \to \widehat{\mathcal{C}}_\Lambda\) is called the completion of \(\mathcal{F}\).

If \(\mathcal{F}\) is a category cofibered in groupoids over \(\mathcal C_\Lambda\), we have defined \(\widehat{\mathcal{F}}(R)\) for \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\) in terms of the filtration of \(R\) by powers of its maximal ideal. But suppose \(\mathcal{I} = (I_n)\) is a filtration of \(R\) by ideals inducing the \(\mathfrak{m}_R\)-adic topology. We define \(\widehat{\mathcal{F}}_\mathcal{I}(R)\) to be the category with the following objects and morphisms:

An object is a collection \((\xi_n, f_n)_{n \in \mathbf{N}}\) of objects \(\xi_n\) of \(\mathcal{F}(R/I_n)\) and morphisms \(f_n : \xi_{n + 1} \to \xi_n\) lying over the projections \(R/I_{n + 1} \to R/I_n\).
A morphism \(a : (\xi_n, f_n) \to (\eta_n, g_n)\) consists of a collection \(a_n : \xi_n \to \eta_n\) of morphisms in \(\mathcal{F}(R/I_n)\), such that for every \(n\) the diagram \[\xymatrix{ \xi_{n + 1} \ar[r]^{f_n} \ar[d]_{a_{n + 1}} & \xi_n \ar[d]^{a_n} \\ \eta_{n + 1} \ar[r]^{g_n} & \eta_n }\] commutes.

Lemma

In the situation above, \(\widehat{\mathcal{F}}_\mathcal{I}(R)\) is equivalent to the category \(\widehat{\mathcal{F}}(R)\).

Proof

An equivalence \(\widehat{\mathcal{F}}_\mathcal{I}(R) \to \widehat{\mathcal{F}}(R)\) can be defined as follows. For each \(n\), let \(m(n)\) be the least \(m\) that \(I_m \subset \mathfrak m_R^n\). Given an object \((\xi_n, f_n)\) of \(\widehat{\mathcal{F}}_\mathcal{I}(R)\), let \(\eta_n\) be the pushforward of \(\xi_{m(n)}\) along \(R/I_{m(n)} \to R/\mathfrak m_R^n\). Let \(g_n : \eta_{n + 1} \to \eta_n\) be the unique morphism of \(\mathcal{F}\) lying over \(R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n\) such that \[\xymatrix{ \xi_{m(n + 1)} \ar[rrr]_{f_{m(n)} \circ \ldots \circ f_{m(n + 1) - 1}} \ar[d] & & & \xi_{m(n)} \ar[d] \\ \eta_{n + 1} \ar[rrr]^{g_n} & & & \eta_n }\] commutes (existence and uniqueness is guaranteed by the axioms of a cofibred category). The functor \(\widehat{\mathcal{F}}_\mathcal{I}(R) \to \widehat{\mathcal{F}}(R)\) sends \((\xi_n, f_n)\) to \((R, \eta_n, g_n)\). We omit the verification that this is indeed an equivalence of categories.

Remark

Let \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) be a category cofibered in groupoids. Suppose that for each \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\) we are given a filtration \(\mathcal{I}_R\) of \(R\) by ideals. If \(\mathcal{I}_R\) induces the \(\mathfrak m_R\)-adic topology on \(R\) for all \(R\), then one can define a category \(\widehat{\mathcal{F}}_\mathcal{I}\) by mimicking the definition of \(\widehat{\mathcal{F}}\). This category comes equipped with a morphism \(\widehat{p}_\mathcal{I} : \widehat{\mathcal{F}}_\mathcal{I} \to \widehat{\mathcal{C}}_\Lambda\) making it into a category cofibered in groupoids such that \(\widehat{\mathcal{F}}_\mathcal{I}(R)\) is isomorphic to \(\widehat{\mathcal{F}}_{\mathcal{I}_R}(R)\) as defined above. The categories cofibered in groupoids \(\widehat{\mathcal{F}}_\mathcal{I}\) and \(\widehat{\mathcal{F}}\) are equivalent, by using over an object \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\) the equivalence of Lemma 06H6.

Remark

Let \(F: \mathcal{C}_\Lambda \to \textit{Sets}\) be a functor. Identifying functors with cofibered sets, the completion of \(F\) is the functor \(\widehat{F} : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}\) given by \(\widehat{F}(S) = \lim F(S/\mathfrak{m}_S^{n})\). This agrees with the definition in Schlessinger’s paper [Sch].

Remark

Let \(\mathcal{F}\) be a category cofibred in groupoids over \(\mathcal{C}_\Lambda\). We claim that there is a canonical equivalence \[can : \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \longrightarrow \mathcal{F}.\] Namely, let \(A \in \Ob(\mathcal{C}_\Lambda)\) and let \((A, \xi_n, f_n)\) be an object of \(\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda}(A)\). Since \(A\) is Artinian there is a minimal \(m \in \mathbf{N}\) such that \(\mathfrak m_A^m = 0\). Then \(can\) sends \((A, \xi_n, f_n)\) to \(\xi_m\). This functor is an equivalence of categories cofibered in groupoids by Categories, Lemma 003Z because it is an equivalence on all fibre categories by Lemma 06H6 and the fact that the \(\mathfrak m_A\)-adic topology on a local Artinian ring \(A\) comes from the zero ideal. We will frequently identify \(\mathcal{F}\) with a full subcategory of \(\widehat{\mathcal{F}}\) via a quasi-inverse to the functor \(can\).

Remark

Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a morphism of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\). Then there is an induced morphism \(\widehat{\varphi}: \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}\) of categories cofibered in groupoids over \(\widehat{\mathcal{C}}_\Lambda\). It sends an object \(\xi = (R, \xi_n, f_n)\) of \(\widehat{\mathcal{F}}\) to \((R, \varphi(\xi_n), \varphi(f_n))\), and it sends a morphism \((a_0 : R \to S, a_n : \xi_n \to \eta_n)\) between objects \(\xi\) and \(\eta\) of \(\widehat{\mathcal{F}}\) to \((a_0 : R \to S, \varphi(a_n) : \varphi(\xi_n) \to \varphi(\eta_n))\). Finally, if \(t : \varphi \to \varphi'\) is a \(2\)-morphism between \(1\)-morphisms \(\varphi, \varphi': \mathcal{F} \to \mathcal{G}\) of categories cofibred in groupoids, then we obtain a \(2\)-morphism \(\widehat{t} : \widehat{\varphi} \to \widehat{\varphi}'\). Namely, for \(\xi = (R, \xi_n, f_n)\) as above we set \(\widehat{t}_\xi = (t_{\varphi(\xi_n)})\). Hence completion defines a functor between \(2\)-categories \[\widehat{~} : \text{Cof}(\mathcal{C}_\Lambda) \longrightarrow \text{Cof}(\widehat{\mathcal{C}}_\Lambda)\] from the \(2\)-category of categories cofibred in groupoids over \(\mathcal{C}_\Lambda\) to the \(2\)-category of categories cofibred in groupoids over \(\widehat{\mathcal{C}}_\Lambda\).

Remark

We claim the completion functor of Remark 06H9 and the restriction functor \(|_{\mathcal{C}_\Lambda} : \text{Cof}(\widehat{\mathcal{C}}_\Lambda) \to \text{Cof}(\mathcal{C}_\Lambda)\) of Remarks 06GK (06GR) are “2-adjoint” in the following precise sense. Let \(\mathcal{F} \in \Ob(\text{Cof}(\mathcal{C}_\Lambda))\) and let \(\mathcal{G} \in \Ob(\text{Cof}(\widehat{\mathcal{C}}_\Lambda))\). Then there is an equivalence of categories \[\Phi : \Mor_{\mathcal{C}_\Lambda}( \mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \longrightarrow \Mor_{\widehat{\mathcal{C}}_\Lambda}(\mathcal{G}, \widehat{\mathcal{F}})\] To describe this equivalence, we define canonical morphisms \(\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}}\) and \(\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\) as follows

Let \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda))\) and let \(\xi\) be an object of the fiber category \(\mathcal{G}(R)\). Choose a pushforward \(\xi \to \xi_n\) of \(\xi\) to \(R/\mathfrak m_R^n\) for each \(n \in \mathbf{N}\), and let \(f_n : \xi_{n + 1} \to \xi_n\) be the induced morphism. Then \(\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}}\) sends \(\xi\) to \((R, \xi_n, f_n)\).

This is the equivalence \(can : \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\) of Remark 06SJ.

Having said this, the equivalence \(\Phi : \Mor_{\mathcal{C}_\Lambda}( \mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \to \Mor_{\widehat{\mathcal{C}}_\Lambda}(\mathcal{G}, \widehat{\mathcal{F}})\) sends a morphism \(\varphi : \mathcal{G}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\) to \[\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda}} \xrightarrow{\widehat{\varphi}} \widehat{\mathcal{F}}\] There is a quasi-inverse \(\Psi : \Mor_{\widehat{\mathcal{C}}_\Lambda}( \mathcal{G}, \widehat{\mathcal{F}}) \to \Mor_{\mathcal{C}_\Lambda}( \mathcal{G}|_{\mathcal{C}_\Lambda}, \mathcal{F})\) to \(\Phi\) which sends \(\psi : \mathcal{G} \to \widehat{\mathcal{F}}\) to \[\mathcal{G}|_{\mathcal{C}_\Lambda} \xrightarrow{\psi|_{\mathcal{C}_\Lambda}} \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \to \mathcal{F}.\] We omit the verification that \(\Phi\) and \(\Psi\) are quasi-inverse. We also do not address functoriality of \(\Phi\) (because it would lead into 3-category territory which we want to avoid at all cost).

Remark

For a category \(\mathcal{C}\) we denote by \(\text{CofSet}(\mathcal{C})\) the category of cofibered sets over \(\mathcal{C}\). It is a \(1\)-category isomorphic the category of functors \(\mathcal{C} \to \textit{Sets}\). See Remarks 06GK (06GP). The completion and restriction functors restrict to functors \(\widehat{~} : \text{CofSet}(\mathcal{C}_\Lambda) \to \text{CofSet}(\widehat{\mathcal{C}}_\Lambda)\) and \(|_{\mathcal{C}_\Lambda} : \text{CofSet}(\widehat{\mathcal{C}}_\Lambda) \to \text{CofSet}(\mathcal{C}_\Lambda)\) which we denote by the same symbols. As functors on the categories of cofibered sets, completion and restriction are adjoints in the usual 1-categorical sense: the same construction as in Remark 06HA defines a functorial bijection \[\Mor_{\mathcal{C}_\Lambda}(G|_{\mathcal{C}_\Lambda}, F) \longrightarrow \Mor_{\widehat{\mathcal{C}}_\Lambda}(G, \widehat{F})\] for \(F \in \Ob(\text{CofSet}(\mathcal{C}_\Lambda))\) and \(G \in \Ob(\text{CofSet}(\widehat{\mathcal{C}}_\Lambda))\). Again the map \(\widehat{F}|_{\mathcal{C}_\Lambda} \to F\) is an isomorphism.

Remark

Let \(G : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}\) be a functor that commutes with limits. Then the map \(G \to \widehat{G|_{\mathcal{C}_\Lambda}}\) described in Remark 06HA is an isomorphism. Indeed, if \(S\) is an object of \(\widehat{\mathcal{C}}_\Lambda\), then we have canonical bijections \[\widehat{G|_{\mathcal{C}_\Lambda}}(S) = \lim_n G(S/\mathfrak{m}_S^n) = G(\lim_n S/\mathfrak{m}_S^n) = G(S).\] In particular, if \(R\) is an object of \(\widehat{\mathcal{C}}_\Lambda\) then \(\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}\) because the representable functor \(\underline{R}\) commutes with limits by definition of limits.

Remark

Let \(R\) be an object of \(\widehat{\mathcal{C}}_\Lambda\). It defines a functor \(\underline{R}: \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}\) as described in Remarks 06GK (06GQ). As usual we identify this functor with the associated cofibered set. If \(\mathcal{F}\) is a cofibered category over \(\mathcal{C}_\Lambda\), then there is an equivalence of categories [06SK]\[\begin{equation} \Mor_{\mathcal{C}_\Lambda}( \underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \longrightarrow \widehat{\mathcal{F}}(R). \end{equation}\] It is given by the composition \[\Mor_{\mathcal{C}_\Lambda}( \underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F}) \xrightarrow{\Phi} \Mor_{\widehat{\mathcal{C}}_\Lambda}( \underline{R}, \widehat{\mathcal{F}}) \xrightarrow{\sim} \widehat{\mathcal{F}}(R)\] where \(\Phi\) is as in Remark 06HA and the second equivalence comes from the 2-Yoneda lemma (the cofibered analogue of Categories, Lemma 004B). Explicitly, the equivalence sends a morphism \(\varphi : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\) to the formal object \((R, \varphi(R \to R/\mathfrak{m}_R^n), \varphi(f_n))\) in \(\widehat{\mathcal{F}}(R)\), where \(f_n : R/\mathfrak m_R^{n + 1} \to R/\mathfrak m_R^n\) is the projection.

Assume a choice of pushforwards for \(\mathcal{F}\) has been made. Given any \(\xi \in \Ob(\widehat{\mathcal{F}}(R))\) we construct an explicit \(\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\) which maps to \(\xi\) under (06SK). Namely, say \(\xi = (R, \xi_n, f_n)\). An object \(\alpha\) in \(\underline{R}|_{\mathcal{C}_\Lambda}\) is the same thing as a morphism \(\alpha : R \to A\) of \(\widehat{\mathcal{C}}_\Lambda\) with \(A\) Artinian. Let \(m \in \mathbf{N}\) be minimal such that \(\mathfrak m_A^m = 0\). Then \(\alpha\) factors through a unique \(\alpha_m : R/\mathfrak m_R^m \to A\) and we can set \(\underline{\xi}(\alpha) = \alpha_{m, *}\xi_m\). We omit the description of \(\underline{\xi}\) on morphisms and we omit the proof that \(\underline{\xi}\) maps to \(\xi\) via (06SK).

Assume a choice of pushforwards for \(\widehat{\mathcal{F}}\) has been made. In this case the proof of Categories, Lemma 004B gives an explicit quasi-inverse \[\iota : \widehat{\mathcal{F}}(R) \longrightarrow \Mor_{\widehat{\mathcal{C}}_\Lambda}( \underline{R}, \widehat{\mathcal{F}})\] to the 2-Yoneda equivalence which takes \(\xi\) to the morphism \(\iota(\xi) : \underline{R} \to \widehat{\mathcal{F}}\) sending \(f \in \underline{R}(S) = \Mor_{\mathcal{C}_\Lambda}(R, S)\) to \(f_*\xi\). A quasi-inverse to (06SK) is then \[\widehat{\mathcal{F}}(R) \xrightarrow{\iota} \Mor_{\widehat{\mathcal{C}}_\Lambda}( \underline{R}, \widehat{\mathcal{F}}) \xrightarrow{\Psi} \Mor_{\mathcal{C}_\Lambda}( \underline{R}|_{\mathcal{C}_\Lambda}, \mathcal{F})\] where \(\Psi\) is as in Remark 06HA. Given \(\xi \in \Ob(\widehat{\mathcal{F}}(R))\) we have \(\Psi(\iota(\xi)) \cong \underline{\xi}\) where \(\underline{\xi}\) is as in the previous paragraph, because both are mapped to \(\xi\) under the equivalence of categories (06SK). Using \(\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda}}\) (see Remark 06HE) and unwinding the definitions of \(\Phi\) and \(\Psi\) we conclude that \(\iota(\xi)\) is isomorphic to the completion of \(\underline{\xi}\).

Remark

Let \(\mathcal{F}\) be a category cofibred in groupoids over \(\mathcal{C}_\Lambda\). Let \(\xi = (R, \xi_n, f_n)\) and \(\eta = (S, \eta_n, g_n)\) be formal objects of \(\mathcal{F}\). Let \(a = (a_n) : \xi \to \eta\) be a morphism of formal objects, i.e., a morphism of \(\widehat{\mathcal{F}}\). Let \(f = \widehat{p}(a) = a_0 : R \to S\) be the projection of \(a\) in \(\widehat{\mathcal{C}}_\Lambda\). Then we obtain a \(2\)-commutative diagram \[\xymatrix{ \underline{R}|_{\mathcal{C}_\Lambda} \ar[rd]_{\underline{\xi}} & & \underline{S}|_{\mathcal{C}_\Lambda} \ar[ll]^f \ar[ld]^{\underline{\eta}} \\ & \mathcal{F} }\] where \(\underline{\xi}\) and \(\underline{\eta}\) are the morphisms constructed in Remark 06HC. To see this let \(\alpha : S \to A\) be an object of \(\underline{S}|_{\mathcal{C}_\Lambda}\) (see loc. cit.). Let \(m \in \mathbf{N}\) be minimal such that \(\mathfrak m_A^m = 0\). We get a commutative diagram \[\xymatrix{ R \ar[d]^f \ar[r] & R/\mathfrak m_R^m \ar[d]_{f_m} \ar[rd]^{\beta_m} \\ S \ar[r] & S/\mathfrak m_S^m \ar[r]^{\alpha_m} & A }\] such that the bottom arrows compose to give \(\alpha\). Then \(\underline{\eta}(\alpha) = \alpha_{m, *}\eta_m\) and \(\underline{\xi}(\alpha \circ f) = \beta_{m, *}\xi_m\). The morphism \(a_m : \xi_m \to \eta_m\) lies over \(f_m\) hence we obtain a canonical morphism \[\underline{\xi}(\alpha \circ f) = \beta_{m, *}\xi_m \longrightarrow \underline{\eta}(\alpha) = \alpha_{m, *}\eta_m\] lying over \(\text{id}_A\) such that \[\xymatrix{ \xi_m \ar[r] \ar[d]^{a_m} & \beta_{m, *}\xi_m \ar[d] \\ \eta_m \ar[r] & \alpha_{m, *}\eta_m }\] commutes by the axioms of a category cofibred in groupoids. This defines a transformation of functors \(\underline{\xi} \circ f \to \underline{\eta}\) which witnesses the 2-commutativity of the first diagram of this remark.

Remark

According to Remark 06HC, giving a formal object \(\xi\) of \(\mathcal{F}\) is equivalent to giving a prorepresentable functor \(U : \mathcal{C}_\Lambda \to \textit{Sets}\) and a morphism \(U \to \mathcal{F}\).

Smooth morphisms

In this section we discuss smooth morphisms of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\).

Definition

Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a morphism of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\). We say \(\varphi\) is smooth if it satisfies the following condition: Let \(B \to A\) be a surjective ring map in \(\mathcal{C}_\Lambda\). Let \(y \in \Ob(\mathcal{G}(B)), x \in \Ob(\mathcal{F}(A))\), and \(y \to \varphi(x)\) be a morphism lying over \(B \to A\). Then there exists \(x' \in \Ob(\mathcal{F}(B))\), a morphism \(x' \to x\) lying over \(B \to A\), and a morphism \(\varphi(x') \to y\) lying over \(\text{id}: B \to B\), such that the diagram \[\xymatrix{ \varphi(x') \ar[r] \ar[dr] & y \ar[d] \\ & \varphi(x) }\] commutes.

Lemma

Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a morphism of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\). Then \(\varphi\) is smooth if the condition in Definition 06HG is assumed to hold only for small extensions \(B \to A\).

Proof

Let \(B \to A\) be a surjective ring map in \(\mathcal{C}_\Lambda\). Let \(y \in \Ob(\mathcal{G}(B))\), \(x \in \Ob(\mathcal{F}(A))\), and \(y \to \varphi(x)\) be a morphism lying over \(B \to A\). By Lemma 06GE we can factor \(B \to A\) into small extensions \(B = B_n \to B_{n-1} \to \ldots \to B_0 = A\). We argue by induction on \(n\). If \(n = 1\) the result is true by assumption. If \(n > 1\), then denote \(f : B = B_n \to B_{n - 1}\) and denote \(g : B_{n - 1} \to B_0 = A\). Choose a pushforward \(y \to f_* y\) of \(y\) along \(f\), so that the morphism \(y \to \varphi(x)\) factors as \(y \to f_* y \to \varphi(x)\). By the induction hypothesis we can find \(x_{n - 1} \to x\) lying over \(g : B_{n - 1} \to A\) and \(a : \varphi(x_{n - 1}) \to f_*y\) lying over \(\text{id} : B_{n - 1} \to B_{n - 1}\) such that \[\xymatrix{ \varphi(x_{n - 1}) \ar[r]_-a \ar[dr] & f_*y \ar[d] \\ & \varphi(x) }\] commutes. We can apply the assumption to the composition \(y \to \varphi(x_{n - 1})\) of \(y \to f_*y\) with \(a^{-1} : f_*y \to \varphi(x_{n - 1})\). We obtain \(x_n \to x_{n - 1}\) lying over \(B_n \to B_{n - 1}\) and \(\varphi(x_n) \to y\) lying over \(\text{id} : B_n \to B_n\) so that the diagram \[\xymatrix{ \varphi(x_n) \ar[r] \ar[d] & y \ar[d] \\ \varphi(x_{n - 1}) \ar[r]^-a \ar[dr] & f_*y \ar[d] \\ & \varphi(x) }\] commutes. Then the composition \(x_n \to x_{n - 1} \to x\) and \(\varphi(x_n) \to y\) are the morphisms required by the definition of smoothness.

Remark

Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a morphism of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\). Let \(B \to A\) be a ring map in \(\mathcal{C}_\Lambda\). Choices of pushforwards along \(B \to A\) for objects in the fiber categories \(\mathcal{F}(B)\) and \(\mathcal{G}(B)\) determine functors \(\mathcal{F}(B) \to \mathcal{F}(A)\) and \(\mathcal{G}(B) \to \mathcal{G}(A)\) fitting into a \(2\)-commutative diagram \[\xymatrix{ \mathcal{F}(B) \ar[r]^{\varphi} \ar[d] & \mathcal{G}(B) \ar[d] \\ \mathcal{F}(A) \ar[r]^{\varphi} & \mathcal{G}(A) . }\] Hence there is an induced functor \(\mathcal{F}(B) \to \mathcal{F}(A) \times_{\mathcal{G}(A)} \mathcal{G}(B)\). Unwinding the definitions shows that \(\varphi : \mathcal{F} \to \mathcal{G}\) is smooth if and only if this induced functor is essentially surjective whenever \(B \to A\) is surjective (or equivalently, by Lemma 06HH, whenever \(B \to A\) is a small extension).

Remark

The characterization of smooth morphisms in Remark 06HI is analogous to Schlessinger’s notion of a smooth morphism of functors, cf. [Sch, Definition 2.2.]. In fact, when \(\mathcal{F}\) and \(\mathcal{G}\) are cofibered in sets then our notion is equivalent to Schlessinger’s. Namely, in this case let \(F, G : \mathcal{C}_\Lambda \to \textit{Sets}\) be the corresponding functors, see Remarks 06GK (06GP). Then \(F \to G\) is smooth if and only if for every surjection of rings \(B \to A\) in \(\mathcal{C}_\Lambda\) the map \(F(B) \to F(A) \times_{G(A)} G(B)\) is surjective.

Remark

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). Then the morphism \(\mathcal{F} \to \overline{\mathcal{F}}\) is smooth. Namely, suppose that \(f : B \to A\) is a ring map in \(\mathcal{C}_\Lambda\). Let \(x \in \Ob(\mathcal{F}(A))\) and let \(\overline{y} \in \overline{\mathcal{F}}(B)\) be the isomorphism class of \(y \in \Ob(\mathcal{F}(B))\) such that \(\overline{f_*y} = \overline{x}\). Then we simply take \(x' = y\), the implied morphism \(x' = y \to x\) over \(B \to A\), and the equality \(\overline{x'} = \overline{y}\) as the solution to the problem posed in Definition 06HG.

If \(R \to S\) is a ring map \(\widehat{\mathcal{C}}_\Lambda\), then there is an induced morphism \(\underline{S} \to \underline{R}\) between the functors \(\underline{S}, \underline{R}: \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}\). In this situation, smoothness of the restriction \(\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}\) is a familiar notion:

Lemma

Let \(R \to S\) be a ring map in \(\widehat{\mathcal{C}}_\Lambda\). Then the induced morphism \(\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}\) is smooth if and only if \(S\) is a power series ring over \(R\).

Proof

Assume \(S\) is a power series ring over \(R\). Say \(S = R[[x_1, \ldots, x_n]]\). Smoothness of \(\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}\) means the following (see Remark 06HJ): Given a surjective ring map \(B \to A\) in \(\mathcal{C}_\Lambda\), a ring map \(R \to B\), a ring map \(S \to A\) such that the solid diagram \[\xymatrix{ S \ar[r] \ar@{..>}[rd] & A \\ R \ar[u] \ar[r] & B \ar[u] }\] is commutative then a dotted arrow exists making the diagram commute. (Note the similarity with Algebra, Definition 00TI.) To construct the dotted arrow choose elements \(b_i \in B\) whose images in \(A\) are equal to the images of \(x_i\) in \(A\). Note that \(b_i \in \mathfrak m_B\) as \(x_i\) maps to an element of \(\mathfrak m_A\). Hence there is a unique \(R\)-algebra map \(R[[x_1, \ldots, x_n]] \to B\) which maps \(x_i\) to \(b_i\) and which can serve as our dotted arrow.

Conversely, assume \(\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}\) is smooth. Let \(x_1, \ldots, x_n \in S\) be elements whose images form a basis in the relative cotangent space \(\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2)\) of \(S\) over \(R\). Set \(T = R[[X_1, \ldots, X_n]]\). Note that both \[S/(\mathfrak m_R S + \mathfrak m_S^2) \cong R/\mathfrak m_R[x_1, \ldots, x_n]/(x_ix_j)\] and \[T/(\mathfrak m_R T + \mathfrak m_T^2) \cong R/\mathfrak m_R[X_1, \ldots, X_n]/(X_iX_j).\] Let \(S/(\mathfrak m_R S + \mathfrak m_S^2) \to T/(\mathfrak m_R T + \mathfrak m_T^2)\) be the local \(R\)-algebra isomorphism given by mapping the class of \(x_i\) to the class of \(X_i\). Let \(f_1 : S \to T/(\mathfrak m_R T + \mathfrak m_T^2)\) be the composition \(S \to S/(\mathfrak m_R S + \mathfrak m_S^2) \to T/(\mathfrak m_R T + \mathfrak m_T^2)\). The assumption that \(\underline{S}|_{\mathcal{C}_\Lambda} \to \underline{R}|_{\mathcal{C}_\Lambda}\) is smooth means we can lift \(f_1\) to a map \(f_2 : S \to T/\mathfrak{m}_T^2\), then to a map \(f_3 : S \to T/\mathfrak{m}_T^3\), and so on, for all \(n \geq 1\). Thus we get an induced map \(f : S \to T = \lim T/\mathfrak m_T^n\) of local \(R\)-algebras. By our choice of \(f_1\), the map \(f\) induces an isomorphism \(\mathfrak m_S/(\mathfrak m_R S + \mathfrak m_S^2) \to \mathfrak m_T/(\mathfrak m_R T + \mathfrak m_T^2)\) of relative cotangent spaces. Hence \(f\) is surjective by Lemma 06GZ (where we think of \(f\) as a map in \(\widehat{\mathcal{C}}_R\)). Choose preimages \(y_i \in S\) of \(X_i \in T\) under \(f\). As \(T\) is a power series ring over \(R\) there exists a local \(R\)-algebra homomorphism \(s : T \to S\) mapping \(X_i\) to \(y_i\). By construction \(f \circ s = \text{id}\). Then \(s\) is injective. But \(s\) induces an isomorphism on relative cotangent spaces since \(f\) does, so it is also surjective by Lemma 06GZ again. Hence \(s\) and \(f\) are isomorphisms.

Smooth morphisms satisfy the following functorial properties.

Lemma

Let \(\varphi : \mathcal{F} \to \mathcal{G}\) and \(\psi : \mathcal{G} \to \mathcal{H}\) be morphisms of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\).

If \(\varphi\) and \(\psi\) are smooth, then \(\psi \circ \varphi\) is smooth.

If \(\varphi\) is essentially surjective and \(\psi \circ \varphi\) is smooth, then \(\psi\) is smooth.

If \(\mathcal{G}' \to \mathcal{G}\) is a morphism of categories cofibered in groupoids and \(\varphi\) is smooth, then \(\mathcal{F} \times_\mathcal{G} \mathcal{G}' \to \mathcal{G}'\) is smooth.

Proof

Statements (1) and (2) follow immediately from the definitions. Proof of (3) omitted. Hints: use the formulation of smoothness given in Remark 06HI and use that \(\mathcal{F} \times_\mathcal{G} \mathcal{G}'\) is the \(2\)-fibre product, see Remarks 06GK (06SI).

Lemma

Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a smooth morphism of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\). Assume \(\varphi : \mathcal{F}(k) \to \mathcal{G}(k)\) is essentially surjective. Then \(\varphi : \mathcal{F} \to \mathcal{G}\) and \(\widehat{\varphi} : \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}\) are essentially surjective.

Proof

Let \(y\) be an object of \(\mathcal{G}\) lying over \(A \in \Ob(\mathcal{C}_\Lambda)\). Let \(y \to y_0\) be a pushforward of \(y\) along \(A \to k\). By the assumption on essential surjectivity of \(\varphi : \mathcal{F}(k) \to \mathcal{G}(k)\) there exist an object \(x_0\) of \(\mathcal{F}\) lying over \(k\) and an isomorphism \(y_0 \to \varphi(x_0)\). Smoothness of \(\varphi\) implies there exists an object \(x\) of \(\mathcal{F}\) over \(A\) whose image \(\varphi(x)\) is isomorphic to \(y\). Thus \(\varphi : \mathcal{F} \to \mathcal{G}\) is essentially surjective.

Let \(\eta = (R, \eta_n, g_n)\) be an object of \(\widehat{\mathcal{G}}\). We construct an object \(\xi\) of \(\widehat{\mathcal{F}}\) with an isomorphism \(\eta \to \varphi(\xi)\). By the assumption on essential surjectivity of \(\varphi : \mathcal{F}(k) \to \mathcal{G}(k)\), there exists a morphism \(\eta_1 \to \varphi(\xi_1)\) in \(\mathcal{G}(k)\) for some \(\xi_1 \in \Ob(\mathcal{F}(k))\). The morphism \(\eta_2 \xrightarrow{g_1} \eta_1 \to \varphi(\xi_1)\) lies over the surjective ring map \(R/\mathfrak m_R^2 \to k\), hence by smoothness of \(\varphi\) there exists \(\xi_2 \in \Ob(\mathcal{F}(R/\mathfrak m_R^2))\), a morphism \(f_1: \xi_2 \to \xi_1\) lying over \(R/\mathfrak m_R^2 \to k\), and a morphism \(\eta_2 \to \varphi(\xi_2)\) such that \[\xymatrix{ \varphi(\xi_2) \ar[r]^{\varphi(f_1)} & \varphi(\xi_{1}) \\ \eta_2 \ar[u] \ar[r]^{g_1} & \eta_1 \ar[u] \\ }\] commutes. Continuing in this way we construct an object \(\xi = (R, \xi_n, f_n)\) of \(\widehat{\mathcal{F}}\) and a morphism \(\eta \to \varphi(\xi) = (R, \varphi(\xi_n), \varphi(f_n))\) in \(\widehat{\mathcal{G}}(R)\).

Later we are interested in producing smooth morphisms from prorepresentable functors to predeformation categories \(\mathcal{F}\). By the discussion in Remark 06HC these morphisms correspond to certain formal objects of \(\mathcal{F}\). More precisely, these are the so-called versal formal objects of \(\mathcal{F}\).

Definition

Let \(\mathcal{F}\) be a category cofibered in groupoids. Let \(\xi\) be a formal object of \(\mathcal{F}\) lying over \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\). We say \(\xi\) is versal if the corresponding morphism \(\underline{\xi}: \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\) of Remark 06HC is smooth.

Remark

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal C_\Lambda\), and let \(\xi\) be a formal object of \(\mathcal{F}\). It follows from the definition of smoothness that versality of \(\xi\) is equivalent to the following condition: If \[\xymatrix{ & y \ar[d] \\ \xi \ar[r] & x }\] is a diagram in \(\widehat{\mathcal{F}}\) such that \(y \to x\) lies over a surjective map \(B \to A\) of Artinian rings (we may assume it is a small extension), then there exists a morphism \(\xi \to y\) such that \[\xymatrix{ & y \ar[d] \\ \xi \ar[r] \ar[ur] & x }\] commutes. In particular, the condition that \(\xi\) be versal does not depend on the choices of pushforwards made in the construction of \(\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\) in Remark 06HC.

Lemma

Let \(\mathcal{F}\) be a predeformation category. Let \(\xi\) be a versal formal object of \(\mathcal{F}\). For any formal object \(\eta\) of \(\widehat{\mathcal{F}}\), there exists a morphism \(\xi \to \eta\).

Proof

By assumption the morphism \(\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\) is smooth. Then \(\iota(\xi) : \underline{R} \to \widehat{\mathcal{F}}\) is the completion of \(\underline{\xi}\), see Remark 06HC. By Lemma 06HN there exists an object \(f\) of \(\underline{R}\) such that \(\iota(\xi)(f) = \eta\). Then \(f\) is a ring map \(f : R \to S\) in \(\widehat{\mathcal{C}}_\Lambda\). And \(\iota(\xi)(f) = \eta\) means that \(f_*\xi \cong \eta\) which means exactly that there is a morphism \(\xi \to \eta\) lying over \(f\).

Smooth or unobstructed categories

Let \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) be a category cofibered in groupoids. We can consider \(\mathcal{C}_\Lambda\) as a category cofibered in groupoids over \(\mathcal{C}_\Lambda\) using the identity functor. In this way \(p : \mathcal{F} \longrightarrow \mathcal{C}_\Lambda\) becomes a morphism of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\).

Definition

Let \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) be a category cofibered in groupoids. We say \(\mathcal{F}\) is smooth or unobstructed if its structure morphism \(p\) is smooth in the sense of Definition 06HG.

This is the “absolute” notion of smoothness for a category cofibered in groupoids over \(\mathcal{C}_\Lambda\), although it would be more correct to say that \(\mathcal{F}\) is smooth over \(\Lambda\). One has to be careful with the phrase “\(\mathcal{F}\) is unobstructed”: it may happen that \(\mathcal{F}\) has an obstruction theory with nonvanishing obstruction spaces even though \(\mathcal{F}\) is smooth.

Remark

Suppose \(\mathcal{F}\) is a predeformation category admitting a smooth morphism \(\varphi : \mathcal U \to \mathcal{F}\) from a predeformation category \(\mathcal{U}\). Then by Lemma 06HN \(\varphi\) is essentially surjective, so by Lemma 06HM \(p: \mathcal{F} \to \mathcal{C}_\Lambda\) is smooth if and only if the composition \(\mathcal U \xrightarrow{\varphi} \mathcal{F} \xrightarrow{p} \mathcal{C}_\Lambda\) is smooth, i.e. \(\mathcal{F}\) is smooth if and only if \(\mathcal{U}\) is smooth.

Lemma

Let \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\). The following are equivalent

\(\underline{R}|_{\mathcal{C}_\Lambda}\) is smooth,

\(\Lambda \to R\) is formally smooth in the \(\mathfrak m_R\)-adic topology,

\(\Lambda \to R\) is flat and \(R \otimes_\Lambda k'\) is geometrically regular over \(k'\), and

\(\Lambda \to R\) is flat and \(k' \to R \otimes_\Lambda k'\) is formally smooth in the \(\mathfrak m_R\)-adic topology.

In the classical case, these are also equivalent to

\(R\) is isomorphic to \(\Lambda[[x_1, \ldots, x_n]]\) for some \(n\).

Proof

Smoothness of \(p : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{C}_\Lambda\) means that given \(B \to A\) surjective in \(\mathcal{C}_\Lambda\) and given \(R \to A\) we can find the dotted arrow in the diagram \[\xymatrix{ R \ar[r] \ar@{-->}[rd] & A \\ \Lambda \ar[r] \ar[u] & B \ar[u] }\] This is certainly true if \(\Lambda \to R\) is formally smooth in the \(\mathfrak m_R\)-adic topology, see More on Algebra, Definitions 07NI and 07EB. Conversely, if this holds, then we see that \(\Lambda \to R\) is formally smooth in the \(\mathfrak m_R\)-adic topology by More on Algebra, Lemma 0DYG. Thus (1) and (2) are equivalent.

The equivalence of (2), (3), and (4) is More on Algebra, Proposition 07NQ. The equivalence with (5) follows for example from Lemma 06HL and the fact that \(\mathcal{C}_\Lambda\) is the same as \(\underline{\Lambda}|_{\mathcal{C}_\Lambda}\) in the classical case.

Lemma

Let \(\mathcal{F}\) be a predeformation category. Let \(\xi\) be a versal formal object of \(\mathcal{F}\) lying over \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\). The following are equivalent

\(\mathcal{F}\) is unobstructed, and

\(\Lambda \to R\) is formally smooth in the \(\mathfrak m_R\)-adic topology.

In the classical case these are also equivalent to

\(R \cong \Lambda[[x_1, \ldots, x_n]]\) for some \(n\).

Proof

If (1) holds, i.e., if \(\mathcal{F}\) is unobstructed, then the composition \[\underline{R}|_{\mathcal{C}_\Lambda} \xrightarrow{\underline{\xi}} \mathcal{F} \to \mathcal{C}_\Lambda\] is smooth, see Lemma 06HM. Hence we see that (2) holds by Lemma 0DYL. Conversely, if (2) holds, then the composition is smooth and moreover the first arrow is essentially surjective by Lemma 06HT. Hence we find that the second arrow is smooth by Lemma 06HM which means that \(\mathcal{F}\) is unobstructed by definition. The equivalence with (3) in the classical case follows from Lemma 0DYL.

Lemma

There exists an \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\) such that the equivalent conditions of Lemma 0DYL hold and moreover \(H_1(L_{k/\Lambda}) = \mathfrak m_R/\mathfrak m_R^2\) and \(\Omega_{R/\Lambda} \otimes_R k = \Omega_{k/\Lambda}\).

Proof

In the classical case we choose \(R = \Lambda\). More generally, if the residue field extension \(k/k'\) is separable, then there exists a unique finite étale extension \(\Lambda^\wedge \to R\) (Algebra, Lemmas 04GM and 04GK) of the completion \(\Lambda^\wedge\) of \(\Lambda\) inducing the extension \(k/k'\) on residue fields.

In the general case we proceed as follows. Choose a smooth \(\Lambda\)-algebra \(P\) and a \(\Lambda\)-algebra surjection \(P \to k\). (For example, let \(P\) be a polynomial algebra.) Denote \(\mathfrak m_P\) the kernel of \(P \to k\). The Jacobi-Zariski sequence, see (06S7) and Algebra, Lemma 00S2, is an exact sequence \[0 \to H_1(\NL_{k/\Lambda}) \to \mathfrak m_P/\mathfrak m_P^2 \to \Omega_{P/\Lambda} \otimes_P k \to \Omega_{k/\Lambda} \to 0\] We have the \(0\) on the left because \(P/k\) is smooth, hence \(\NL_{P/\Lambda}\) is quasi-isomorphic to a finite projective module placed in degree \(0\), hence \(H_1(\NL_{P/\Lambda} \otimes_P k) = 0\). Suppose \(f \in \mathfrak m_P\) maps to a nonzero element of \(\Omega_{P/\Lambda} \otimes_P k\). Setting \(P' = P/(f)\) we have a \(\Lambda\)-algebra surjection \(P' \to k\). Observe that \(P'\) is smooth at \(\mathfrak m_{P'}\): this follows from More on Morphisms, Lemma 057C. Thus after replacing \(P\) by a principal localization of \(P'\), we see that \(\dim(\mathfrak m_P/\mathfrak m_P^2)\) decreases. Repeating finitely many times, we may assume the map \(\mathfrak m_P/\mathfrak m_P^2 \to \Omega_{P/\Lambda} \otimes_P k\) is zero so that the exact sequence breaks into isomorphisms \(H_1(L_{k/\Lambda}) = \mathfrak m_P/\mathfrak m_P^2\) and \(\Omega_{P/\Lambda} \otimes_P k = \Omega_{k/\Lambda}\).

Let \(R\) be the \(\mathfrak m_P\)-adic completion of \(P\). Then \(R\) is an object of \(\widehat{\mathcal{C}}_\Lambda\). Namely, it is a complete local Noetherian ring (see Algebra, Lemma 0316) and its residue field is identified with \(k\). We claim that \(R\) works.

First observe that the map \(P \to R\) induces isomorphisms \(\mathfrak m_P/\mathfrak m_P^2 = \mathfrak m_R/\mathfrak m_R^2\) and \(\Omega_{P/\Lambda} \otimes_P k = \Omega_{R/\Lambda} \otimes_R k\). This is true because both \(\mathfrak m_P/\mathfrak m_P^2\) and \(\Omega_{P/\Lambda} \otimes_P k\) only depend on the \(\Lambda\)-algebra \(P/\mathfrak m_P^2\), see Algebra, Lemma 02HQ, the same holds for \(R\) and we have \(P/\mathfrak m_P^2 = R/\mathfrak m_R^2\). Using the functoriality of the Jacobi-Zariski sequence (06S8) we deduce that \(H_1(L_{k/\Lambda}) = \mathfrak m_R/\mathfrak m_R^2\) and \(\Omega_{R/\Lambda} \otimes_R k = \Omega_{k/\Lambda}\) as the same is true for \(P\).

Finally, since \(\Lambda \to P\) is smooth we see that \(\Lambda \to P\) is formally smooth by Algebra, Proposition 00TN. Then \(\Lambda \to P\) is formally smooth for the \(\mathfrak m_P\)-adic topology by More on Algebra, Lemma 07EC. This property is inherited by the completion \(R\) by More on Algebra, Lemma 07ED and the proof is complete. In fact, it turns out that whenever \(\underline{R}|_{\mathcal{C}_\Lambda}\) is smooth, then \(R\) is isomorphic to a completion of a smooth algebra over \(\Lambda\), but we won’t use this.

Example

Here is a more explicit example of an \(R\) as in Lemma 06SM. Let \(p\) be a prime number and let \(n \in \mathbf{N}\). Let \(\Lambda = \mathbf{F}_p(t_1, t_2, \ldots, t_n)\) and let \(k = \mathbf{F}_p(x_1, \ldots, x_n)\) with map \(\Lambda \to k\) given by \(t_i \mapsto x_i^p\). Then we can take \[R = \Lambda[x_1, \ldots, x_n]^\wedge_{(x_1^p - t_1, \ldots, x_n^p - t_n)}\] We cannot do “better” in this example, i.e., we cannot approximate \(\mathcal{C}_\Lambda\) by a smaller smooth object of \(\widehat{\mathcal{C}}_\Lambda\) (one can argue that the dimension of \(R\) has to be at least \(n\) since the map \(\Omega_{R/\Lambda} \otimes_R k \to \Omega_{k/\Lambda}\) is surjective). We will discuss this phenomenon later in more detail.

Schlessinger’s conditions

In the following we often consider fibre products \(A_1 \times_A A_2\) of rings in the category \(\mathcal{C}_\Lambda\). We have seen in Example 06S4 that such a fibre product may not always be an object of \(\mathcal{C}_\Lambda\). However, in virtually all cases below one of the two maps \(A_i \to A\) is surjective and \(A_1 \times_A A_2\) will be an object of \(\mathcal{C}_\Lambda\) by Lemma 06GH. We will use this result without further mention.

We denote by \(k[\epsilon]\) the ring of dual numbers over \(k\). More generally, for a \(k\)-vector space \(V\), we denote by \(k[V]\) the \(k\)-algebra whose underlying vector space is \(k \oplus V\) and whose multiplication is given by \((a, v) \cdot (a', v') = (aa', av' + a'v)\). When \(V = k\), \(k[V]\) is the ring of dual numbers over \(k\). For any finite dimensional \(k\)-vector space \(V\) the ring \(k[V]\) is in \(\mathcal{C}_\Lambda\).

Definition

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal C_\Lambda\). We define conditions (S1) and (S2) on \(\mathcal{F}\) as follows:

Every diagram in \(\mathcal{F}\) \[\vcenter{ \xymatrix{ & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & A_2 \ar[d] \\ A_1 \ar[r] & A } }\] in \(\mathcal{C}_\Lambda\) with \(A_2 \to A\) surjective can be completed to a commutative diagram \[\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A. } }\]

The condition of (S1) holds for diagrams in \(\mathcal{F}\) lying over a diagram in \(\mathcal{C}_\Lambda\) of the form \[\xymatrix{ & k[\epsilon] \ar[d] \\ A \ar[r] & k. }\] Moreover, if we have two commutative diagrams in \(\mathcal{F}\) \[\vcenter{ \xymatrix{ y \ar[r]_c \ar[d]_a & x_\epsilon \ar[d]^e \\ x \ar[r]^d & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r]_{c'} \ar[d]_{a'} & x_\epsilon \ar[d]^e \\ x \ar[r]^d & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\ A \ar[r] & k } }\] then there exists a morphism \(b : y \to y'\) in \(\mathcal{F}(A \times_k k[\epsilon])\) such that \(a = a' \circ b\).

We can partly explain the meaning of conditions (S1) and (S2) in terms of fibre categories. Suppose that \(f_1 : A_1 \to A\) and \(f_2 : A_2 \to A\) are ring maps in \(\mathcal{C}_\Lambda\) with \(f_2\) surjective. Denote \(p_i : A_1 \times_A A_2 \to A_i\) the projection maps. Assume a choice of pushforwards for \(\mathcal{F}\) has been made. Then the commutative diagram of rings translates into a \(2\)-commutative diagram \[\xymatrix{ \mathcal{F}(A_1 \times_A A_2) \ar[r]_-{p_{2, *}} \ar[d]_{p_{1, *}} & \mathcal{F}(A_2) \ar[d]^{f_{2, *}} \\ \mathcal{F}(A_1) \ar[r]^{f_{1, *}} & \mathcal{F}(A) }\] of fibre categories whence a functor [06SP]\[\begin{equation} \mathcal{F}(A_1 \times_A A_2) \to \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2) \end{equation}\] into the \(2\)-fibre product of categories. Condition (S1) requires that this functor be essentially surjective. The first part of condition (S2) requires that this functor be a essentially surjective if \(f_2\) equals the map \(k[\epsilon] \to k\). Moreover in this case, the second part of (S2) implies that two objects which become isomorphic in the target are isomorphic in the source (but it is not equivalent to this statement). The advantage of stating the conditions as in the definition is that no choices have to be made.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal C_\Lambda\). Then \(\mathcal{F}\) satisfies (S1) if the condition of (S1) is assumed to hold only when \(A_2 \to A\) is a small extension.

Proof

Proof omitted. Hints: apply Lemma 06GE and use induction similar to the proof of Lemma 06HH.

Remark

When \(\mathcal{F}\) is cofibered in sets, conditions (S1) and (S2) are exactly conditions (H1) and (H2) from Schlessinger’s paper [Sch]. Namely, for a functor \(F: \mathcal{C}_\Lambda \to \textit{Sets}\), conditions (S1) and (S2) state:

If \(A_1 \to A\) and \(A_2 \to A\) are maps in \(\mathcal{C}_\Lambda\) with \(A_2 \to A\) surjective, then the induced map \(F(A_1 \times_A A_2) \to F(A_1) \times_{F(A)} F(A_2)\) is surjective.

If \(A \to k\) is a map in \(\mathcal{C}_\Lambda\), then the induced map \(F(A \times_k k[\epsilon]) \to F(A) \times_{F(k)} F(k[\epsilon])\) is bijective.

The injectivity of the map \(F(A \times_k k[\epsilon]) \to F(A) \times_{F(k)} F(k[\epsilon])\) comes from the second part of condition (S2) and the fact that morphisms are identities.

Lemma

Let \(\mathcal{F}\) be a category cofibred in groupoids over \(\mathcal{C}_\Lambda\). If \(\mathcal{F}\) satisfies (S2), then the condition of (S2) also holds when \(k[\epsilon]\) is replaced by \(k[V]\) for any finite dimensional \(k\)-vector space \(V\).

Proof

In the case that \(\mathcal{F}\) is cofibred in sets, i.e., corresponds to a functor \(F : \mathcal{C}_\Lambda \to \textit{Sets}\) this follows from the description of (S2) for \(F\) in Remark 06HY and the fact that \(k[V] \cong k[\epsilon] \times_k \ldots \times_k k[\epsilon]\) with \(\dim_k V\) factors. The case of functors is what we will use in the rest of this chapter.

We prove the general case by induction on \(\dim(V)\). If \(\dim(V) = 1\), then \(k[V] \cong k[\epsilon]\) and the result holds by assumption. If \(\dim(V) > 1\) we write \(V = V' \oplus k\epsilon\). Pick a diagram \[\vcenter{ \xymatrix{ & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & k[V] \ar[d] \\ A \ar[r] & k } }\] Choose a morphism \(x_V \to x_{V'}\) lying over \(k[V] \to k[V']\) and a morphism \(x_V \to x_\epsilon\) lying over \(k[V] \to k[\epsilon]\). Note that the morphism \(x_V \to x_0\) factors as \(x_V \to x_{V'} \to x_0\) and as \(x_V \to x_\epsilon \to x_0\). By induction hypothesis we can find a diagram \[\vcenter{ \xymatrix{ y' \ar[d] \ar[r] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V'] \ar[d] \ar[r] & k[V'] \ar[d] \\ A \ar[r] & k } }\] This gives us a commutative diagram \[\vcenter{ \xymatrix{ & x_\epsilon \ar[d] \\ y' \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & k[\epsilon] \ar[d] \\ A \times_k k[V'] \ar[r] & k } }\] Hence by (S2) we get a commutative diagram \[\vcenter{ \xymatrix{ y \ar[d] \ar[r] & x_\epsilon \ar[d] \\ y' \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ (A \times_k k[V']) \times_k k[\epsilon] \ar[d] \ar[r] & k[\epsilon] \ar[d] \\ A \times_k k[V'] \ar[r] & k } }\] Note that \((A \times_k k[V']) \times_k k[\epsilon] = A \times_k k[V' \oplus k\epsilon] = A \times_k k[V]\). We claim that \(y\) fits into the correct commutative diagram. To see this we let \(y \to y_V\) be a morphism lying over \(A \times_k k[V] \to k[V]\). We can factor the morphisms \(y \to y' \to x_{V'}\) and \(y \to x_\epsilon\) through the morphism \(y \to y_V\) (by the axioms of categories cofibred in groupoids). Hence we see that both \(y_V\) and \(x_V\) fit into commutative diagrams \[\vcenter{ \xymatrix{ y_V \ar[r] \ar[d] & x_\epsilon \ar[d] \\ x_{V'} \ar[r] & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ x_V \ar[r] \ar[d] & x_\epsilon \ar[d] \\ x_{V'} \ar[r] & x_0 } }\] and hence by the second part of (S2) there exists an isomorphism \(y_V \to x_V\) compatible with \(y_V \to x_{V'}\) and \(x_V \to x_{V'}\) and in particular compatible with the maps to \(x_0\). The composition \(y \to y_V \to x_V\) then fits into the required commutative diagram \[\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V] \ar[d] \ar[r] & k[V] \ar[d] \\ A \ar[r] & k } }\] In this way we see that the first part of \((S2)\) holds with \(k[\epsilon]\) replaced by \(k[V]\).

To prove the second part suppose given two commutative diagrams \[\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r] \ar[d] & x_V \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V] \ar[d] \ar[r] & k[V] \ar[d] \\ A \ar[r] & k } }\] We will use the morphisms \(x_V \to x_{V'} \to x_0\) and \(x_V \to x_\epsilon \to x_0\) introduced in the first paragraph of the proof. Choose morphisms \(y \to y_{V'}\) and \(y' \to y'_{V'}\) lying over \(A \times_k k[V] \to A \times_k k[V']\). The axioms of a cofibred category imply we can find commutative diagrams \[\vcenter{ \xymatrix{ y_{V'} \ar[r] \ar[d] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y'_{V'} \ar[r] \ar[d] & x_{V'} \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[V'] \ar[d] \ar[r] & k[V'] \ar[d] \\ A \ar[r] & k } }\] By induction hypothesis we obtain an isomorphism \(b : y_{V'} \to y'_{V'}\) compatible with the morphisms \(y_{V'} \to x\) and \(y'_{V'} \to x\), in particular compatible with the morphisms to \(x_0\). Then we have commutative diagrams \[\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_\epsilon \ar[d] \\ y'_{V'} \ar[r] & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r] \ar[d] & x_\epsilon \ar[d] \\ y'_{V'} \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[d] \ar[r] & k[\epsilon] \ar[d] \\ A \ar[r] & k } }\] where the morphism \(y \to y'_{V'}\) is the composition \(y \to y_{V'} \xrightarrow{b} y'_{V'}\) and where the morphisms \(y \to x_\epsilon\) and \(y' \to x_\epsilon\) are the compositions of the maps \(y \to x_V\) and \(y' \to x_V\) with the morphism \(x_V \to x_\epsilon\). Then the second part of (S2) guarantees the existence of an isomorphism \(y \to y'\) compatible with the maps to \(y'_{V'}\), in particular compatible with the maps to \(x\) (because \(b\) was compatible with the maps to \(x\)).

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\).

If \(\mathcal{F}\) satisfies (S1), then so does \(\overline{\mathcal{F}}\).

If \(\mathcal{F}\) satisfies (S2), then so does \(\overline{\mathcal{F}}\) provided at least one of the following conditions is satisfied

\(\mathcal{F}\) is a predeformation category,

the category \(\mathcal{F}(k)\) is a set or a setoid, or

for any morphism \(x_\epsilon \to x_0\) of \(\mathcal{F}\) lying over \(k[\epsilon] \to k\) the pushforward map \(\text{Aut}_{k[\epsilon]}(x_\epsilon) \to \text{Aut}_k(x_0)\) is surjective.

Proof

Assume \(\mathcal{F}\) has (S1). Suppose we have ring maps \(f_i : A_i \to A\) in \(\mathcal{C}_\Lambda\) with \(f_2\) surjective. Let \(x_i \in \mathcal{F}(A_i)\) such that the pushforwards \(f_{1, *}(x_1)\) and \(f_{2, *}(x_2)\) are isomorphic. Then we can denote \(x\) an object of \(\mathcal{F}\) over \(A\) isomorphic to both of these and we obtain a diagram as in (S1). Hence we find an object \(y\) of \(\mathcal{F}\) over \(A_1 \times_A A_2\) whose pushforward to \(A_1\), resp. \(A_2\) is isomorphic to \(x_1\), resp. \(x_2\). In this way we see that (S1) holds for \(\overline{\mathcal{F}}\).

Assume \(\mathcal{F}\) has (S2). The first part of (S2) for \(\overline{\mathcal{F}}\) follows as in the argument above. The second part of (S2) for \(\overline{\mathcal{F}}\) signifies that the map \[\overline{\mathcal{F}}(A \times_k k[\epsilon]) \to \overline{\mathcal{F}}(A) \times_{\overline{\mathcal{F}}(k)} \overline{\mathcal{F}}(k[\epsilon])\] is injective for any ring \(A\) in \(\mathcal{C}_\Lambda\). Suppose that \(y, y' \in \mathcal{F}(A \times_k k[\epsilon])\). Using the axioms of cofibred categories we can choose commutative diagrams \[\vcenter{ \xymatrix{ y \ar[r]_c \ar[d]_a & x_\epsilon \ar[d]^e \\ x \ar[r]^d & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r]_{c'} \ar[d]_{a'} & x'_\epsilon \ar[d]^{e'} \\ x' \ar[r]^{d'} & x'_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[d] \ar[r] & k[\epsilon] \ar[d] \\ A \ar[r] & k } }\] Assume that there exist isomorphisms \(\alpha : x \to x'\) in \(\mathcal{F}(A)\) and \(\beta : x_\epsilon \to x'_\epsilon\) in \(\mathcal{F}(k[\epsilon])\). This also means there exists an isomorphism \(\gamma : x_0 \to x'_0\) compatible with \(\alpha\). To prove (S2) for \(\overline{\mathcal{F}}\) we have to show that there exists an isomorphism \(y \to y'\) in \(\mathcal{F}(A \times_k k[\epsilon])\). By (S2) for \(\mathcal{F}\) such a morphism will exist if we can choose the isomorphisms \(\alpha\) and \(\beta\) and \(\gamma\) such that \[\xymatrix{ x \ar[d]^\alpha \ar[r] & x_0 \ar[d]^\gamma & x_\epsilon \ar[d]^\beta \ar[l]^e \\ x' \ar[r] & x'_0 & x'_\epsilon \ar[l]_{e'} }\] is commutative (because then we can replace \(x\) by \(x'\) and \(x_\epsilon\) by \(x'_\epsilon\) in the previous displayed diagram). The left hand square commutes by our choice of \(\gamma\). We can factor \(e' \circ \beta\) as \(\gamma' \circ e\) for some second map \(\gamma' : x_0 \to x'_0\). Now the question is whether we can arrange it so that \(\gamma = \gamma'\)? This is clear if \(\mathcal{F}(k)\) is a set, or a setoid. Moreover, if \(\text{Aut}_{k[\epsilon]}(x_\epsilon) \to \text{Aut}_k(x_0)\) is surjective, then we can adjust the choice of \(\beta\) by precomposing with an automorphism of \(x_\epsilon\) whose image is \(\gamma^{-1} \circ \gamma'\) to make things work.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). Let \(x_0 \in \Ob(\mathcal{F}(k))\). Let \(\mathcal{F}_{x_0}\) be the category cofibred in groupoids over \(\mathcal{C}_\Lambda\) constructed in Remark 06GU.

If \(\mathcal{F}\) satisfies (S1), then so does \(\mathcal{F}_{x_0}\).

If \(\mathcal{F}\) satisfies (S2), then so does \(\mathcal{F}_{x_0}\).

Proof

Any diagram as in Definition 06HW in \(\mathcal{F}_{x_0}\) gives rise to a diagram in \(\mathcal{F}\) and the output of condition (S1) or (S2) for this diagram in \(\mathcal{F}\) can be viewed as an output for \(\mathcal{F}_{x_0}\) as well.

Lemma

Let \(p: \mathcal{F} \to \mathcal{C}_\Lambda\) be a category cofibered in groupoids. Consider a diagram of \(\mathcal{F}\) \[\vcenter{ \xymatrix{ y \ar[r] \ar[d]_a & x_\epsilon \ar[d]_e \\ x \ar[r]^d & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\ A \ar[r] & k. } }\] in \(\mathcal{C}_\Lambda\). Assume \(\mathcal{F}\) satisfies (S2). Then there exists a morphism \(s : x \to y\) with \(a \circ s = \text{id}_x\) if and only if there exists a morphism \(s_\epsilon : x \to x_\epsilon\) with \(e \circ s_\epsilon = d\).

Proof

The “only if” direction is clear. Conversely, assume there exists a morphism \(s_\epsilon : x \to x_\epsilon\) with \(e \circ s_\epsilon = d\). Note that \(p(s_\epsilon) : A \to k[\epsilon]\) is a ring map compatible with the map \(A \to k\). Hence we obtain \[\sigma = (\text{id}_A, p(s_\epsilon)) : A \to A \times_k k[\epsilon].\] Choose a pushforward \(x \to \sigma_*x\). By construction we can factor \(s_\epsilon\) as \(x \to \sigma_*x \to x_\epsilon\). Moreover, as \(\sigma\) is a section of \(A \times_k k[\epsilon] \to A\), we get a morphism \(\sigma_*x \to x\) such that \(x \to \sigma_*x \to x\) is \(\text{id}_x\). Because \(e \circ s_\epsilon = d\) we find that the diagram \[\xymatrix{ \sigma_*x \ar[r] \ar[d] & x_\epsilon \ar[d]_e \\ x \ar[r]^d & x_0 }\] is commutative. Hence by (S2) we obtain a morphism \(\sigma_*x \to y\) such that \(\sigma_*x \to y \to x\) is the given map \(\sigma_*x \to x\). The solution to the problem is now to take \(a : x \to y\) equal to the composition \(x \to \sigma_*x \to y\).

Lemma

Consider a commutative diagram in a predeformation category \(\mathcal{F}\) \[\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_2 \ar[d]^{a_2} \\ x_1 \ar[r]^{a_1} & x } } \quad\text{lying over} \vcenter{ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d]^{f_2} \\ A_1 \ar[r]^{f_1} & A } }\] in \(\mathcal{C}_\Lambda\) where \(f_2 : A_2 \to A\) is a small extension. Assume there is a map \(h : A_1 \to A_2\) such that \(f_2 = f_1 \circ h\). Let \(I = \Ker(f_2)\). Consider the ring map \[g : A_1 \times_A A_2 \longrightarrow k[I] = k \oplus I, \quad (u, v) \longmapsto \overline{u} \oplus (v - h(u))\] Choose a pushforward \(y \to g_*y\). Assume \(\mathcal{F}\) satisfies (S2). If there exists a morphism \(x_1 \to g_*y\), then there exists a morphism \(b: x_1 \to x_2\) such that \(a_1 = a_2 \circ b\).

Proof

Note that \(\text{id}_{A_1} \times g : A_1 \times_A A_2 \to A_1 \times_k k[I]\) is an isomorphism and that \(k[I] \cong k[\epsilon]\). Hence we have a diagram \[\vcenter{ \xymatrix{ y \ar[r] \ar[d] & g_*y \ar[d] \\ x_1 \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A_1 \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\ A_1 \ar[r] & k. } }\] where \(x_0\) is an object of \(\mathcal{F}\) lying over \(k\) (every object of \(\mathcal{F}\) has a unique morphism to \(x_0\), see discussion following Definition 06GS). If we have a morphism \(x_1 \to g_*y\) then Lemma 06IS provides us with a section \(s : x_1 \to y\) of the map \(y \to x_1\). Composing this with the map \(y \to x_2\) we obtain \(b : x_1 \to x_2\) which has the property that \(a_1 = a_2 \circ b\) because the diagram of the lemma commutes and because \(s\) is a section.

Tangent spaces of functors

Let \(R\) be a ring. We write \(\text{Mod}_R\) for the category of \(R\)-modules and \(\text{Mod}^{fg}_R\) for the category of finitely generated \(R\)-modules.

Definition

Let \(L: \text{Mod}^{fg}_R \to \text{Mod}_R\), resp. \(L: \text{Mod}_R \to \text{Mod}_R\) be a functor. We say that \(L\) is \(R\)-linear if for every pair of objects \(M, N\) of \(\text{Mod}^{fg}_R\), resp. \(\text{Mod}_R\) the map \[L : \Hom_R(M, N) \longrightarrow \Hom_R(L(M), L(N))\] is a map of \(R\)-modules.

Remark

One can define the notion of an \(R\)-linearity for any functor between categories enriched over \(\text{Mod}_R\). We made the definition specifically for functors \(L: \text{Mod}^{fg}_R \to \text{Mod}_R\) and \(L: \text{Mod}_R \to \text{Mod}_R\) because these are the cases that we have needed so far.

Remark

If \(L: \text{Mod}^{fg}_R \to \text{Mod}_R\) is an \(R\)-linear functor, then \(L\) preserves finite products and sends the zero module to the zero module, see Homology, Lemma 0105. On the other hand, if a functor \(\text{Mod}^{fg}_R \to \textit{Sets}\) preserves finite products and sends the zero module to a one element set, then it has a unique lift to a \(R\)-linear functor, see Lemma 06I6.

Lemma

Let \(L: \text{Mod}^{fg}_R \to \textit{Sets}\), resp. \(L: \text{Mod}_R \to \textit{Sets}\) be a functor. Suppose \(L(0)\) is a one element set and \(L\) preserves finite products. Then there exists a unique \(R\)-linear functor \(\widetilde{L} : \text{Mod}^{fg}_R \to \text{Mod}_R\), resp. \(\widetilde{L} : \text{Mod}_R \to \text{Mod}_R\), such that \[\vcenter{ \xymatrix{ & \text{Mod}_R \ar[dr]^{\text{forget}} & \\ \text{Mod}^{fg}_R \ar[ur]^{\widetilde{L}} \ar[rr]^{L} & & \textit{Sets} } } \quad\text{resp.}\quad \vcenter{ \xymatrix{ & \text{Mod}_R \ar[dr]^{\text{forget}} & \\ \text{Mod}_R \ar[ur]^{\widetilde{L}} \ar[rr]^{L} & & \textit{Sets} } }\] commutes.

Proof

We only prove this in case \(L: \text{Mod}^{fg}_R \to \textit{Sets}\). Let \(M\) be a finitely generated \(R\)-module. We define \(\widetilde{L}(M)\) to be the set \(L(M)\) with the following \(R\)-module structure.

Multiplication: If \(r \in R\), multiplication by \(r\) on \(L(M)\) is defined to be the map \(L(M) \to L(M)\) induced by the multiplication map \(r \cdot : M \to M\).

Addition: The sum map \(M \times M \to M: (m_1, m_2) \mapsto m_1 + m_2\) induces a map \(L(M \times M) \to L(M)\). By assumption \(L(M \times M)\) is canonically isomorphic to \(L(M) \times L(M)\). Addition on \(L(M)\) is defined by the map \(L(M) \times L(M) \cong L(M \times M) \to L(M)\).

Zero: There is a unique map \(0 \to M\). The zero element of \(L(M)\) is the image of \(L(0) \to L(M)\).

We omit the verification that this defines an \(R\)-module \(\widetilde{L}(M)\), the unique such that is \(R\)-linearly functorial in \(M\).

Lemma

Let \(L_1, L_2: \text{Mod}^{fg}_R \to \textit{Sets}\) be functors that take \(0\) to a one element set and preserve finite products. Let \(t : L_1 \to L_2\) be a morphism of functors. Then \(t\) induces a morphism \(\widetilde{t} : \widetilde{L}_1 \to \widetilde{L}_2\) between the functors guaranteed by Lemma 06I6, which is given simply by \(\widetilde{t}_M = t_M: \widetilde{L}_1(M) \to \widetilde{L}_2(M)\) for each \(M \in \Ob(\text{Mod}^{fg}_R)\). In other words, \(t_M: \widetilde{L}_1(M) \to \widetilde{L}_2(M)\) is a map of \(R\)-modules.

Proof

Omitted.

In the case \(R = K\) is a field, a \(K\)-linear functor \(L : \text{Mod}^{fg}_K \to \text{Mod}_K\) is determined by its value \(L(K)\).

Lemma

Let \(K\) be a field. Let \(L: \text{Mod}^{fg}_K \to \text{Mod}_K\) be a \(K\)-linear functor. Then \(L\) is isomorphic to the functor \(L(K) \otimes_K - : \text{Mod}^{fg}_K \to \text{Mod}_K\).

Proof

For \(V \in \Ob(\text{Mod}^{fg}_K)\), the isomorphism \(L(K) \otimes_K V \to L(V)\) is given on pure tensors by \(x \otimes v \mapsto L(f_v)(x)\), where \(f_v: K \to V\) is the \(K\)-linear map sending \(1 \mapsto v\). When \(V = K\), this is the isomorphism \(L(K) \otimes_K K \to L(K)\) given by multiplication by \(K\). For general \(V\), it is an isomorphism by the case \(V = K\) and the fact that \(L\) commutes with finite products (Remark 06I5).

For a ring \(R\) and an \(R\)-module \(M\), let \(R[M]\) be the \(R\)-algebra whose underlying \(R\)-module is \(R \oplus M\) and whose multiplication is given by \((r, m) \cdot (r', m') = (rr', rm' + r'm)\). When \(M = R\) this is the ring of dual numbers over \(R\), which we denote by \(R[\epsilon]\).

Now let \(S\) be a ring and assume \(R\) is an \(S\)-algebra. Then the assignment \(M \mapsto R[M]\) determines a functor \(\text{Mod}_R \to S\text{-Alg}/R\), where \(S\text{-Alg}/R\) denotes the category of \(S\)-algebras over \(R\). Note that \(S\text{-Alg}/R\) admits finite products: if \(A_1 \to R\) and \(A_2 \to R\) are two objects, then \(A_1 \times_R A_2\) is a product.

Lemma

Let \(R\) be an \(S\)-algebra. Then the functor \(\text{Mod}_R \to S\text{-Alg}/R\) described above preserves finite products.

Proof

This is merely the statement that if \(M\) and \(N\) are \(R\)-modules, then the map \(R[M \times N] \to R[M] \times_R R[N]\) is an isomorphism in \(S\text{-Alg}/R\).

Lemma

Let \(R\) be an \(S\)-algebra, and let \(\mathcal{C}\) be a strictly full subcategory of \(S\text{-Alg}/R\) containing \(R[M]\) for all \(M \in \Ob(\text{Mod}^{fg}_R)\). Let \(F: \mathcal{C} \to \textit{Sets}\) be a functor. Suppose that \(F(R)\) is a one element set and that for any \(M, N \in \Ob(\text{Mod}^{fg}_R)\), the induced map \[F(R[M] \times_R R[N]) \to F(R[M]) \times F(R[N])\] is a bijection. Then \(F(R[M])\) has a natural \(R\)-module structure for any \(M \in \Ob(\text{Mod}^{fg}_R)\).

Proof

Note that \(R \cong R[0]\) and \(R[M] \times_R R[N] \cong R[M \times N]\) hence \(R\) and \(R[M] \times_R R[N]\) are objects of \(\mathcal{C}\) by our assumptions on \(\mathcal{C}\). Thus the conditions on \(F\) make sense. The functor \(\text{Mod}_R \to S\text{-Alg}/R\) of Lemma 06I9 restricts to a functor \(\text{Mod}^{fg}_R \to \mathcal{C}\) by the assumption on \(\mathcal{C}\). Let \(L\) be the composition \(\text{Mod}^{fg}_R \to \mathcal{C} \to \textit{Sets}\), i.e., \(L(M) = F(R[M])\). Then \(L\) preserves finite products by Lemma 06I9 and the assumption on \(F\). Hence Lemma 06I6 shows that \(L(M) = F(R[M])\) has a natural \(R\)-module structure for any \(M \in \Ob(\text{Mod}^{fg}_R)\).

Definition

Let \(\mathcal{C}\) be a category as in Lemma 06IA. Let \(F : \mathcal{C} \to \textit{Sets}\) be a functor such that \(F(R)\) is a one element set. The tangent space \(TF\) of \(F\) is \(F(R[\epsilon])\).

When \(F : \mathcal{C} \to \textit{Sets}\) satisfies the hypotheses of Lemma 06IA, the tangent space \(TF\) has a natural \(R\)-module structure.

Example

Since \(\mathcal{C}_\Lambda\) contains all \(k[V]\) for finite dimensional vector spaces \(V\) we see that Definition 06IB applies with \(S = \Lambda\), \(R = k\), \(\mathcal{C} = \mathcal{C}_\Lambda\), and \(F : \mathcal{C}_\Lambda \to \textit{Sets}\) a predeformation functor. The tangent space is \(TF = F(k[\epsilon])\).

Example

Let us work out the tangent space of Example 06SR when \(F : \mathcal{C}_\Lambda \to \textit{Sets}\) is a prorepresentable functor, say \(F = \underline{S}|_{\mathcal{C}_\Lambda}\) for \(S \in \Ob(\widehat{\mathcal{C}}_\Lambda)\). Then \(F\) commutes with arbitrary limits and thus satisfies the hypotheses of Lemma 06IA. We compute \[TF = F(k[\epsilon]) = \Mor_{\mathcal{C}_\Lambda}(S, k[\epsilon]) = \text{Der}_\Lambda(S, k)\] and more generally for a finite dimensional \(k\)-vector space \(V\) we have \[F(k[V]) = \Mor_{\mathcal{C}_\Lambda}(S, k[V]) = \text{Der}_\Lambda(S, V).\] Explicitly, a \(\Lambda\)-algebra map \(f : S \to k[V]\) compatible with the augmentations \(q : S \to k\) and \(k[V] \to k\) corresponds to the derivation \(D\) defined by \(s \mapsto f(s) - q(s)\). Conversely, a \(\Lambda\)-derivation \(D : S \to V\) corresponds to \(f : S \to k[V]\) in \(\mathcal{C}_\Lambda\) defined by the rule \(f(s) = q(s) + D(s)\). Since these identifications are functorial we see that the \(k\)-vector spaces structures on \(TF\) and \(\text{Der}_\Lambda(S, k)\) correspond (see Lemma 06I7). It follows that \(\dim_k TF\) is finite by Lemma 06SC.

Example

The computation of Example 06IC simplifies in the classical case. Namely, in this case the tangent space of the functor \(F = \underline{S}|_{\mathcal{C}_\Lambda}\) is simply the relative cotangent space of \(S\) over \(\Lambda\), in a formula \(TF = T_{S/\Lambda}\). In fact, this works more generally when the field extension \(k/k'\) is separable. See Exercises, Exercise 029E.

Lemma

Let \(F, G: \mathcal{C} \to \textit{Sets}\) be functors satisfying the hypotheses of Lemma 06IA. Let \(t : F \to G\) be a morphism of functors. For any \(M \in \Ob(\text{Mod}^{fg}_R)\), the map \(t_{R[M]}: F(R[M]) \to G(R[M])\) is a map of \(R\)-modules, where \(F(R[M])\) and \(G(R[M])\) are given the \(R\)-module structure from Lemma 06IA. In particular, \(t_{R[\epsilon]} : TF \to TG\) is a map of \(R\)-modules.

Proof

Follows from Lemma 06I7.

Example

Suppose that \(f : R \to S\) is a ring map in \(\widehat{\mathcal{C}}_\Lambda\). Set \(F = \underline{R}|_{\mathcal{C}_\Lambda}\) and \(G = \underline{S}|_{\mathcal{C}_\Lambda}\). The ring map \(f\) induces a transformation of functors \(G \to F\). By Lemma 06ID we get a \(k\)-linear map \(TG \to TF\). This is the map \[TG = \text{Der}_\Lambda(S, k) \longrightarrow \text{Der}_\Lambda(R, k) = TF\] as follows from the canonical identifications \(F(k[V]) = \text{Der}_\Lambda(R, V)\) and \(G(k[V]) = \text{Der}_\Lambda(S, V)\) of Example 06IC and the rule for computing the map on tangent spaces.

Lemma

Let \(F: \mathcal{C} \to \textit{Sets}\) be a functor satisfying the hypotheses of Lemma 06IA. Assume \(R = K\) is a field. Then \(F(K[V]) \cong TF \otimes_K V\) for any finite dimensional \(K\)-vector space \(V\).

Proof

Follows from Lemma 06I8.

Tangent spaces of predeformation categories

We will define tangent spaces of predeformation functors using the general Definition 06IB. We have spelled this out in Example 06SR. It applies to predeformation categories by looking at the associated functor of isomorphism classes.

Definition

Let \(\mathcal{F}\) be a predeformation category. The tangent space \(T \mathcal{F}\) of \(\mathcal{F}\) is the set \(\overline{\mathcal{F}}(k[\epsilon])\) of isomorphism classes of objects in the fiber category \(\mathcal F(k[\epsilon])\).

Thus \(T \mathcal{F}\) is nothing but the tangent space of the associated functor \(\overline{\mathcal{F}}: \mathcal{C}_\Lambda \to \textit{Sets}\). It has a natural vector space structure when \(\mathcal{F}\) satisfies (S2), or, in fact, as long as \(\overline{\mathcal{F}}\) does.

Lemma

Let \(\mathcal{F}\) be a predeformation category such that \(\overline{\mathcal{F}}\) satisfies (S2)². Then \(T \mathcal{F}\) has a natural \(k\)-vector space structure. For any finite dimensional vector space \(V\) we have \(\overline{\mathcal{F}}(k[V]) = T\mathcal{F} \otimes_k V\) functorially in \(V\).

Proof

Let us write \(F = \overline{\mathcal{F}} : \mathcal{C}_\Lambda \to \textit{Sets}\). This is a predeformation functor and \(F\) satisfies (S2). By Lemma 06HZ (and the translation of Remark 06HY) we see that \[F(A \times_k k[V]) \longrightarrow F(A) \times F(k[V])\] is a bijection for every finite dimensional vector space \(V\) and every \(A \in \Ob(\mathcal{C}_\Lambda)\). In particular, if \(A = k[W]\) then we see that \(F(k[W] \times_k k[V]) = F(k[W]) \times F(k[V])\). In other words, the hypotheses of Lemma 06IA hold and we see that \(TF = T \mathcal{F}\) has a natural \(k\)-vector space structure. The final assertion follows from Lemma 06IE.

A morphism of predeformation categories induces a map on tangent spaces.

Definition

Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a morphism predeformation categories. The differential \(d \varphi : T \mathcal{F} \to T \mathcal{G}\) of \(\varphi\) is the map obtained by evaluating the morphism of functors \(\overline{\varphi}: \overline{\mathcal{F}} \to \overline{\mathcal{G}}\) at \(A = k[\epsilon]\).

Lemma

Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a morphism of predeformation categories. Assume \(\overline{\mathcal{F}}\) and \(\overline{\mathcal{G}}\) both satisfy (S2). Then \(d \varphi : T \mathcal{F} \to T \mathcal{G}\) is \(k\)-linear.

Proof

In the proof of Lemma 06IH we have seen that \(\overline{\mathcal{F}}\) and \(\overline{\mathcal{G}}\) satisfy the hypotheses of Lemma 06IA. Hence the lemma follows from Lemma 06ID.

Remark

We can globalize the notions of tangent space and differential to arbitrary categories cofibered in groupoids as follows. Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\), and let \(x \in \Ob(\mathcal{F}(k))\). As in Remark 06GU, we get a predeformation category \(\mathcal{F}_x\). We define \[T_x\mathcal{F} = T\mathcal{F}_x\] to be the tangent space of \(\mathcal{F}\) at \(x\). If \(\varphi : \mathcal{F} \to \mathcal{G}\) is a morphism of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\) and \(x \in \Ob(\mathcal{F}(k))\), then there is an induced morphism \(\varphi_x: \mathcal{F}_x \to \mathcal{G}_{\varphi(x)}\). We define the differential \(d_x \varphi : T_x \mathcal{F} \to T_{\varphi(x)} \mathcal{G}\) of \(\varphi\) at \(x\) to be the map \(d \varphi_x: T \mathcal{F}_x \to T \mathcal{G}_{\varphi(x)}\). If both \(\mathcal{F}\) and \(\mathcal{G}\) satisfy (S2) then all of these tangent spaces have a natural \(k\)-vector space structure and all the differentials \(d_x \varphi : T_x \mathcal{F} \to T_{\varphi(x)} \mathcal{G}\) are \(k\)-linear (use Lemmas 06SQ and 06IJ).

The following observations are uninteresting in the classical case or when \(k/k'\) is a separable field extension, because then \(\text{Der}_\Lambda(k, k)\) and \(\text{Der}_\Lambda(k, V)\) are zero. There is a canonical identification \[\Mor_{\mathcal{C}_\Lambda}(k, k[\epsilon]) = \text{Der}_\Lambda(k, k).\] Namely, for \(D \in \text{Der}_\Lambda(k, k)\) let \(f_D : k \to k[\epsilon]\) be the map \(a \mapsto a + D(a)\epsilon\). More generally, given a finite dimensional vector space \(V\) over \(k\) we have \[\Mor_{\mathcal{C}_\Lambda}(k, k[V]) = \text{Der}_\Lambda(k, V)\] and we will use the same notation \(f_D\) for the map associated to the derivation \(D\). We also have \[\Mor_{\mathcal{C}_\Lambda}(k[W], k[V]) = \Hom_k(V, W) \oplus \text{Der}_\Lambda(k, V)\] where \((\varphi, D)\) corresponds to the map \(f_{\varphi, D} : a + w \mapsto a + \varphi(w) + D(a)\). We will sometimes write \(f_{1, D} : a + v \to a + v + D(a)\) for the automorphism of \(k[V]\) determined by the derivation \(D : k \to V\). Note that \(f_{1, D} \circ f_{1, D'} = f_{1, D + D'}\).

Let \(\mathcal{F}\) be a predeformation category over \(\mathcal{C}_\Lambda\). Let \(x_0 \in \Ob(\mathcal{F}(k))\). By the above there is a canonical map \[\gamma_V : \text{Der}_\Lambda(k, V) \longrightarrow \overline{\mathcal{F}}(k[V])\] defined by \(D \mapsto f_{D, *}(x_0)\). Moreover, there is an action \[a_V : \text{Der}_\Lambda(k, V) \times \overline{\mathcal{F}}(k[V]) \longrightarrow \overline{\mathcal{F}}(k[V])\] defined by \((D, x) \mapsto f_{1, D, *}(x)\). These two maps are compatible, i.e., \(f_{1, D, *}f_{D', *}x_0 = f_{D + D', *}x_0\) as follows from a computation of the compositions of these maps. Note that the maps \(\gamma_V\) and \(a_V\) are independent of the choice of \(x_0\) as there is a unique \(x_0\) up to isomorphism.

Lemma

Let \(\mathcal{F}\) be a predeformation category over \(\mathcal{C}_\Lambda\). If \(\overline{\mathcal{F}}\) has (S2) then the maps \(\gamma_V\) are \(k\)-linear and we have \(a_V(D, x) = x + \gamma_V(D)\).

Proof

In the proof of Lemma 06IH we have seen that the functor \(V \mapsto \overline{\mathcal{F}}(k[V])\) transforms \(0\) to a singleton and products to products. The same is true of the functor \(V \mapsto \text{Der}_\Lambda(k, V)\). Hence \(\gamma_V\) is linear by Lemma 06I7. Let \(D : k \to V\) be a \(\Lambda\)-derivation. Set \(D_1 : k \to V^{\oplus 2}\) equal to \(a \mapsto (D(a), 0)\). Then \[\xymatrix{ k[V \times V] \ar[r]_{+} \ar[d]^{f_{1, D_1}} & k[V] \ar[d]^{f_{1, D}} \\ k[V \times V] \ar[r]^{+} & k[V] }\] commutes. Unwinding the definitions and using that \(\overline{F}(V \times V) = \overline{F}(V) \times \overline{F}(V)\) this means that \(a_D(x_1) + x_2 = a_D(x_1 + x_2)\) for all \(x_1, x_2 \in \overline{F}(V)\). Thus it suffices to show that \(a_V(D, 0) = 0 + \gamma_V(D)\) where \(0 \in \overline{F}(V)\) is the zero vector. By definition this is the element \(f_{0, *}(x_0)\). Since \(f_D = f_{1, D} \circ f_0\) the desired result follows.

A special case of the constructions above are the map [06SV]\[\begin{equation} \gamma : \text{Der}_\Lambda(k, k) \longrightarrow T\mathcal{F} \end{equation}\] and the action [06SW]\[\begin{equation} a : \text{Der}_\Lambda(k, k) \times T\mathcal{F} \longrightarrow T\mathcal{F} \end{equation}\] defined for any predeformation category \(\mathcal{F}\). Note that if \(\varphi : \mathcal{F} \to \mathcal{G}\) is a morphism of predeformation categories, then we get commutative diagrams \[\vcenter{ \xymatrix{ \text{Der}_\Lambda(k, k) \ar[r]_-\gamma \ar[rd]_\gamma & T\mathcal{F} \ar[d]_{d\varphi} \\ & T\mathcal{G} } } \quad\text{and}\quad \vcenter{ \xymatrix{ \text{Der}_\Lambda(k, k) \times T\mathcal{F} \ar[r]_-a \ar[d]_{1 \times d\varphi} & T\mathcal{F} \ar[d]^{d\varphi} \\ \text{Der}_\Lambda(k, k) \times T\mathcal{G} \ar[r]^-a & T\mathcal{G} } }\]

Versal formal objects

The existence of a versal formal object forces \(\mathcal{F}\) to have property (S1).

Lemma

Let \(\mathcal{F}\) be a predeformation category. Assume \(\mathcal{F}\) has a versal formal object. Then \(\mathcal{F}\) satisfies (S1).

Proof

Let \(\xi\) be a versal formal object of \(\mathcal{F}\). Let \[\xymatrix{ & x_2 \ar[d] \\ x_1 \ar[r] & x }\] be a diagram in \(\mathcal{F}\) such that \(x_2 \to x\) lies over a surjective ring map. Since the natural morphism \(\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda} \xrightarrow{\sim} \mathcal{F}\) is an equivalence (see Remark 06SJ), we can consider this diagram also as a diagram in \(\widehat{\mathcal{F}}\). By Lemma 06HT there exists a morphism \(\xi \to x_1\), so by Remark 06HS we also get a morphism \(\xi \to x_2\) making the diagram \[\xymatrix{ \xi \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r] & x }\] commute. If \(x_1 \to x\) and \(x_2 \to x\) lie above ring maps \(A_1 \to A\) and \(A_2 \to A\) then taking the pushforward of \(\xi\) to \(A_1 \times_A A_2\) gives an object \(y\) as required by (S1).

In the case that our cofibred category satisfies (S1) and (S2) we can characterize the versal formal objects as follows.

Lemma

Let \(\mathcal{F}\) be a predeformation category satisfying (S1) and (S2). Let \(\xi\) be a formal object of \(\mathcal{F}\) corresponding to \(\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\), see Remark 06HC. Then \(\xi\) is versal if and only if the following two conditions hold:

the map \(d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to T\mathcal{F}\) on tangent spaces is surjective, and

given a diagram in \(\widehat{\mathcal{F}}\) \[\vcenter{ \xymatrix{ & y \ar[d] \\ \xi \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & B \ar[d]^{f} \\ R \ar[r] & A } }\] in \(\widehat{\mathcal{C}}_\Lambda\) with \(B \to A\) a small extension of Artinian rings, then there exists a ring map \(R \to B\) such that \[\xymatrix{ & B \ar[d]^{f} \\ R \ar[ur] \ar[r] & A }\] commutes.

Proof

If \(\xi\) is versal then (1) holds by Lemma 06HN and (2) holds by Remark 06HS. Assume (1) and (2) hold. By Remark 06HS we must show that given a diagram in \(\widehat{\mathcal{F}}\) as in (2), there exists \(\xi \to y\) such that \[\xymatrix{ & y \ar[d] \\ \xi \ar[ur] \ar[r] & x }\] commutes. Let \(b : R \to B\) be the map guaranteed by (2). Denote \(y' = b_*\xi\) and choose a factorization \(\xi \to y' \to x\) lying over \(R \to B \to A\) of the given morphism \(\xi \to x\). By (S1) we obtain a commutative diagram \[\vcenter{ \xymatrix{ z \ar[r] \ar[d] & y \ar[d] \\ y' \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ B \times_A B \ar[d] \ar[r] & B \ar[d]^{f} \\ B \ar[r]^{f} & A . } }\] Set \(I = \Ker(f)\). Let \(\overline{g} : B \times_A B \to k[I]\) be the ring map \((u, v) \mapsto \overline{u} \oplus (v - u)\), cf. Lemma 06IT. By (1) there exists a morphism \(\xi \to \overline{g}_*z\) which lies over a ring map \(i : R \to k[\epsilon]\). Choose an Artinian quotient \(b_1 : R \to B_1\) such that both \(b : R \to B\) and \(i : R \to k[\epsilon]\) factor through \(R \to B_1\), i.e., giving \(h : B_1 \to B\) and \(i' : B_1 \to k[\epsilon]\). Choose a pushforward \(y_1 = b_{1, *}\xi\), a factorization \(\xi \to y_1 \to y'\) lying over \(R \to B_1 \to B\) of \(\xi \to y'\), and a factorization \(\xi \to y_1 \to \overline{g}_*z\) lying over \(R \to B_1 \to k[\epsilon]\) of \(\xi \to \overline{g}_*z\). Applying (S1) once more we obtain \[\vcenter{ \xymatrix{ z_1 \ar[r] \ar[d] & z \ar[r] \ar[d] & y \ar[d] \\ y_1 \ar[r] & y' \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ B_1 \times_A B \ar[d] \ar[r] & B \times_A B \ar[r] \ar[d] & B \ar[d]^{f} \\ B_1 \ar[r] & B \ar[r] & A . } }\] Note that the map \(g : B_1 \times_A B \to k[I]\) of Lemma 06IT (defined using \(h\)) is the composition of \(B_1 \times_A B \to B \times_A B\) and the map \(\overline{g}\) above. By construction there exists a morphism \(y_1 \to g_*z_1 \cong \overline{g}_*z\)! Hence Lemma 06IT applies (to the outer rectangles in the diagrams above) to give a morphism \(y_1 \to y\) and precomposing with \(\xi \to y_1\) gives the desired morphism \(\xi \to y\).

If \(\mathcal{F}\) has property (S1) then the “largest quotient where a lift exists” exists. Here is a precise statement.

Lemma

Let \(\mathcal{F}\) be a category cofibred in groupoids over \(\mathcal{C}_\Lambda\) which has (S1). Let \(B \to A\) be a surjection in \(\mathcal{C}_\Lambda\) with kernel \(I\) annihilated by \(\mathfrak m_B\). Let \(x \in \mathcal{F}(A)\). The set of ideals \[\mathcal{J} = \{ J \subset I \mid \text{there exists an }y \to x\text{ lying over }B/J \to A\}\] has a smallest element.

Proof

Note that \(\mathcal{J}\) is nonempty as \(I \in \mathcal{J}\). Also, if \(J \in \mathcal{J}\) and \(J \subset J' \subset I\) then \(J' \in \mathcal{J}\) because we can pushforward the object \(y\) to an object \(y'\) over \(B/J'\). Let \(J\) and \(K\) be elements of the displayed set. We claim that \(J \cap K \in \mathcal{J}\) which will prove the lemma. Since \(I\) is a \(k\)-vector space we can find an ideal \(J \subset J' \subset I\) such that \(J \cap K = J' \cap K\) and such that \(J' + K = I\). By the above we may replace \(J\) by \(J'\) and assume that \(J + K = I\). In this case \[A/(J \cap K) = A/J \times_{A/I} A/K.\] Hence the existence of an element \(z \in \mathcal{F}(A/(J \cap K))\) mapping to \(x\) follows, via (S1), from the existence of the elements we have assumed exist over \(A/J\) and \(A/K\).

We will improve on the following result later.

Lemma

Let \(\mathcal{F}\) be a category cofibred in groupoids over \(\mathcal{C}_\Lambda\). Assume the following conditions hold:

\(\mathcal{F}\) is a predeformation category.

\(\mathcal{F}\) satisfies (S1).

\(\mathcal{F}\) satisfies (S2).

\(\dim_k T\mathcal{F}\) is finite.

Then \(\mathcal{F}\) has a versal formal object.

Proof

Assume (1), (2), (3), and (4) hold. Choose an object \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\) such that \(\underline{R}|_{\mathcal{C}_\Lambda}\) is smooth. See Lemma 06SM. Let \(r = \dim_k T\mathcal{F}\) and put \(S = R[[X_1, \ldots, X_r]]\).

We are going to inductively construct for \(n \geq 2\) pairs \((J_n, f_{n - 1} : \xi_n \to \xi_{n - 1})\) where \(J_n \subset S\) is an decreasing sequence of ideals and \(f_{n - 1} : \xi_n \to \xi_{n - 1}\) is a morphism of \(\mathcal{F}\) lying over the projection \(S/J_n \to S/J_{n - 1}\).

Step 1. Let \(J_1 = \mathfrak m_S\). Let \(\xi_1\) be the unique (up to unique isomorphism) object of \(\mathcal{F}\) over \(k = S/J_1 = S/\mathfrak m_S\)

Step 2. Let \(J_2 = \mathfrak m_S^2 + \mathfrak{m}_R S\). Then \(S/J_2 = k[V]\) with \(V = kX_1 \oplus \ldots \oplus kX_r\) By (S2) for \(\overline{\mathcal{F}}\) we get a bijection \[\overline{\mathcal{F}}(S/J_2) \longrightarrow T\mathcal{F} \otimes_k V,\] see Lemmas 06I0 and 06IH. Choose a basis \(\theta_1, \ldots, \theta_r\) for \(T\mathcal{F}\) and set \(\xi_2 = \sum \theta_i \otimes X_i \in \Ob(\mathcal{F}(S/J_2))\). The point of this choice is that \[d\xi_2 : \Mor_{\mathcal{C}_\Lambda}(S/J_2, k[\epsilon]) \longrightarrow T\mathcal{F}\] is surjective. Let \(f_1 : \xi_2 \to \xi_1\) be the unique morphism.

Induction step. Assume \((J_n, f_{n - 1} : \xi_n \to \xi_{n - 1})\) has been constructed for some \(n \geq 2\). There is a minimal element \(J_{n + 1}\) of the set of ideals \(J \subset S\) satisfying: (a) \(\mathfrak m_S J_n \subset J \subset J_n\) and (b) there exists a morphism \(\xi_{n + 1} \to \xi_n\) lying over \(S/J \to S/J_n\), see Lemma 06SZ. Let \(f_n : \xi_{n + 1} \to \xi_n\) be any morphism of \(\mathcal{F}\) lying over \(S/J_{n + 1} \to S/J_n\).

Set \(J = \bigcap J_n\). Set \(\overline{S} = S/J\). Set \(\overline{J}_n = J_n/J\). By Lemma 06SE the sequence of ideals \((\overline{J}_n)\) induces the \(\mathfrak m_{\overline{S}}\)-adic topology on \(\overline{S}\). Since \((\xi_n, f_n)\) is an object of \(\widehat{\mathcal{F}}_\mathcal{I}(\overline{S})\), where \(\mathcal{I}\) is the filtration \((\overline{J}_n)\) of \(\overline{S}\), we see that \((\xi_n, f_n)\) induces an object \(\xi\) of \(\widehat{\mathcal{F}}(\overline{S})\). see Lemma 06H6.

We prove \(\xi\) is versal. For versality it suffices to check conditions (1) and (2) of Lemma 06IU. Condition (1) follows from our choice of \(\xi_2\) in Step 2 above. Suppose given a diagram in \(\widehat{\mathcal{F}}\) \[\vcenter{ \xymatrix{ & y \ar[d] \\ \xi \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & B \ar[d]^{f} \\ \overline{S} \ar[r] & A } }\] in \(\widehat{\mathcal{C}}_\Lambda\) with \(f: B \to A\) a small extension of Artinian rings. We have to show there is a map \(\overline{S} \to B\) fitting into the diagram on the right. Choose \(n\) such that \(\overline{S} \to A\) factors through \(\overline{S} \to S/J_n\). This is possible as the sequence \((\overline{J}_n)\) induces the \(\mathfrak m_{\overline{S}}\)-adic topology as we saw above. The pushforward of \(\xi\) along \(\overline{S} \to S/J_n\) is \(\xi_n\). We may factor \(\xi \to x\) as \(\xi \to \xi_n \to x\) hence we get a diagram in \(\mathcal{F}\) \[\vcenter{ \xymatrix{ & y \ar[d] \\ \xi_n \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & B \ar[d]^{f} \\ S/J_n \ar[r] & A . } }\] To check condition (2) of Lemma 06IU it suffices to complete the diagram \[\xymatrix{ S/J_{n + 1} \ar[d] \ar@{-->}[r] & B \ar[d]^{f} \\ S/J_n \ar[r] & A }\] or equivalently, to complete the diagram \[\xymatrix{ & S/J_n \times_A B \ar[d]^{p_1} \\ S/J_{n + 1} \ar@{-->}[ur] \ar[r] & S/J_n. }\] If \(p_1\) has a section we are done. If not, by Lemma 06GH (2) \(p_1\) is a small extension, so by Lemma 06H0 (4) \(p_1\) is an essential surjection. Recall that \(S = R[[X_1, \ldots, X_r]]\) and that we chose \(R\) such that \(\underline{R}|_{\mathcal{C}_\Lambda}\) is smooth. Hence there exists a map \(h : R \to B\) lifting the map \(R \to S \to S/J_n \to A\). By the universal property of a power series ring there is an \(R\)-algebra map \(h : S = R[[X_1, \ldots, X_r]] \to B\) lifting the given map \(S \to S/J_n \to A\). This induces a map \(g: S \to S/J_n \times_A B\) making the solid square in the diagram \[\xymatrix{ S \ar[d] \ar[r]_-g & S/J_n \times_A B \ar[d]^{p_1} \\ S/J_{n + 1} \ar@{-->}[ur] \ar[r] & S/J_n }\] commute. Then \(g\) is a surjection since \(p_1\) is an essential surjection. We claim the ideal \(K = \Ker(g)\) of \(S\) satisfies conditions (a) and (b) of the construction of \(J_{n + 1}\) in the induction step above. Namely, \(K \subset J_n\) is clear and \(\mathfrak m_SJ_n \subset K\) as \(p_1\) is a small extension; this proves (a). By (S1) applied to \[\xymatrix{ & y \ar[d] \\ \xi_n \ar[r] & x, }\] there exists a lifting of \(\xi_n\) to \(S/K \cong S/J_n \times_A B\), so (b) holds. Since \(J_{n + 1}\) was the minimal ideal with properties (a) and (b) this implies \(J_{n + 1} \subset K\). Thus the desired map \(S/J_{n+1} \to S/K \cong S/J_n \times_A B\) exists.

Remark

Let \(F : \mathcal{C}_\Lambda \to \textit{Sets}\) be a predeformation functor satisfying (S1) and (S2). The condition \(\dim_k TF < \infty\) is precisely condition (H3) from Schlessinger’s paper. Recall that (S1) and (S2) correspond to conditions (H1) and (H2), see Remark 06HY. Thus Lemma 06IW tells us \[(H1) + (H2) + (H3) \Rightarrow \text{ there exists a versal formal object}\] for predeformation functors. We will make the link with hulls in Remark 06IY.

Minimal versal formal objects

We do a little bit of work to try and understand (non)uniqueness of versal formal objects. It turns out that if a predeformation category has a versal formal object, then it has a minimal versal formal object and any two such are isomorphic. Moreover, all versal formal objects are “more or less” the same up to replacing the base ring by a power series extension.

Let \(\mathcal{F}\) be a category cofibred in groupoids over \(\mathcal{C}_\Lambda\). For every object \(x\) of \(\mathcal{F}\) lying over \(A \in \Ob(\mathcal{C}_\Lambda)\) consider the category \(\mathcal{S}_x\) with objects \[\Ob(\mathcal{S}_x) = \{x' \to x \mid x' \to x\text{ lies over }A' \subset A\}\] and morphisms are morphisms over \(x\). For every \(y \to x\) in \(\mathcal{F}\) lying over \(f : B \to A\) in \(\mathcal{C}_\Lambda\) there is a functor \(f_* : \mathcal{S}_y \to \mathcal{S}_x\) defined as follows: Given \(y' \to y\) lying over \(B' \subset B\) set \(A' = f(B')\) and let \(y' \to x'\) be over \(B' \to f(B')\) be the pushforward of \(y'\). By the axioms of a category cofibred in groupoids we obtain a unique morphism \(x' \to x\) lying over \(f(B') \to A\) such that \[\xymatrix{ y' \ar[d] \ar[r] & x' \ar[d] \\ y \ar[r] & x }\] commutes. Then \(x' \to x\) is an object of \(\mathcal{S}_x\). We say an object \(x' \to x\) of \(\mathcal{S}_x\) is minimal if any morphism \((x'_1 \to x) \to (x' \to x)\) in \(\mathcal{S}_x\) is an isomorphism, i.e., \(x'\) and \(x'_1\) are defined over the same subring of \(A\). Since \(A\) has finite length as a \(\Lambda\)-module we see that minimal objects always exist.

Lemma

Let \(\mathcal{F}\) be a category cofibred in groupoids over \(\mathcal{C}_\Lambda\) which has (S1).

For \(y \to x\) in \(\mathcal{F}\) a minimal object in \(\mathcal{S}_y\) maps to a minimal object of \(\mathcal{S}_x\).

For \(y \to x\) in \(\mathcal{F}\) lying over a surjection \(f : B \to A\) in \(\mathcal{C}_\Lambda\) every minimal object of \(\mathcal{S}_x\) is the image of a minimal object of \(\mathcal{S}_y\).

Proof

Proof of (1). Say \(y \to x\) lies over \(f : B \to A\). Let \(y' \to y\) lying over \(B' \subset B\) be a minimal object of \(\mathcal{S}_y\). Let \[\vcenter{ \xymatrix{ y' \ar[d] \ar[r] & x' \ar[d] \\ y \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ B' \ar[d] \ar[r] & f(B') \ar[d] \\ B \ar[r] & A } }\] be as in the construction of \(f_*\) above. Suppose that \((x'' \to x) \to (x' \to x)\) is a morphism of \(\mathcal{S}_x\) with \(x'' \to x'\) lying over \(A'' \subset f(B')\). By (S1) there exists \(y'' \to y'\) lying over \(B' \times_{f(B')} A'' \to B'\). Since \(y' \to y\) is minimal we conclude that \(B' \times_{f(B')} A'' \to B'\) is an isomorphism, which implies that \(A'' = f(B')\), i.e., \(x' \to x\) is minimal.

Proof of (2). Suppose \(f : B \to A\) is surjective and \(y \to x\) lies over \(f\). Let \(x' \to x\) be a minimal object of \(\mathcal{S}_x\) lying over \(A' \subset A\). By (S1) there exists \(y' \to y\) lying over \(B' = f^{-1}(A') = B \times_A A' \to B\) whose image in \(\mathcal{S}_x\) is \(x' \to x\). So \(f_*(y' \to y) = x' \to x\). Choose a morphism \((y'' \to y) \to (y' \to y)\) in \(\mathcal{S}_y\) with \(y'' \to y\) a minimal object (this is possible by the remark on lengths above the lemma). Then \(f_*(y'' \to y)\) is an object of \(\mathcal{S}_x\) which maps to \(x' \to x\) (by functoriality of \(f_*\)) hence is isomorphic to \(x' \to x\) by minimality of \(x' \to x\).

Lemma

Let \(\mathcal{F}\) be a category cofibred in groupoids over \(\mathcal{C}_\Lambda\) which has (S1). Let \(\xi\) be a versal formal object of \(\mathcal{F}\) lying over \(R\). There exists a morphism \(\xi' \to \xi\) lying over \(R' \subset R\) with the following minimality properties

for every \(f : R \to A\) with \(A \in \Ob(\mathcal{C}_\Lambda)\) the pushforwards \[\vcenter{ \xymatrix{ \xi' \ar[d] \ar[r] & x' \ar[d] \\ \xi \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ R' \ar[d] \ar[r] & f(R') \ar[d] \\ R \ar[r] & A } }\] produce a minimal object \(x' \to x\) of \(\mathcal{S}_x\), and

for any morphism of formal objects \(\xi'' \to \xi'\) the corresponding morphism \(R'' \to R'\) is surjective.

Proof

Write \(\xi = (R, \xi_n, f_n)\). Set \(R'_1 = k\) and \(\xi'_1 = \xi_1\). Suppose that we have constructed minimal objects \(\xi'_m \to \xi_m\) of \(\mathcal{S}_{\xi_m}\) lying over \(R'_m \subset R/\mathfrak m_R^m\) for \(m \leq n\) and morphisms \(f'_m : \xi'_{m + 1} \to \xi'_m\) compatible with \(f_m\) for \(m \leq n - 1\). By Lemma 06T1 (2) there exists a minimal object \(\xi'_{n + 1} \to \xi_{n + 1}\) lying over \(R'_{n + 1} \subset R/\mathfrak m_R^{n + 1}\) whose image is \(\xi'_n \to \xi_n\) over \(R'_n \subset R/\mathfrak m_R^n\). This produces the commutative diagram \[\xymatrix{ \xi'_{n + 1} \ar[r]_{f'_n} \ar[d] & \xi'_n \ar[d] \\ \xi_{n + 1} \ar[r]^{f_n} & \xi_n }\] by construction. Moreover the ring map \(R'_{n + 1} \to R'_n\) is surjective. Set \(R' = \lim_n R'_n\). Then \(R' \to R\) is injective.

However, it isn’t a priori clear that \(R'\) is Noetherian. To prove this we use that \(\xi\) is versal. Namely, versality implies that there exists a morphism \(\xi \to \xi'_n\) in \(\widehat{\mathcal{F}}\), see Lemma 06HT. The corresponding map \(R \to R'_n\) has to be surjective (as \(\xi'_n \to \xi_n\) is minimal in \(\mathcal{S}_{\xi_n}\)). Thus the dimensions of the cotangent spaces are bounded and Lemma 06SF implies \(R'\) is Noetherian, i.e., an object of \(\widehat{\mathcal{C}}_\Lambda\). By Lemma 06H6 (plus the result on filtrations of Lemma 06SF) the sequence of elements \(\xi'_n\) defines a formal object \(\xi'\) over \(R'\) and we have a map \(\xi' \to \xi\).

By construction (1) holds for \(R \to R/\mathfrak m_R^n\) for each \(n\). Since each \(R \to A\) as in (1) factors through \(R \to R/\mathfrak m_R^n \to A\) we see that (1) for \(x' \to x\) over \(f(R) \subset A\) follows from the minimality of \(\xi'_n \to \xi_n\) over \(R'_n \to R/\mathfrak m_R^n\) by Lemma 06T1 (1).

If \(R'' \to R'\) as in (2) is not surjective, then \(R'' \to R' \to R'_n\) would not be surjective for some \(n\) and \(\xi'_n \to \xi_n\) wouldn’t be minimal, a contradiction. This contradiction proves (2).

Lemma

Let \(\mathcal{F}\) be a category cofibred in groupoids over \(\mathcal{C}_\Lambda\) which has (S1). Let \(\xi\) be a versal formal object of \(\mathcal{F}\) lying over \(R\). Let \(\xi' \to \xi\) be a morphism of formal objects lying over \(R' \subset R\) as constructed in Lemma 06T2. Then \[R \cong R'[[x_1, \ldots, x_r]]\] is a power series ring over \(R'\). Moreover, \(\xi'\) is a versal formal object too.

Proof

By Lemma 06HT there exists a morphism \(\xi \to \xi'\). By Lemma 06T2 the corresponding map \(f : R \to R'\) induces a surjection \(f|_{R'} : R' \to R'\). This is an isomorphism by Algebra, Lemma 06RN. Hence \(I = \Ker(f)\) is an ideal of \(R\) such that \(R = R' \oplus I\). Let \(x_1, \ldots, x_n \in I\) be elements which form a basis for \(I/\mathfrak m_RI\). Consider the map \(S = R'[[X_1, \ldots, X_r]] \to R\) mapping \(X_i\) to \(x_i\). For every \(n \geq 1\) we get a surjection of Artinian \(R'\)-algebras \(B = S/\mathfrak m_S^n \to R/\mathfrak m_R^n = A\). Denote \(y \in \Ob(\mathcal{F}(B)\), resp. \(x \in \Ob(\mathcal{F}(A))\) the pushforward of \(\xi'\) along \(R' \to S \to B\), resp. \(R' \to S \to A\). Note that \(x\) is also the pushforward of \(\xi\) along \(R \to A\) as \(\xi\) is the pushforward of \(\xi'\) along \(R' \to R\). Thus we have a solid diagram \[\vcenter{ \xymatrix{ & y \ar[d] \\ \xi \ar[r] \ar@{..>}[ru] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & S/\mathfrak m_S^n \ar[d] \\ R \ar[r] \ar@{..>}[ru] & R/\mathfrak m_R^n } }\] Because \(\xi\) is versal, using Remark 06HS we obtain the dotted arrows fitting into these diagrams. In particular, the maps \(S/\mathfrak m_S^n \to R/\mathfrak m_R^n\) have sections \(h_n : R/\mathfrak m_R^n \to S/\mathfrak m_S^n\). It follows from Lemma 06SG that \(S \to R\) is an isomorphism.

As \(\xi\) is a pushforward of \(\xi'\) along \(R' \to R\) we obtain from Remark 06SL a commutative diagram \[\xymatrix{ \underline{R}|_{\mathcal{C}_\Lambda} \ar[rr] \ar[rd]_{\underline{\xi}} & & \underline{R'}|_{\mathcal{C}_\Lambda} \ar[ld]^{\underline{\xi'}} \\ & \mathcal{F} }\] Since \(R' \to R\) has a left inverse (namely \(R \to R/I = R'\)) we see that \(\underline{R}|_{\mathcal{C}_\Lambda} \to \underline{R'}|_{\mathcal{C}_\Lambda}\) is essentially surjective. Hence by Lemma 06HM we see that \(\underline{\xi'}\) is smooth, i.e., \(\xi'\) is a versal formal object.

Motivated by the preceding lemmas we make the following definition.

Definition

Let \(\mathcal{F}\) be a predeformation category. We say a versal formal object \(\xi\) of \(\mathcal{F}\) is minimal³ if for any morphism of formal objects \(\xi' \to \xi\) the underlying map on rings is surjective. Sometimes a minimal versal formal object is called miniversal.

The work in this section shows this definition is reasonable. First of all, the existence of a versal formal object implies that \(\mathcal{F}\) has (S1). Then the preceding lemmas show there exists a minimal versal formal object. Finally, any two minimal versal formal objects are isomorphic. Here is a summary of our results (with detailed proofs).

Lemma

Let \(\mathcal{F}\) be a predeformation category which has a versal formal object. Then

\(\mathcal{F}\) has a minimal versal formal object,

minimal versal objects are unique up to isomorphism, and

any versal object is the pushforward of a minimal versal object along a power series ring extension.

Proof

Suppose \(\mathcal{F}\) has a versal formal object \(\xi\) over \(R\). Then it satisfies (S1), see Lemma 06SY. Let \(\xi' \to \xi\) over \(R' \subset R\) be any of the morphisms constructed in Lemma 06T2. By Lemma 06T3 we see that \(\xi'\) is versal, hence it is a minimal versal formal object (by construction). This proves (1). Also, \(R \cong R'[[x_1, \ldots, x_n]]\) which proves (3).

Suppose that \(\xi_i/R_i\) are two minimal versal formal objects. By Lemma 06HT there exist morphisms \(\xi_1 \to \xi_2\) and \(\xi_2 \to \xi_1\). The corresponding ring maps \(f : R_1 \to R_2\) and \(g : R_2 \to R_1\) are surjective by minimality. Hence the compositions \(g \circ f : R_1 \to R_1\) and \(f \circ g : R_2 \to R_2\) are isomorphisms by Algebra, Lemma 06RN. Thus \(f\) and \(g\) are isomorphisms whence the maps \(\xi_1 \to \xi_2\) and \(\xi_2 \to \xi_1\) are isomorphisms (because \(\widehat{\mathcal{F}}\) is cofibred in groupoids by Lemma 06H4). This proves (2) and finishes the proof of the lemma.

Miniversal formal objects and tangent spaces

The general notion of minimality introduced in Definition 06T4 can sometimes be deduced from the behaviour on tangent spaces. Let \(\xi\) be a formal object of the predeformation category \(\mathcal{F}\) and let \(\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\) be the corresponding morphism. Then we can consider the following the condition [06IM]\[\begin{equation} d\underline{\xi} : \text{Der}_\Lambda(R, k) \to T\mathcal{F} \text{ is bijective} \end{equation}\] and the condition [06T6]\[\begin{equation} d\underline{\xi} : \text{Der}_\Lambda(R, k) \to T\mathcal{F} \text{ is bijective on }\text{Der}_\Lambda(k, k)\text{-orbits.} \end{equation}\] Here we are using the identification \(T\underline{R}|_{\mathcal{C}_\Lambda} = \text{Der}_\Lambda(R, k)\) of Example 06IC and the action (06SW) of derivations on the tangent spaces. If \(k' \subset k\) is separable, then \(\text{Der}_\Lambda(k, k) = 0\) and the two conditions are equivalent. It turns out that, in the presence of condition (S2) a versal formal object is minimal if and only if \(\underline{\xi}\) satisfies (06T6). Moreover, if \(\underline{\xi}\) satisfies (06IM), then \(\mathcal{F}\) satisfies (S2).

Lemma

Let \(\mathcal{F}\) be a predeformation category. Let \(\xi\) be a versal formal object of \(\mathcal{F}\) such that (06T6) holds. Then \(\xi\) is a minimal versal formal object. In particular, such \(\xi\) are unique up to isomorphism.

Proof

If \(\xi\) is not minimal, then there exists a morphism \(\xi' \to \xi\) lying over \(R' \to R\) such that \(R = R'[[x_1, \ldots, x_n]]\) with \(n > 0\), see Lemma 06T5. Thus \(d\underline{\xi}\) factors as \[\text{Der}_\Lambda(R, k) \to \text{Der}_\Lambda(R', k) \to T\mathcal{F}\] and we see that (06T6) cannot hold because \(D : f \mapsto \partial/\partial x_1(f) \bmod \mathfrak m_R\) is an element of the kernel of the first arrow which is not in the image of \(\text{Der}_\Lambda(k, k) \to \text{Der}_\Lambda(R, k)\).

Lemma

Let \(\mathcal{F}\) be a predeformation category. Let \(\xi\) be a versal formal object of \(\mathcal{F}\) such that (06IM) holds. Then

\(\mathcal{F}\) satisfies (S1).

\(\mathcal{F}\) satisfies (S2).

\(\dim_k T\mathcal{F}\) is finite.

Proof

Condition (S1) holds by Lemma 06SY. The first part of (S2) holds since (S1) holds. Let \[\vcenter{ \xymatrix{ y \ar[r]_c \ar[d]_a & x_\epsilon \ar[d]^e \\ x \ar[r]^d & x_0 } } \quad\text{and}\quad \vcenter{ \xymatrix{ y' \ar[r]_{c'} \ar[d]_{a'} & x_\epsilon \ar[d]^e \\ x \ar[r]^d & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\ A \ar[r] & k } }\] be diagrams as in the second part of (S2). As above we can find morphisms \(b : \xi \to y\) and \(b' : \xi \to y'\) such that \[\xymatrix{ \xi \ar[r]^{b'} \ar[d]_b & y' \ar[d]^{a'} \\ y \ar[r]^{a} & x }\] commutes. Let \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) denote the structure morphism. Say \(\widehat{p}(\xi) = R\), i.e., \(\xi\) lies over \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\). We see that the pushforward of \(\xi\) via \(p(c) \circ p(b)\) is \(x_\epsilon\) and that the pushforward of \(\xi\) via \(p(c') \circ p(b')\) is \(x_\epsilon\). Since \(\xi\) satisfies (06IM), we see that \(p(c) \circ p(b) = p(c') \circ p(b')\) as maps \(R \to k[\epsilon]\). Hence \(p(b) = p(b')\) as maps from \(R \to A \times_k k[\epsilon]\). Thus we see that \(y\) and \(y'\) are isomorphic to the pushforward of \(\xi\) along this map and we get a unique morphism \(y \to y'\) over \(A \times_k k[\epsilon]\) compatible with \(b\) and \(b'\) as desired.

Finally, by Example 06IC we see \(\dim_k T\mathcal{F} = \dim_k T\underline{R}|_{\mathcal{C}_\Lambda}\) is finite.

Example

There exist predeformation categories which have a versal formal object satisfying (06T6) but which do not satisfy (S2). A quick example is to take \(F = \underline{k[\epsilon]}/G\) where \(G \subset \text{Aut}_{\mathcal{C}_\Lambda}(k[\epsilon])\) is a finite nontrivial subgroup. Namely, the map \(\underline{k[\epsilon]} \to F\) is smooth, but the tangent space of \(F\) does not have a natural \(k\)-vector space structure (as it is a quotient of a \(k\)-vector space by a finite group).

Lemma

Let \(\mathcal{F}\) be a predeformation category satisfying (S2) which has a versal formal object. Then its minimal versal formal object satisfies (06T6).

Proof

Let \(\xi\) be a minimal versal formal object for \(\mathcal{F}\), see Lemma 06T5. Say \(\xi\) lies over \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\). In order to parse (06T6) we point out that \(T\mathcal{F}\) has a natural \(k\)-vector space structure (see Lemma 06IH), that \(d\underline{\xi} : \text{Der}_\Lambda(R, k) \to T\mathcal{F}\) is linear (see Lemma 06IJ), and that the action of \(\text{Der}_\Lambda(k, k)\) is given by addition (see Lemma 06SU). Consider the diagram \[\xymatrix{ & \Hom_k(\mathfrak m_R/\mathfrak m_R^2, k) \\ K \ar[r] & \text{Der}_\Lambda(R, k) \ar[r]^{d\underline{\xi}} \ar[u] & T\mathcal{F} \\ & \text{Der}_\Lambda(k, k) \ar[u] \ar[ru] }\] The vector space \(K\) is the kernel of \(d\underline{\xi}\). Note that the middle column is exact in the middle as it is dual to the sequence (06S6). If (06T6) fails, then we can find a nonzero element \(D \in K\) which does not map to zero in \(\Hom_k(\mathfrak m_R/\mathfrak m_R^2, k)\). This means there exists an \(t \in \mathfrak m_R\) such that \(D(t) = 1\). Set \(R' = \{a \in R \mid D(a) = 0\}\). As \(D\) is a derivation this is a subring of \(R\). Since \(D(t) = 1\) we see that \(R' \to k\) is surjective (compare with the proof of Lemma 06H0). Note that \(\mathfrak m_{R'} = \Ker(D : \mathfrak m_R \to k)\) is an ideal of \(R\) and \(\mathfrak m_R^2 \subset \mathfrak m_{R'}\). Hence \[\mathfrak m_R/\mathfrak m_R^2 = \mathfrak m_{R'}/\mathfrak m_R^2 + k\overline{t}\] which implies that the map \[R'/\mathfrak m_R^2 \times_k k[\epsilon] \to R/\mathfrak m_R^2\] sending \(\epsilon\) to \(\overline{t}\) is an isomorphism. In particular there is a map \(R/\mathfrak m_R^2 \to R'/\mathfrak m_R^2\).

Let \(\xi \to y\) be a morphism lying over \(R \to R/\mathfrak m_R^2\). Let \(y \to x\) be a morphism lying over \(R/\mathfrak m_R^2 \to R'/\mathfrak m_R^2\). Let \(y \to x_\epsilon\) be a morphism lying over \(R/\mathfrak m_R^2 \to k[\epsilon]\). Let \(x_0\) be the unique (up to unique isomorphism) object of \(\mathcal{F}\) over \(k\). By the axioms of a category cofibred in groupoids we obtain a commutative diagram \[\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_\epsilon \ar[d] \\ x \ar[r] & x_0 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ R'/\mathfrak m_R^2 \times_k k[\epsilon] \ar[r] \ar[d] & k[\epsilon] \ar[d] \\ R'/\mathfrak m_R^2 \ar[r] & k. } }\] Because \(D \in K\) we see that \(x_\epsilon\) is isomorphic to \(0 \in \mathcal{F}(k[\epsilon])\), i.e., \(x_\epsilon\) is the pushforward of \(x_0\) via \(k \to k[\epsilon], a \mapsto a\). Hence by Lemma 06IS we see that there exists a morphism \(x \to y\). Since \(\text{length}_\Lambda(R'/\mathfrak m_R^2) < \text{length}_\Lambda(R/\mathfrak m_R^2)\) the corresponding ring map \(R'/\mathfrak m_R^2 \to R/\mathfrak m_R^2\) is not surjective. This contradicts the minimality of \(\xi/R\), see part (1) of Lemma 06T2. This contradiction shows that such a \(D\) cannot exist, hence we win.

Theorem

Let \(\mathcal{F}\) be a predeformation category. Consider the following conditions

\(\mathcal{F}\) has a minimal versal formal object satisfying (06IM),

\(\mathcal{F}\) has a minimal versal formal object satisfying (06T6),

the following conditions hold:

\(\mathcal{F}\) satisfies (S1).

\(\mathcal{F}\) satisfies (S2).

\(\dim_k T\mathcal{F}\) is finite.

We always have \[(1) \Rightarrow (3) \Rightarrow (2).\] If \(k' \subset k\) is separable, then all three are equivalent.

Proof

Lemma 06IV shows that (1) \(\Rightarrow\) (3). Lemmas 06IW and 06T8 show that (3) \(\Rightarrow\) (2). If \(k' \subset k\) is separable then \(\text{Der}_\Lambda(k, k) = 0\) and we see that (06IM) \(=\) (06T6), i.e., (1) is the same as (2).

An alternative proof of (3) \(\Rightarrow\) (1) in the classical case is to add a few words to the proof of Lemma 06IW to see that one can right away construct a versal object which satisfies (06IM) in this case. This avoids the use of Lemma 06IW in the classical case. Details omitted.

Remark

Let \(F : \mathcal{C}_\Lambda \to \textit{Sets}\) be a predeformation functor satisfying (S1) and (S2) and \(\dim_k TF < \infty\). Recall that these conditions correspond to the conditions (H1), (H2), and (H3) from Schlessinger’s paper, see Remark 0D3G. Now, in the classical case (or if \(k' \subset k\) is separable) following Schlessinger we introduce the notion of a hull: a hull is a versal formal object \(\xi \in \widehat{F}(R)\) such that \(d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to TF\) is an isomorphism, i.e., (06IM) holds. Thus Theorem 06IX tells us \[(H1) + (H2) + (H3) \Rightarrow \text{ there exists a hull}\] in the classical case. In other words, our theorem recovers Schlessinger’s theorem on the existence of hulls.

Remark

Let \(\mathcal{F}\) be a predeformation category. Recall that \(\mathcal{F} \to \overline{\mathcal{F}}\) is smooth, see Remark 06HK. Hence if \(\xi \in \widehat{\mathcal{F}}(R)\) is a versal formal object, then the composition \[\underline{R}|_{\mathcal{C}_\Lambda} \longrightarrow \mathcal{F} \longrightarrow \overline{\mathcal{F}}\] is smooth (Lemma 06HM) and we conclude that the image \(\overline{\xi}\) of \(\xi\) in \(\overline{\mathcal{F}}\) is a versal formal object. If (06IM) holds, then \(\overline{\xi}\) induces an isomorphism \(T\underline{R}|_{\mathcal{C}_\Lambda} \to T\overline{\mathcal{F}}\) because \(\mathcal{F} \to \overline{\mathcal{F}}\) identifies tangent spaces. Hence in this case \(\overline{\xi}\) is a hull for \(\overline{\mathcal{F}}\), see Remark 06IY. By Theorem 06IX we can always find such a \(\xi\) if \(k' \subset k\) is separable and \(\mathcal{F}\) is a predeformation category satisfying (S1), (S2), and \(\dim_k T\mathcal{F} < \infty\).

Example

In Lemma 06SM we constructed objects \(R \in \widehat{\mathcal{C}}_\Lambda\) such that \(\underline{R}|_{\mathcal{C}_\Lambda}\) is smooth and such that \[H_1(L_{k/\Lambda}) = \mathfrak m_R/\mathfrak m_R^2 \quad\text{and}\quad \Omega_{R/\Lambda} \otimes_R k = \Omega_{k/\Lambda}\] Let us reinterpret this using the theorem above. Namely, consider \(\mathcal{F} = \mathcal{C}_\Lambda\) as a category cofibred in groupoids over itself (using the identity functor). Then \(\mathcal{F}\) is a predeformation category, satisfies (S1) and (S2), and we have \(T\mathcal{F} = 0\). Thus \(\mathcal{F}\) satisfies condition (3) of Theorem 06IX. The theorem implies that (2) holds, i.e., we can find a minimal versal formal object \(\xi \in \widehat{\mathcal{F}}(S)\) over some \(S \in \widehat{\mathcal{C}}_\Lambda\) satisfying (06T6). Lemma 0DYL shows that \(\Lambda \to S\) is formally smooth in the \(\mathfrak m_S\)-adic topology (because \(\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F} = \mathcal{C}_\Lambda\) is smooth). Now condition (06T6) tells us that \(\text{Der}_\Lambda(S, k) \to 0\) is bijective on \(\text{Der}_\Lambda(k, k)\)-orbits. This means the injection \(\text{Der}_\Lambda(k, k) \to \text{Der}_\Lambda(S, k)\) is also surjective. In other words, we have \(\Omega_{S/\Lambda} \otimes_S k = \Omega_{k/\Lambda}\). Since \(\Lambda \to S\) is formally smooth in the \(\mathfrak m_S\)-adic topology, we can apply More on Algebra, Lemma 0DYH to conclude the exact sequence (06S7) turns into a pair of identifications \[H_1(L_{k/\Lambda}) = \mathfrak m_S/\mathfrak m_S^2 \quad\text{and}\quad \Omega_{S/\Lambda} \otimes_S k = \Omega_{k/\Lambda}\] Reading the argument backwards, we find that the \(R\) constructed in Lemma 06SM carries a minimal versal object. By the uniqueness of minimal versal objects (Lemma 06T5) we also conclude \(R \cong S\), i.e., the two constructions give the same answer.

Rim-Schlessinger conditions and deformation categories

There is a very natural property of categories cofibred in groupoids over \(\mathcal{C}_\Lambda\) which is easy to check in practice and which implies Schlessinger’s properties (S1) and (S2) we have introduced earlier.

Definition

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal C_\Lambda\). We say that \(\mathcal{F}\) satisfies condition (RS) if for every diagram in \(\mathcal{F}\) \[\vcenter{ \xymatrix{ & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & A_2 \ar[d] \\ A_1 \ar[r] & A } }\] in \(\mathcal{C}_\Lambda\) with \(A_2 \to A\) surjective, there exists a fiber product \(x_1 \times_x x_2\) in \(\mathcal{F}\) such that the diagram \[\vcenter{ \xymatrix{ x_1 \times_x x_2 \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad\text{lies over}\quad \vcenter{ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A. } }\]

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\) satisfying (RS). Given a commutative diagram in \(\mathcal{F}\) \[\vcenter{ \xymatrix{ y \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A. } }\] with \(A_2 \to A\) surjective, then it is a fiber square.

Proof

Since \(\mathcal{F}\) satisfies (RS), there exists a fiber product diagram \[\vcenter{ \xymatrix{ x_1 \times_x x_2 \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A. } }\] The induced map \(y \to x_1 \times_x x_2\) lies over \(\text{id} : A_1 \times_A A_1 \to A_1 \times_A A_1\), hence it is an isomorphism.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal C_\Lambda\). Then \(\mathcal{F}\) satisfies (RS) if the condition in Definition 06J2 is assumed to hold only when \(A_2 \to A\) is a small extension.

Proof

Apply Lemma 06GE. The proof is similar to that of Lemma 06HH.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). The following are equivalent

\(\mathcal{F}\) satisfies (RS),

the functor \(\mathcal{F}(A_1 \times_A A_2) \to \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)\) see (06SP) is an equivalence of categories whenever \(A_2 \to A\) is surjective, and

same as in (2) whenever \(A_2 \to A\) is a small extension.

Proof

Assume (1). By Lemma 06J3 we see that every object of \(\mathcal{F}(A_1 \times_A A_2)\) is of the form \(x_1 \times_x x_2\). Moreover \[\Mor_{A_1 \times_A A_2}(x_1 \times_x x_2, y_1 \times_y y_2) = \Mor_{A_1}(x_1, y_1) \times_{\Mor_A(x, y)} \Mor_{A_2}(x_2, y_2).\] Hence we see that \(\mathcal{F}(A_1 \times_A A_2)\) is a \(2\)-fibre product of \(\mathcal{F}(A_1)\) with \(\mathcal{F}(A_2)\) over \(\mathcal{F}(A)\) by Categories, Remark 06RL. In other words, we see that (2) holds.

The implication (2) \(\Rightarrow\) (3) is immediate.

Assume (3). Let \(q_1 : A_1 \to A\) and \(q_2 : A_2 \to A\) be given with \(q_2\) a small extension. We will use the description of the \(2\)-fibre product \(\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)\) from Categories, Remark 06RL. Hence let \(y \in \mathcal{F}(A_1 \times_A A_2)\) correspond to \((x_1, x_2, x, a_1 : x_1 \to x, a_2 : x_2 \to x)\). Let \(z\) be an object of \(\mathcal{F}\) lying over \(C\). Then \[\begin{align*} \Mor_\mathcal{F}(z, y) & = \{(f, \alpha) \mid f : C \to A_1 \times_A A_2, \alpha : f_*z \to y\} \\ & = \{(f_1, f_2, \alpha_1, \alpha_2) \mid f_i : C \to A_i, \ \alpha_i : f_{i, *}z \to x_i, \\ & \quad\quad\quad\quad q_1 \circ f_1 = q_2 \circ f_2, \ q_{1, *} \alpha_1 = q_{2, *}\alpha_2\} \\ & = \Mor_\mathcal{F}(z, x_1) \times_{\Mor_\mathcal{F}(z, x)} \Mor_\mathcal{F}(z, x_2) \end{align*}\] whence \(y\) is a fibre product of \(x_1\) and \(x_2\) over \(x\). Thus we see that \(\mathcal{F}\) satisfies (RS) in case \(A_2 \to A\) is a small extension. Hence (RS) holds by Lemma 06J4.

Remark

When \(\mathcal{F}\) is cofibered in sets, condition (RS) is exactly condition (H4) from Schlessinger’s paper [Sch, Theorem 2.11]. Namely, for a functor \(F: \mathcal{C}_\Lambda \to \textit{Sets}\), condition (RS) states: If \(A_1 \to A\) and \(A_2 \to A\) are maps in \(\mathcal{C}_\Lambda\) with \(A_2 \to A\) surjective, then the induced map \(F(A_1 \times_A A_2) \to F(A_1) \times_{F(A)} F(A_2)\) is bijective.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). The condition (RS) for \(\mathcal{F}\) implies both (S1) and (S2) for \(\mathcal{F}\).

Proof

Using the reformulation of Lemma 06J5 and the explanation of (S1) following Definition 06HW it is immediate that (RS) implies (S1). This proves the first part of (S2). The second part of (S2) follows because Lemma 06J3 tells us that \(y = x_1 \times_{d, x_0, e} x_2 = y'\) if \(y, y'\) are as in the second part of the definition of (S2) in Definition 06HW. (In fact the morphism \(y \to y'\) is compatible with both \(a, a'\) and \(c, c'\)!)

The following lemma is the analogue of Lemma 06I0. Recall that if \(\mathcal{F}\) is a category cofibred in groupoids over \(\mathcal{C}_\Lambda\) and \(x\) is an object of \(\mathcal{F}\) lying over \(A\), then we denote \(\text{Aut}_A(x) = \Mor_A(x, x) = \Mor_{\mathcal{F}(A)}(x, x)\). If \(x' \to x\) is a morphism of \(\mathcal{F}\) lying over \(A' \to A\) then there is a well defined map of groups \(\text{Aut}_{A'}(x') \to \text{Aut}_A(x)\).

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\) satisfying (RS). The following conditions are equivalent:

\(\overline{\mathcal{F}}\) satisfies (RS).

Let \(f_1: A_1 \to A\) and \(f_2: A_2 \to A\) be ring maps in \(\mathcal{C}_\Lambda\) with \(f_2\) surjective. The induced map of sets of isomorphism classes \[\overline{\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)} \to \overline{\mathcal{F}}(A_1) \times_{\overline{\mathcal{F}}(A)} \overline{\mathcal{F}}(A_2)\] is injective.

For every morphism \(x' \to x\) in \(\mathcal{F}\) lying over a surjective ring map \(A' \to A\), the map \(\text{Aut}_{A'}(x') \to \text{Aut}_A(x)\) is surjective.

For every morphism \(x' \to x\) in \(\mathcal{F}\) lying over a small extension \(A' \to A\), the map \(\text{Aut}_{A'}(x') \to \text{Aut}_A(x)\) is surjective.

Proof

We prove that (1) is equivalent to (2) and (2) is equivalent to (3). The equivalence of (3) and (4) follows from Lemma 06GE.

Let \(f_1: A_1 \to A\) and \(f_2: A_2 \to A\) be ring maps in \(\mathcal{C}_\Lambda\) with \(f_2\) surjective. By Remark 06J6 we see \(\overline{\mathcal{F}}\) satisfies (RS) if and only if the map \[\overline{\mathcal{F}}(A_1 \times_A A_2) \to \overline{\mathcal F}(A_1) \times_{\overline{\mathcal{F}}(A)} \overline{\mathcal{F}}(A_2)\] is bijective for any such \(f_1, f_2\). This map is at least surjective since that is the condition of (S1) and \(\overline{\mathcal{F}}\) satisfies (S1) by Lemmas 06J7 and 06I0. Moreover, this map factors as \[\overline{\mathcal{F}}(A_1 \times_A A_2) \longrightarrow \overline{\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)} \longrightarrow \overline{\mathcal{F}}(A_1) \times_{\overline{\mathcal{F}}(A)} \overline{\mathcal{F}}(A_2),\] where the first map is a bijection since \[\mathcal{F}(A_1 \times_A A_2) \longrightarrow \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)\] is an equivalence by (RS) for \(\mathcal{F}\). Hence (1) is equivalent to (2).

Assume (2) holds. Let \(x' \to x\) be a morphism in \(\mathcal{F}\) lying over a surjective ring map \(f: A' \to A\). Let \(a \in \text{Aut}_A(x)\). The objects \[(x', x', a : x \to x), \ (x', x', \text{id} : x \to x)\] of \(\mathcal{F}(A') \times_{\mathcal{F}(A)} \mathcal{F}(A')\) have the same image in \(\overline{\mathcal{F}}(A') \times_{\overline{\mathcal{F}}(A)} \overline{\mathcal{F}}(A')\). By (2) there exists maps \(b_1, b_2 : x' \to x'\) such that \[\xymatrix{ x \ar[r]_a \ar[d]_{f_*b_1} & x \ar[d]^{f_*b_2} \\ x \ar[r]^{\text{id}} & x }\] commutes. Hence \(b_2^{-1} \circ b_1 \in \text{Aut}_{A'}(x')\) has image \(a \in \text{Aut}_A(x)\). Hence (3) holds.

Assume (3) holds. Suppose \[(x_1, x_2, a : (f_1)_*x_1 \to (f_2)_*x_2), \ (x'_1, x'_2, a' : (f_1)_*x'_1 \to (f_2)_*x'_2)\] are objects of \(\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)\) with the same image in \(\overline{\mathcal{F}}(A_1) \times_{\overline{\mathcal{F}}(A)} \overline{\mathcal{F}}(A_2)\). Then there are morphisms \(b_1: x_1 \to x'_1\) in \(\mathcal{F}(A_1)\) and \(b_2: x_2 \to x'_2\) in \(\mathcal F(A_2)\). By (3) we can modify \(b_2\) by an automorphism of \(x_2\) over \(A_2\) so that the diagram \[\xymatrix{ (f_1)_*x_1 \ar[r]_a \ar[d]_{(f_1)_*b_1} & (f_2)_*x_2 \ar[d]^{(f_2)_*b_2} \\ (f_1)_*x'_1 \ar[r]^{a'} & (f_2)_*x'_2. }\] commutes. This proves \((x_1, x_2, a) \cong (x'_1, x'_2, a')\) in \(\overline{\mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)}\). Hence (2) holds.

Finally we define the notion of a deformation category.

Definition

A deformation category is a predeformation category \(\mathcal{F}\) satisfying (RS). A morphism of deformation categories is a morphism of categories over \(\mathcal{C}_\Lambda\).

Remark

We say that a functor \(F: \mathcal{C}_\Lambda \to \textit{Sets}\) is a deformation functor if the associated cofibered set is a deformation category, i.e. if \(F(k)\) is a one element set and \(F\) satisfies (RS). If \(\mathcal{F}\) is a deformation category, then \(\overline{\mathcal{F}}\) is a predeformation functor but not necessarily a deformation functor, as Lemma 06J8 shows.

Example

A prorepresentable functor \(F\) is a deformation functor. Namely, suppose \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\) and \(F(A) = \Mor_{\widehat{\mathcal{C}}_\Lambda}(R, A)\). There is a unique morphism \(R \to k\), so \(F(k)\) is a one element set. Since \[\Hom_\Lambda(R, A_1 \times_A A_2) = \Hom_\Lambda(R, A_1) \times_{\Hom_\Lambda(R, A)} \Hom_\Lambda(R, A_2)\] the same is true for maps in \(\widehat{\mathcal{C}}_\Lambda\) and we see that \(F\) has (RS).

The following is one of our typical remarks on passing from a category cofibered in groupoids to the predeformation category at a point over \(k\): it says that this process preserves (RS).

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). Let \(x_0 \in \Ob(\mathcal{F}(k))\). Let \(\mathcal{F}_{x_0}\) be the category cofibred in groupoids over \(\mathcal{C}_\Lambda\) constructed in Remark 06GU. If \(\mathcal{F}\) satisfies (RS), then so does \(\mathcal{F}_{x_0}\). In particular, \(\mathcal{F}_{x_0}\) is a deformation category.

Proof

Any diagram as in Definition 06J2 in \(\mathcal{F}_{x_0}\) gives rise to a diagram in \(\mathcal{F}\) and the output of (RS) for this diagram in \(\mathcal{F}\) can be viewed as an output for \(\mathcal{F}_{x_0}\) as well.

The following lemma is the analogue of the fact that \(2\)-fibre products of algebraic stacks are algebraic stacks.

Lemma

Let \[\xymatrix{ \mathcal{H} \times_\mathcal{F} \mathcal{G} \ar[r] \ar[d] & \mathcal{G} \ar[d]^g \\ \mathcal{H} \ar[r]^f & \mathcal{F} }\] be \(2\)-fibre product of categories cofibered in groupoids over \(\mathcal{C}_\Lambda\). If \(\mathcal{F}, \mathcal{G}, \mathcal{H}\) all satisfy (RS), then \(\mathcal{H} \times_\mathcal{F} \mathcal{G}\) satisfies (RS).

Proof

If \(A\) is an object of \(\mathcal{C}_\Lambda\), then an object of the fiber category of \(\mathcal{H} \times_\mathcal{F} \mathcal{G}\) over \(A\) is a triple \((u, v, a)\) where \(u \in \mathcal{H}(A)\), \(v \in \mathcal{G}(A)\), and \(a : f(u) \to g(v)\) is a morphism in \(\mathcal{F}(A)\). Consider a diagram in \(\mathcal{H} \times_\mathcal{F} \mathcal{G}\) \[\vcenter{ \xymatrix{ & (u_2, v_2, a_2) \ar[d] \\ (u_1, v_1, a_1) \ar[r] & (u, v, a) } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & A_2 \ar[d] \\ A_1 \ar[r] & A } }\] in \(\mathcal{C}_\Lambda\) with \(A_2 \to A\) surjective. Since \(\mathcal{H}\) and \(\mathcal{G}\) satisfy (RS), there are fiber products \(u_1 \times_u u_2\) and \(v_1 \times_v v_2\) lying over \(A_1 \times_A A_2\). Since \(\mathcal{F}\) satisfies (RS), Lemma 06J3 shows \[\vcenter{ \xymatrix{ f(u_1 \times_u u_2) \ar[r] \ar[d] & f(u_2) \ar[d] \\ f(u_1) \ar[r] & f(u) } } \quad\text{and}\quad \vcenter{ \xymatrix{ g(v_1 \times_v v_2) \ar[r] \ar[d] & g(v_2) \ar[d] \\ g(v_1) \ar[r] & g(v) } }\] are both fiber squares in \(\mathcal{F}\). Thus we can view \(a_1 \times_a a_2\) as a morphism from \(f(u_1 \times_u u_2)\) to \(g(v_1 \times_v v_2)\) over \(A_1 \times_A A_2\). It follows that \[\xymatrix{ (u_1 \times_u u_2, v_1 \times_v v_2, a_{1} \times_a a_2) \ar[d] \ar[r] & (u_2, v_2, a_2) \ar[d] \\ (u_1, v_1, a_1) \ar[r] & (u, v, a) }\] is a fiber square in \(\mathcal{H} \times_\mathcal{F} \mathcal{G}\) as desired.

Lifts of objects

The content of this section is that the tangent space has a principal homogeneous action on the set of lifts along a small extension in the case of a deformation category.

Definition

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). Let \(f: A' \to A\) be a map in \(\mathcal{C}_\Lambda\). Let \(x \in \mathcal{F}(A)\). The category \(\textit{Lift}(x, f)\) of lifts of \(x\) along \(f\) is the category with the following objects and morphisms.

Objects: A lift of \(x\) along \(f\) is a morphism \(x' \to x\) lying over \(f\).

Morphisms: A morphism of lifts from \(a_1 : x'_1 \to x\) to \(a_2 : x'_2 \to x\) is a morphism \(b : x'_1 \to x'_2\) in \(\mathcal{F}(A')\) such that \(a_2 = a_1 \circ b\).

The set \(\text{Lift}(x, f)\) of lifts of \(x\) along \(f\) is the set of isomorphism classes of \(\textit{Lift}(x, f)\).

Remark

When the map \(f: A' \to A\) is clear from the context, we may write \(\textit{Lift}(x, A')\) and \(\text{Lift}(x, A')\) in place of \(\textit{Lift}(x, f)\) and \(\text{Lift}(x, f)\).

Remark

Let \(\mathcal{F}\) be a category cofibred in groupoids over \(\mathcal{C}_\Lambda\). Let \(x_0 \in \Ob(\mathcal{F}(k))\). Let \(V\) be a finite dimensional vector space. Then \(\text{Lift}(x_0, k[V])\) is the set of isomorphism classes of \(\mathcal{F}_{x_0}(k[V])\) where \(\mathcal{F}_{x_0}\) is the predeformation category of objects in \(\mathcal{F}\) lying over \(x_0\), see Remark 06GU. Hence if \(\mathcal{F}\) satisfies (S2), then so does \(\mathcal{F}_{x_0}\) (see Lemma 06SQ) and by Lemma 06IH we see that \[\text{Lift}(x_0, k[V]) = T\mathcal{F}_{x_0} \otimes_k V\] as \(k\)-vector spaces.

Remark

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal C_\Lambda\) satisfying (RS). Let \[\xymatrix{ A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A }\] be a fibre square in \(\mathcal{C}_\Lambda\) such that either \(A_1 \to A\) or \(A_2 \to A\) is surjective. Let \(x \in \Ob(\mathcal{F}(A))\). Given lifts \(x_1 \to x\) and \(x_2 \to x\) of \(x\) to \(A_1\) and \(A_2\), we get by (RS) a lift \(x_1 \times_x x_2 \to x\) of \(x\) to \(A_1 \times_A A_2\). Conversely, by Lemma 06J3 any lift of \(x\) to \(A_1 \times_A A_2\) is of this form. Hence a bijection \[\text{Lift}(x, A_1) \times \text{Lift}(x, A_2) \longrightarrow \text{Lift}(x, A_1 \times_A A_2).\] Similarly, if \(x_1 \to x\) is a fixed lifting of \(x\) to \(A_1\), then there is a bijection \[\text{Lift}(x_1, A_1 \times_A A_2) \longrightarrow \text{Lift}(x, A_2).\] Now let \[\xymatrix{ A_1' \times_A A_2 \ar[r] \ar[d] & A_1 \times_A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1' \ar[r] & A_1 \ar[r] & A }\] be a composition of fibre squares in \(\mathcal{C}_\Lambda\) with both \(A'_1 \to A_1\) and \(A_1 \to A\) surjective. Let \(x_1 \to x\) be a morphism lying over \(A_1 \to A\). Then by the above we have bijections \[\begin{align*} \text{Lift}(x_1, A_1' \times_A A_2) & = \text{Lift}(x_1, A_1') \times \text{Lift}(x_1, A_1 \times_A A_2) \\ & = \text{Lift}(x_1, A_1') \times \text{Lift}(x, A_2). \end{align*}\]

Lemma

Let \(\mathcal{F}\) be a deformation category. Let \(A' \to A\) be a surjective ring map in \(\mathcal{C}_\Lambda\) whose kernel \(I\) is annihilated by \(\mathfrak m_{A'}\). Let \(x \in \Ob(\mathcal{F}(A))\). If \(\text{Lift}(x, A')\) is nonempty, then there is a free and transitive action of \(T\mathcal{F} \otimes_k I\) on \(\text{Lift}(x, A')\).

Proof

Consider the ring map \(g : A' \times_A A' \to k[I]\) defined by the rule \(g(a_1, a_2) = \overline{a_1} \oplus a_2 - a_1\) (compare with Lemma 06IT). There is an isomorphism \[A' \times_A A' \xrightarrow{\sim} A' \times_k k[I]\] given by \((a_1, a_2) \mapsto (a_1, g(a_1, a_2))\). This isomorphism commutes with the projections to \(A'\) on the first factor, and hence with the projections of \(A' \times_A A'\) and \(A' \times_k k[I]\) to \(A\). Thus there is a bijection [06TA]\[\begin{equation} \text{Lift}(x, A' \times_A A') \longrightarrow \text{Lift}(x, A' \times_k k[I]) \end{equation}\] By Remark 06JH there is a bijection [06TB]\[\begin{equation} \text{Lift}(x, A') \times \text{Lift}(x, A') \longrightarrow \text{Lift}(x, A' \times_A A') \end{equation}\] There is a commutative diagram \[\xymatrix{ A' \times_k k[I] \ar[r] \ar[d] & A \times_k k[I] \ar[r] \ar[d] & k[I] \ar[d] \\ A' \ar[r] & A \ar[r] & k. }\] Thus if we choose a pushforward \(x \to x_0\) of \(x\) along \(A \to k\), we obtain by the end of Remark 06JH a bijection [06TC]\[\begin{equation} \text{Lift}(x, A' \times_k k[I]) \longrightarrow \text{Lift}(x, A') \times \text{Lift}(x_0, k[I]) \end{equation}\] Composing (06TB), (06TA), and (06TC) we get a bijection \[\Phi : \text{Lift}(x, A') \times \text{Lift}(x, A') \longrightarrow \text{Lift}(x, A') \times \text{Lift}(x_0, k[I]).\] This bijection commutes with the projections on the first factors. By Remark 06JG we see that \(\text{Lift}(x_0, k[I]) = T\mathcal{F} \otimes_k I\). If \(\text{pr}_2\) is the second projection of \(\text{Lift}(x, A') \times \text{Lift}(x, A')\), then we get a map \[a = \text{pr}_2 \circ \Phi^{-1} : \text{Lift}(x, A') \times (T\mathcal{F} \otimes_k I) \longrightarrow \text{Lift}(x, A').\] Unwinding all the above we see that \(a(x' \to x, \theta)\) is the unique lift \(x'' \to x\) such that \(g_*(x', x'') = \theta\) in \(\text{Lift}(x_0, k[I]) = T\mathcal{F} \otimes_k I\). To see this is an action of \(T\mathcal{F} \otimes_k I\) on \(\text{Lift}(x, A')\) we have to show the following: if \(x', x'', x'''\) are lifts of \(x\) and \(g_*(x', x'') = \theta\), \(g_*(x'', x''') = \theta'\), then \(g_*(x', x''') = \theta + \theta'\). This follows from the commutative diagram \[\xymatrix{ A' \times_A A' \times_A A' \ar[rrrrr]_-{(a_1, a_2, a_3) \mapsto (g(a_1, a_2), g(a_2, a_3))} \ar[rrrrrd]_{(a_1, a_2, a_3) \mapsto g(a_1, a_3)} & & & & & k[I] \times_k k[I] = k[I \times I] \ar[d]^{+} \\ & & & & & k[I] }\] The action is free and transitive because \(\Phi\) is bijective.

Remark

The action of Lemma 06JI is functorial. Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a morphism of deformation categories. Let \(A' \to A\) be a surjective ring map whose kernel \(I\) is annihilated by \(\mathfrak m_{A'}\). Let \(x \in \Ob(\mathcal{F}(A))\). In this situation \(\varphi\) induces the vertical arrows in the following commutative diagram \[\xymatrix{ \text{Lift}(x, A') \times (T\mathcal{F} \otimes_k I) \ar[d]_{(\varphi, d\varphi \otimes \text{id}_I)} \ar[r] & \text{Lift}(x, A') \ar[d]^\varphi \\ \text{Lift}(\varphi(x), A') \times (T\mathcal{G} \otimes_k I) \ar[r] & \text{Lift}(\varphi(x), A') }\] The commutativity follows as each of the maps (06TB), (06TA), and (06TC) of the proof of Lemma 06JI gives rise to a similar commutative diagram.

Schlessinger’s theorem on prorepresentable functors

We deduce Schlessinger’s theorem characterizing prorepresentable functors on \(\mathcal{C}_\Lambda\).

Lemma

Let \(F, G: \mathcal{C}_\Lambda \to \textit{Sets}\) be deformation functors. Let \(\varphi : F \to G\) be a smooth morphism which induces an isomorphism \(d\varphi : TF \to TG\) of tangent spaces. Then \(\varphi\) is an isomorphism.

Proof

We prove \(F(A) \to G(A)\) is a bijection for all \(A \in \Ob(\mathcal{C}_\Lambda)\) by induction on \(\text{length}_A(A)\). For \(A = k\) the statement follows from the assumption that \(F\) and \(G\) are deformation functors. Suppose that the statement holds for rings of length less than \(n\) and let \(A'\) be a ring of length \(n\). Choose a small extension \(f : A' \to A\). We have a commutative diagram \[\xymatrix{ F(A') \ar[r] \ar[d]_{F(f)} & G(A') \ar[d]^{G(f)} \\ F(A) \ar[r]^{\sim} & G(A) }\] where the map \(F(A) \to G(A)\) is a bijection. By smoothness of \(F \to G\), \(F(A') \to G(A')\) is surjective (Lemma 06HN). Thus we can check bijectivity by checking it on fibers \(F(f)^{-1}(x) \to G(f)^{-1}(\varphi(x))\) for \(x \in F(A)\) such that \(F(f)^{-1}(x)\) is nonempty. These fibers are precisely \(\text{Lift}(x, A')\) and \(\text{Lift}(\varphi(x), A')\) and by assumption we have an isomorphism \(d\varphi \otimes \text{id} : TF \otimes_k \Ker(f) \to TG \otimes_k \Ker(f)\). Thus, by Lemma 06JI and Remark 06JJ, for \(x \in F(A)\) such that \(F(f)^{-1}(x)\) is nonempty the map \(F(f)^{-1}(x) \to G(f)^{-1}(\varphi(x))\) is a map of sets commuting with free transitive actions by \(TF \otimes_k \Ker(f)\). Hence it is bijective.

Note that in case \(k' \subset k\) is separable condition (c) in the theorem below is empty.

Theorem

Let \(F: \mathcal{C}_\Lambda \to \textit{Sets}\) be a functor. Then \(F\) is prorepresentable if and only if (a) \(F\) is a deformation functor, (b) \(\dim_k TF\) is finite, and (c) \(\gamma : \text{Der}_\Lambda(k, k) \to TF\) is injective.

Proof

Assume \(F\) is prorepresentable by \(R \in \widehat{\mathcal{C}}_\Lambda\). We see \(F\) is a deformation functor by Example 06JB. We see \(\dim_k TF\) is finite by Example 06IC. Finally, \(\text{Der}_\Lambda(k, k) \to TF\) is identified with \(\text{Der}_\Lambda(k, k) \to \text{Der}_\Lambda(R, k)\) by Example 06ST which is injective because \(R \to k\) is surjective.

Conversely, assume (a), (b), and (c) hold. By Lemma 06J7 we see that (S1) and (S2) hold. Hence by Theorem 06IX there exists a minimal versal formal object \(\xi\) of \(F\) such that (06T6) holds. Say \(\xi\) lies over \(R\). The map \[d\underline{\xi} : \text{Der}_\Lambda(R, k) \to T\mathcal{F}\] is bijective on \(\text{Der}_\Lambda(k, k)\)-orbits. Since the action of \(\text{Der}_\Lambda(k, k)\) on the left hand side is free by (c) and Lemma 06SU we see that the map is bijective. Thus we see that \(\underline{\xi}\) is an isomorphism by Lemma 06JL.

Infinitesimal automorphisms

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). Given a morphism \(x' \to x\) in \(\mathcal{F}\) lying over \(A' \to A\), there is an induced homomorphism \[\text{Aut}_{A'}(x') \to \text{Aut}_A(x).\] Lemma 06J8 says that the cokernel of this homomorphism determines whether condition (RS) on \(\mathcal{F}\) passes to \(\overline{\mathcal{F}}\). In this section we study the kernel of this homomorphism. We will see that it also gives a measure of how far \(\mathcal{F}\) is from \(\overline{\mathcal{F}}\).

Definition

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal C_\Lambda\). Let \(x' \to x\) be a morphism in \(\mathcal{F}\) lying over \(A' \to A\). The kernel \[\text{Inf}(x'/x) = \Ker(\text{Aut}_{A'}(x') \to \text{Aut}_A(x))\] is the group of infinitesimal automorphisms of \(x'\) over \(x\).

Definition

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal C_\Lambda\). Let \(x_0 \in \Ob(\mathcal{F}(k))\). Assume a choice of pushforward \(x_0 \to x_0'\) of \(x_0\) along the map \(k \to k[\epsilon], a \mapsto a\) has been made. Then there is a unique map \(x'_0 \to x_0\) such that \(x_0 \to x_0' \to x_0\) is the identity on \(x_0\). Then \[\text{Inf}_{x_0}(\mathcal F) = \text{Inf}(x'_0/x_0)\] is the group of infinitesimal automorphisms of \(x_0\)

Remark

Up to canonical isomorphism \(\text{Inf}_{x_0}(\mathcal{F})\) does not depend on the choice of pushforward \(x_0 \to x_0'\) because any two pushforwards are canonically isomorphic. Moreover, if \(y_0 \in \mathcal{F}(k)\) and \(x_0 \cong y_0\) in \(\mathcal{F}(k)\), then \(\text{Inf}_{x_0}(\mathcal{F}) \cong \text{Inf}_{y_0}(\mathcal{F})\) where the isomorphism depends (only) on the choice of an isomorphism \(x_0 \to y_0\). In particular, \(\text{Aut}_k(x_0)\) acts on \(\text{Inf}_{x_0}(\mathcal{F})\).

Remark

Assume \(\mathcal{F}\) is a predeformation category. Then

for \(x_0 \in \Ob(\mathcal{F}(k))\) the automorphism group \(\text{Aut}_k(x_0)\) is trivial and hence \(\text{Inf}_{x_0}(\mathcal{F}) = \text{Aut}_{k[\epsilon]}(x'_0)\), and

for \(x_0, y_0 \in \Ob(\mathcal{F}(k))\) there is a unique isomorphism \(x_0 \to y_0\) and hence a canonical identification \(\text{Inf}_{x_0}(\mathcal{F}) = \text{Inf}_{y_0}(\mathcal{F})\).

Since \(\mathcal{F}(k)\) is nonempty, choosing \(x_0 \in \Ob(\mathcal{F}(k))\) and setting \[\text{Inf}(\mathcal{F}) = \text{Inf}_{x_0}(\mathcal{F})\] we get a well defined group of infinitesimal automorphisms of \(\mathcal{F}\). With this notation we have \(\text{Inf}(\mathcal{F}_{x_0}) = \text{Inf}_{x_0}(\mathcal{F})\). Please compare with the equality \(T\mathcal{F}_{x_0} = T_{x_0}\mathcal{F}\) in Remark 06IK.

We will see that \(\text{Inf}_{x_0}(\mathcal{F})\) has a natural \(k\)-vector space structure when \(\mathcal{F}\) satisfies (RS). At the same time, we will see that if \(\mathcal{F}\) satisfies (RS), then the infinitesimal automorphisms \(\text{Inf}(x'/x)\) of a morphism \(x' \to x\) lying over a small extension are governed by \(\text{Inf}_{x_0}(\mathcal{F})\), where \(x_0\) is a pushforward of \(x\) to \(\mathcal{F}(k)\). In order to do this, we introduce the automorphism functor for any object \(x \in \Ob(\mathcal{F})\) as follows.

Definition

Let \(p : \mathcal{F} \to \mathcal{C}\) be a category cofibered in groupoids over an arbitrary base category \(\mathcal{C}\). Assume a choice of pushforwards has been made. Let \(x \in \Ob(\mathcal{F})\) and let \(U = p(x)\). Let \(U/\mathcal{C}\) denote the category of objects under \(U\). The automorphism functor of \(x\) is the functor \(\mathit{Aut}(x) : U/\mathcal{C} \to \textit{Sets}\) sending an object \(f : U \to V\) to \(\text{Aut}_V(f_*x)\) and sending a morphism \[\xymatrix{ V' \ar[rr] & & V\\ & U \ar[ul]^{f'} \ar[ur]_f & }\] to the homomorphism \(\text{Aut}_{V'}(f'_*x) \to \text{Aut}_V(f_*x)\) coming from the unique morphism \(f'_*x \to f_*x\) lying over \(V' \to V\) and compatible with \(x \to f'_*x\) and \(x \to f_*x\).

We will be concerned with the automorphism functors of objects in a category cofibered in groupoids \(\mathcal{F}\) over \(\mathcal{C}_\Lambda\). If \(A \in \Ob(\mathcal{C}_\Lambda)\), then the category \(A/\mathcal{C}_\Lambda\) is nothing but the category \(\mathcal{C}_A\), i.e. the category defined in Section 06GB where we take \(\Lambda = A\) and \(k = A/\mathfrak m_A\). Hence the automorphism functor of an object \(x \in \Ob(\mathcal{F}(A))\) is a functor \(\mathit{Aut}(x) : \mathcal{C}_A \to \textit{Sets}\).

The following lemma could be deduced from Lemma 06L4 by thinking about the “inertia” of a category cofibred in groupoids, see for example Stacks, Section 036X and Categories, Section 04Z2. However, it is easier to see it directly.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\) satisfying (RS). Let \(x \in \Ob(\mathcal{F}(A))\). Then \(\mathit{Aut}(x): \mathcal{C}_A \to \textit{Sets}\) satisfies (RS).

Proof

It follows that \(\mathit{Aut}(x)\) satisfies (RS) from the fully faithfulness of the functor \(\mathcal{F}(A_1 \times_A A_2) \to \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)\) in Lemma 06J5.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\) satisfying (RS). Let \(x \in \Ob(\mathcal{F}(A))\). Let \(x_0\) be a pushforward of \(x\) to \(\mathcal{F}(k)\).

\(T_{\text{id}_{x_0}} \mathit{Aut}(x)\) has a natural \(k\)-vector space structure such that addition agrees with composition in \(T_{\text{id}_{x_0}} \mathit{Aut}(x)\). In particular, composition in \(T_{\text{id}_{x_0}} \mathit{Aut}(x)\) is commutative.

There is a canonical isomorphism \(T_{\text{id}_{x_0}} \mathit{Aut}(x) \to T_{\text{id}_{x_0}} \mathit{Aut}(x_0)\) of \(k\)-vector spaces.

Proof

We apply Remark 06GU to the functor \(\mathit{Aut}(x) : \mathcal{C}_A \to \textit{Sets}\) and the element \(\text{id}_{x_0} \in \mathit{Aut}(x)(k)\) to get a predeformation functor \(F = \mathit{Aut}(x)_{\text{id}_{x_0}}\). By Lemmas 06JU and 06JC \(F\) is a deformation functor. By definition \(T_{\text{id}_{x_0}} \mathit{Aut}(x) = TF = F(k[\epsilon])\) which has a natural \(k\)-vector space structure specified by Lemma 06IA.

Addition is defined as the composition \[F(k[\epsilon]) \times F(k[\epsilon]) \longrightarrow F(k[\epsilon] \times_k k[\epsilon]) \longrightarrow F(k[\epsilon])\] where the first map is the inverse of the bijection guaranteed by (RS) and the second is induced by the \(k\)-algebra map \(k[\epsilon] \times_k k[\epsilon] \to k[\epsilon]\) which maps \((\epsilon, 0)\) and \((0, \epsilon)\) to \(\epsilon\). If \(A \to B\) is a ring map in \(\mathcal{C}_\Lambda\), then \(F(A) \to F(B)\) is a homomorphism where \(F(A) = \mathit{Aut}(x)_{\text{id}_{x_0}}(A)\) and \(F(B) = \mathit{Aut}(x)_{\text{id}_{x_0}}(B)\) are groups under composition. We conclude that \(+ : F(k[\epsilon]) \times F(k[\epsilon])\to F(k[\epsilon])\) is a homomorphism where \(F(k[\epsilon])\) is regarded as a group under composition. With \(\text{id} \in F(k[\epsilon])\) the unit element we see that \(+(v, \text{id}) = +(\text{id}, v) = v\) for any \(v \in F(k[\epsilon])\) because \((\text{id}, v)\) is the pushforward of \(v\) along the ring map \(k[\epsilon] \to k[\epsilon] \times_k k[\epsilon]\) with \(\epsilon \mapsto (\epsilon, 0)\). In general, given a group \(G\) with multiplication \(\circ\) and \(+ : G \times G \to G\) is a homomorphism such that \(+(g, 1) = +(1, g) = g\), where \(1\) is the identity of \(G\), then \(+ = \circ\). This shows addition in the \(k\)-vector space structure on \(F(k[\epsilon])\) agrees with composition.

Finally, (2) is a matter of unwinding the definitions. Namely \(T_{\text{id}_{x_0}} \mathit{Aut}(x)\) is the set of automorphisms \(\alpha\) of the pushforward of \(x\) along \(A \to k \to k[\epsilon]\) which are trivial modulo \(\epsilon\). On the other hand \(T_{\text{id}_{x_0}} \mathit{Aut}(x_0)\) is the set of automorphisms of the pushforward of \(x_0\) along \(k \to k[\epsilon]\) which are trivial modulo \(\epsilon\). Since \(x_0\) is the pushforward of \(x\) along \(A \to k\) the result is clear.

Remark

We point out some basic relationships between infinitesimal automorphism groups, liftings, and tangent spaces to automorphism functors. Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). Let \(x' \to x\) be a morphism lying over a ring map \(A' \to A\). Then from the definitions we have an equality \[\text{Inf}(x'/x) = \text{Lift}(\text{id}_x, A')\] where the liftings are of \(\text{id}_x\) as an object of \(\mathit{Aut}(x')\). If \(x_0 \in \Ob(\mathcal{F}(k))\) and \(x'_0\) is the pushforward to \(\mathcal{F}(k[\epsilon])\), then applying this to \(x'_0 \to x_0\) we get \[\text{Inf}_{x_0}(\mathcal{F}) = \text{Lift}(\text{id}_{x_0}, k[\epsilon]) = T_{\text{id}_{x_0}} \mathit{Aut}(x_0),\] the last equality following directly from the definitions.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\) satisfying (RS). Let \(x_0 \in \Ob(\mathcal{F}(k))\). Then \(\text{Inf}_{x_0}(\mathcal{F})\) is equal as a set to \(T_{\text{id}_{x_0}} \mathit{Aut}(x_0)\), and so has a natural \(k\)-vector space structure such that addition agrees with composition of automorphisms.

Proof

The equality of sets is as in the end of Remark 06JW and the statement about the vector space structure follows from Lemma 06JV.

Lemma

Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be a morphism of categories cofibred in groupoids over \(\mathcal{C}_\Lambda\) satisfying (RS). Let \(x_0 \in \Ob(\mathcal{F}(k))\). Then \(\varphi\) induces a \(k\)-linear map \(\text{Inf}_{x_0}(\mathcal{F}) \to \text{Inf}_{\varphi(x_0)}(\mathcal{G})\).

Proof

It is clear that \(\varphi\) induces a morphism from \(\mathit{Aut}(x_0) \to \mathit{Aut}(\varphi(x_0))\) which maps the identity to the identity. Hence this follows from the result for tangent spaces, see Lemma 06IJ.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\) satisfying (RS). Let \(x' \to x\) be a morphism lying over a surjective ring map \(A' \to A\) with kernel \(I\) annihilated by \(\mathfrak m_{A'}\). Let \(x_0\) be a pushforward of \(x\) to \(\mathcal{F}(k)\). Then \(\text{Inf}(x'/x)\) has a free and transitive action by \(T_{\text{id}_{x_0}} \mathit{Aut}(x') \otimes_k I = \text{Inf}_{x_0}(\mathcal{F}) \otimes_k I\).

Proof

This is just the analogue of Lemma 06JI in the setting of automorphism sheaves. To be precise, we apply Remark 06GU to the functor \(\mathit{Aut}(x') : \mathcal{C}_{A'} \to \textit{Sets}\) and the element \(\text{id}_{x_0} \in \mathit{Aut}(x)(k)\) to get a predeformation functor \(F = \mathit{Aut}(x')_{\text{id}_{x_0}}\). By Lemmas 06JU and 06JC \(F\) is a deformation functor. Hence Lemma 06JI gives a free and transitive action of \(TF \otimes_k I\) on \(\text{Lift}(\text{id}_x, A')\), because as \(\text{Lift}(\text{id}_x, A')\) is a group it is always nonempty. Note that we have equalities of vector spaces \[TF = T_{\text{id}_{x_0}} \mathit{Aut}(x') \otimes_k I = \text{Inf}_{x_0}(\mathcal{F}) \otimes_k I\] by Lemma 06JV. The equality \(\text{Inf}(x'/x) = \text{Lift}(\text{id}_x, A')\) of Remark 06JW finishes the proof.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\) satisfying (RS). Let \(x' \to x\) be a morphism in \(\mathcal{F}\) lying over a surjective ring map. Let \(x_0\) be a pushforward of \(x\) to \(\mathcal{F}(k)\). If \(\text{Inf}_{x_0}(\mathcal{F}) = 0\) then \(\text{Inf}(x'/x) = 0\).

Proof

Follows from Lemmas 06GE and 06JY.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\) satisfying (RS). Let \(x_0 \in \Ob(\mathcal{F}(k))\). Then \(\text{Inf}_{x_0}(\mathcal{F}) = 0\) if and only if the natural morphism \(\mathcal{F}_{x_0} \to \overline{\mathcal{F}_{x_0}}\) of categories cofibered in groupoids is an equivalence.

Proof

The morphism \(\mathcal{F}_{x_0} \to \overline{\mathcal{F}_{x_0}}\) is an equivalence if and only if \(\mathcal{F}_{x_0}\) is fibered in setoids, cf. Categories, Section 04S9 (a setoid is by definition a groupoid in which the only automorphism of any object is the identity). We prove that \(\text{Inf}_{x_0}(\mathcal{F}) = 0\) if and only if this condition holds for \(\mathcal{F}_{x_0}\). Obviously if \(\mathcal{F}_{x_0}\) is fibered in setoids then \(\text{Inf}_{x_0}(\mathcal{F}) = 0\). Conversely assume \(\text{Inf}_{x_0}(\mathcal{F}) = 0\). Let \(A\) be an object of \(\mathcal{C}_\Lambda\). Then by Lemma 06JZ, \(\text{Inf}(x/x_0) = 0\) for any object \(x \to x_0\) of \(\mathcal{F}_{x_0}(A)\). Since by definition \(\text{Inf}(x/x_0)\) equals the group of automorphisms of \(x \to x_0\) in \(\mathcal{F}_{x_0}(A)\), this proves \(\mathcal{F}_{x_0}(A)\) is a setoid.

Applications

We collect some results on deformation categories we will use later.

Lemma

Let \(f : \mathcal{H} \to \mathcal{F}\) and \(g : \mathcal{G} \to \mathcal{F}\) be \(1\)-morphisms of deformation categories. Then

\(\mathcal{W} = \mathcal{H} \times_\mathcal{F} \mathcal{G}\) is a deformation category, and

we have a \(6\)-term exact sequence of vector spaces \[0 \to \text{Inf}(\mathcal{W}) \to \text{Inf}(\mathcal{H}) \oplus \text{Inf}(\mathcal{G}) \to \text{Inf}(\mathcal{F}) \to T\mathcal{W} \to T\mathcal{H} \oplus T\mathcal{G} \to T\mathcal{F}\]

Proof

Part (1) follows from Lemma 06L4 and the fact that \(\mathcal{W}(k)\) is the fibre product of two setoids with a unique isomorphism class over a setoid with a unique isomorphism class.

Part (2). Let \(w_0 \in \Ob(\mathcal{W}(k))\) and let \(x_0, y_0, z_0\) be the image of \(w_0\) in \(\mathcal{F}, \mathcal{H}, \mathcal{G}\). Then \(\text{Inf}(\mathcal{W}) = \text{Inf}_{w_0}(\mathcal{W})\) and similarly for \(\mathcal{H}\), \(\mathcal{G}\), and \(\mathcal{F}\), see Remark 06JS. We apply Lemmas 06IJ and 07W6 to get all the linear maps except for the “boundary map” \(\delta : \text{Inf}_{x_0}(\mathcal{F}) \to T\mathcal{W}\). We will insert suitable signs later.

Construction of \(\delta\). Choose a pushforward \(w_0 \to w'_0\) along \(k \to k[\epsilon]\). Denote \(x'_0, y'_0, z'_0\) the images of \(w'_0\) in \(\mathcal{F}, \mathcal{H}, \mathcal{G}\). In particular we obtain isomorphisms \(b' : f(y'_0) \to x'_0\) and \(c' : x'_0 \to g(z'_0)\). Denote \(b : f(y_0) \to x_0\) and \(c : x_0 \to g(z_0)\) the pushforwards along \(k[\epsilon] \to k\). Observe that this means \(w'_0 = (k[\epsilon], y'_0, z'_0, c' \circ b')\) and \(w_0 = (k, y_0, z_0, c \circ b)\) in terms of the explicit form of the fibre product of categories, see Remarks 06GK (06SI). Given \(\alpha : x'_0 \to x'_0\) we set \(\delta(\alpha) = (k[\epsilon], y'_0, z'_0, c' \circ \alpha \circ b')\) which is indeed an object of \(\mathcal{W}\) over \(k[\epsilon]\) and comes with a morphism \((k[\epsilon], y'_0, z'_0, c' \circ \alpha \circ b') \to w_0\) over \(k[\epsilon] \to k\) as \(\alpha\) pushes forward to the identity over \(k\). More generally, for any \(k\)-vector space \(V\) we can define a map \[\text{Lift}(\text{id}_{x_0}, k[V]) \longrightarrow \text{Lift}(w_0, k[V])\] using exactly the same formulae. This construction is functorial in the vector space \(V\) (details omitted). Hence \(\delta\) is \(k\)-linear by an application of Lemma 06I7.

Having constructed these maps it is straightforward to show the sequence is exact. Injectivity of the first map comes from the fact that \(f \times g : \mathcal{W} \to \mathcal{H} \times \mathcal{G}\) is faithful. If \((\beta, \gamma) \in \text{Inf}_{y_0}(\mathcal{H}) \oplus \text{Inf}_{z_0}(\mathcal{G})\) map to the same element of \(\text{Inf}_{x_0}(\mathcal{F})\) then \((\beta, \gamma)\) defines an automorphism of \(w'_0 = (k[\epsilon], y'_0, z'_0, c' \circ b')\) whence exactness at the second spot. If \(\alpha\) as above gives the trivial deformation \((k[\epsilon], y'_0, z'_0, c' \circ \alpha \circ b')\) of \(w_0\), then the isomorphism \(w'_0 = (k[\epsilon], y'_0, z'_0, c' \circ b') \to (k[\epsilon], y'_0, z'_0, c' \circ \alpha \circ b')\) produces a pair \((\beta, \gamma)\) which is a preimage of \(\alpha\). If \(w = (k[\epsilon], y, z, \phi)\) is a deformation of \(w_0\) such that \(y'_0 \cong y\) and \(z \cong z'_0\) then the map \[f(y'_0) \to f(y) \xrightarrow{\phi} g(z) \to g(z'_0)\] is an \(\alpha\) which maps to \(w\) under \(\delta\). Finally, if \(y\) and \(z\) are deformations of \(y_0\) and \(z_0\) and there exists an isomorphism \(\phi : f(y) \to g(z)\) of deformations of \(f(y_0) = x_0 = g(z_0)\) then we get a preimage \(w = (k[\epsilon], y, z, \phi)\) of \((x, y)\) in \(T\mathcal{W}\). This finishes the proof.

Lemma

Let \(\mathcal{H}_1 \to \mathcal{G}\), \(\mathcal{H}_2 \to \mathcal{G}\), and \(\mathcal{G} \to \mathcal{F}\) be maps of categories cofibred in groupoids over \(\mathcal{C}_\Lambda\). Assume

\(\mathcal{F}\) and \(\mathcal{G}\) are deformation categories,

\(T\mathcal{G} \to T\mathcal{F}\) is injective, and

\(\text{Inf}(\mathcal{G}) \to \text{Inf}(\mathcal{F})\) is surjective.

Then \(\mathcal{H}_1 \times_\mathcal{G} \mathcal{H}_2 \to \mathcal{H}_1 \times_\mathcal{F} \mathcal{H}_2\) is smooth.

Proof

Denote \(p_i : \mathcal{H}_i \to \mathcal{G}\) and \(q : \mathcal{G} \to \mathcal{F}\) be the given maps. Let \(A' \to A\) be a small extension in \(\mathcal{C}_\Lambda\). An object of \(\mathcal{H}_1 \times_\mathcal{F} \mathcal{H}_2\) over \(A'\) is a triple \((x'_1, x'_2, a')\) where \(x'_i\) is an object of \(\mathcal{H}_i\) over \(A'\) and \(a' : q(p_1(x'_1)) \to q(p_2(x'_2))\) is a morphism of the fibre category of \(\mathcal{F}\) over \(A'\). By pushforward along \(A' \to A\) we get \((x_1, x_2, a)\). Lifting this to an object of \(\mathcal{H}_1 \times_\mathcal{G} \mathcal{H}_2\) over \(A\) means finding a morphism \(b : p_1(x_1) \to p_2(x_2)\) over \(A\) with \(q(b) = a\). Thus we have to show that we can lift \(b\) to a morphism \(b' : p_1(x'_1) \to p_2(x'_2)\) whose image under \(q\) is \(a'\).

Observe that we can think of \[p_1(x'_1) \to p_1(x_1) \xrightarrow{b} p_2(x_2) \quad\text{and}\quad p_2(x'_2) \to p_2(x_2)\] as two objects of \(\textit{Lift}(p_2(x_2), A' \to A)\). The functor \(q\) sends these objects to the two objects \[q(p_1(x'_1)) \to q(p_1(x_1)) \xrightarrow{b} q(p_2(x_2)) \quad\text{and}\quad q(p_2(x'_2)) \to q(p_2(x_2))\] of \(\textit{Lift}(q(p_2(x_2)), A' \to A)\) which are isomorphic using the map \(a' : q(p_1(x'_1)) \to q(p_2(x'_2))\). On the other hand, the functor \[q : \textit{Lift}(p_2(x_2), A' \to A) \to \textit{Lift}(q(p_2(x_2)), A' \to A)\] defines an injection on isomorphism classes by Lemma 06JI and our assumption on tangent spaces. Thus we see that there is a morphism \(b' : p_1(x_1') \to p_2(x'_2)\) whose pushforward to \(A\) is \(b\). However, we may need to adjust our choice of \(b'\) to achieve \(q(b') = a'\). For this it suffices to see that \(q : \text{Inf}(p_2(x'_2)/p_2(x_2)) \to \text{Inf}(q(p_2(x'_2))/q(p_2(x_2)))\) is surjective. This follows from our assumption on infinitesimal automorphisms and Lemma 06JY.

Lemma

Let \(f : \mathcal{F} \to \mathcal{G}\) be a map of deformation categories. Let \(x_0 \in \Ob(\mathcal{F}(k))\) with image \(y_0 \in \Ob(\mathcal{G}(k))\). If

the map \(T\mathcal{F} \to T\mathcal{G}\) is surjective, and

for every small extension \(A' \to A\) in \(\mathcal{C}_\Lambda\) and \(x \in \mathcal{F}(A)\) with image \(y \in \mathcal{G}(A)\) if there is a lift of \(y\) to \(A'\), then there is a lift of \(x\) to \(A'\),

then \(\mathcal{F} \to \mathcal{G}\) is smooth (and vice versa).

Proof

Let \(A' \to A\) be a small extension. Let \(x \in \mathcal{F}(A)\). Let \(y' \to f(x)\) be a morphism in \(\mathcal{G}\) over \(A' \to A\). Consider the functor \(\text{Lift}(A', x) \to \text{Lift}(A', f(x))\) induced by \(f\). We have to show that there exists an object \(x' \to x\) of \(\text{Lift}(A', x)\) mapping to \(y' \to f(x)\), see Lemma 06HH. By condition (2) we know that \(\text{Lift}(A', x)\) is not the empty category. By condition (2) and Lemma 06JI we conlude that the map on isomorphism classes is surjective as desired.

Lemma

Let \(\mathcal{F} \to \mathcal{G} \to \mathcal{H}\) be maps of categories cofibred in groupoids over \(\mathcal{C}_\Lambda\). If

\(\mathcal{F}\), \(\mathcal{G}\) are deformation categories

the map \(T\mathcal{F} \to T\mathcal{G}\) is surjective, and

\(\mathcal{F} \to \mathcal{H}\) is smooth.

Then \(\mathcal{F} \to \mathcal{G}\) is smooth.

Proof

Let \(A' \to A\) be a small extension in \(\mathcal{C}_\Lambda\) and let \(x \in \mathcal{F}(A)\) with image \(y \in \mathcal{G}(A)\). Assume there is a lift \(y' \in \mathcal{G}(A')\). According to Lemma 0DYP all we have to do is check that \(x\) has a lift too. Take the image \(z' \in \mathcal{H}(A')\) of \(y'\). Since \(\mathcal{F} \to \mathcal{H}\) is smooth, there is an \(x' \in \mathcal{F}(A')\) mapping to both \(x \in \mathcal{F}(A)\) and \(z' \in \mathcal{H}(A')\), see Definition 06HG. This finishes the proof.

Groupoids in functors on an arbitrary category

We begin with generalities on groupoids in functors on an arbitrary category. In the next section we will pass to the category \(\mathcal{C}_\Lambda\). For clarity we shall sometimes refer to an ordinary groupoid, i.e., a category whose morphisms are all isomorphisms, as a groupoid category.

Definition

Let \(\mathcal{C}\) be a category. The category of groupoids in functors on \(\mathcal{C}\) is the category with the following objects and morphisms.

Objects: A groupoid in functors on \(\mathcal{C}\) is a quintuple \((U, R, s, t, c)\) where \(U, R : \mathcal{C} \to \textit{Sets}\) are functors and \(s, t : R \to U\) and \(c : R \times_{s, U, t} R \to R\) are morphisms with the following property: For any object \(T\) of \(\mathcal{C}\), the quintuple \[(U(T), R(T), s, t, c)\] is a groupoid category.

Morphisms: A morphism \((U, R, s, t, c) \to (U', R', s', t', c')\) of groupoids in functors on \(\mathcal{C}\) consists of morphisms \(U \to U'\) and \(R \to R'\) with the following property: For any object \(T\) of \(\mathcal{C}\), the induced maps \(U(T) \to U'(T)\) and \(R(T) \to R'(T)\) define a functor between groupoid categories \[(U(T), R(T), s, t, c) \to (U'(T), R'(T), s', t', c').\]

Remark

A groupoid in functors on \(\mathcal{C}\) amounts to the data of a functor \(\mathcal{C} \to \textit{Groupoids}\), and a morphism of groupoids in functors on \(\mathcal{C}\) amounts to a morphism of the corresponding functors \(\mathcal{C} \to \textit{Groupoids}\) (where \(\textit{Groupoids}\) is regarded as a 1-category). However, for our purposes it is more convenient to use the terminology of groupoids in functors. In fact, thinking of a groupoid in functors as the corresponding functor \(\mathcal{C} \to \textit{Groupoids}\), or equivalently as the category cofibered in groupoids associated to that functor, can lead to confusion (Remark 06KJ).

Remark

Let \((U, R, s, t, c)\) be a groupoid in functors on a category \(\mathcal{C}\). There are unique morphisms \(e : U \to R\) and \(i : R \to R\) such that for every object \(T\) of \(\mathcal{C}\), \(e: U(T) \to R(T)\) sends \(x \in U(T)\) to the identity morphism on \(x\) and \(i: R(T) \to R(T)\) sends \(a \in U(T)\) to the inverse of \(a\) in the groupoid category \((U(T), R(T), s, t, c)\). We will sometimes refer to \(s\), \(t\), \(c\), \(e\), and \(i\) as “source”, “target”, “composition”, “identity”, and “inverse”.

Definition

Let \(\mathcal{C}\) be a category. A groupoid in functors on \(\mathcal{C}\) is representable if it is isomorphic to one of the form \((\underline{U}, \underline{R}, s, t, c)\) where \(U\) and \(R\) are objects of \(\mathcal{C}\) and the pushout \(R \amalg_{s, U, t} R\) exists.

Remark

Hence a representable groupoid in functors on \(\mathcal{C}\) is given by objects \(U\) and \(R\) of \(\mathcal{C}\) and morphisms \(s, t : U \to R\) and \(c : R \to R \amalg_{s, U, t} R\) such that \((\underline{U}, \underline{R}, s, t, c)\) satisfies the condition of Definition 06K3. The reason for requiring the existence of the pushout \(R \amalg_{s, U, t} R\) is so that the composition morphism \(c\) is defined at the level of morphisms in \(\mathcal{C}\). This requirement will always be satisfied below when we consider representable groupoids in functors on \(\widehat{\mathcal{C}}_\Lambda\), since by Lemma 06SB the category \(\widehat{\mathcal{C}}_\Lambda\) admits pushouts.

Remark

We will say “let \((\underline{U}, \underline{R}, s, t, c)\) be a groupoid in functors on \(\mathcal{C}\)” to mean that we have a representable groupoid in functors. Thus this means that \(U\) and \(R\) are objects of \(\mathcal{C}\), there are morphisms \(s, t : U \to R\), the pushout \(R \amalg_{s, U, t} R\) exists, there is a morphism \(c : R \to R \amalg_{s, U, t} R\), and \((\underline{U}, \underline{R}, s, t, c)\) is a groupoid in functors on \(\mathcal{C}\).

We introduce notation for restriction of groupoids in functors. This will be relevant below in situations where we restrict from \(\widehat{\mathcal C}_\Lambda\) to \(\mathcal{C}_\Lambda\).

Definition

Let \((U, R, s, t, c)\) be a groupoid in functors on a category \(\mathcal{C}\). Let \(\mathcal{C}'\) be a subcategory of \(\mathcal{C}\). The restriction \((U, R, s, t, c)|_{\mathcal{C}'}\) of \((U, R, s, t, c)\) to \(\mathcal{C}'\) is the groupoid in functors on \(\mathcal{C}'\) given by \((U|_{\mathcal{C}'}, R|_{\mathcal C'}, s|_{\mathcal{C}'}, t|_{\mathcal{C}'}, c|_{\mathcal{C}'})\).

Remark

In the situation of Definition 06K9, we often denote \(s|_{\mathcal{C}'}, t|_{\mathcal{C}'}, c|_{\mathcal{C}'}\) simply by \(s, t, c\).

Definition

Let \((U, R, s, t, c)\) be a groupoid in functors on a category \(\mathcal{C}\).

The assignment \(T \mapsto (U(T), R(T), s, t, c)\) determines a functor \(\mathcal{C} \to \textit{Groupoids}\). The quotient category cofibered in groupoids \([U/R] \to \mathcal{C}\) is the category cofibered in groupoids over \(\mathcal{C}\) associated to this functor (as in Remarks 06GK (06GN)).

The quotient morphism \(U \to [U/R]\) is the morphism of categories cofibered in groupoids over \(\mathcal{C}\) defined by the rules

\(x \in U(T)\) maps to the object \((T, x) \in \Ob([U/R](T))\), and

\(x \in U(T)\) and \(f : T \to T'\) give rise to the morphism \((f, \text{id}_{U(f)(x)}): (T, x) \to (T, U(f)(x))\) lying over \(f : T \to T'\).

Groupoids in functors on the base category

In this section we discuss groupoids in functors on \(\mathcal{C}_\Lambda\). Our eventual goal is to show that prorepresentable groupoids in functors on \(\mathcal{C}_\Lambda\) serve as “presentations” for well-behaved deformation categories in the same way that smooth groupoids in algebraic spaces serve as presentations for algebraic stacks, cf. Algebraic Stacks, Section 04T3.

Definition

A groupoid in functors on \(\mathcal{C}_\Lambda\) is prorepresentable if it is isomorphic to \((\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda}\) for some representable groupoid in functors \((\underline{R_0}, \underline{R_1}, s, t, c)\) on the category \(\widehat{\mathcal{C}}_\Lambda\).

Let \((U, R, s, t, c)\) be a groupoid in functors on \(\mathcal{C}_\Lambda\). Taking completions, we get a quintuple \((\widehat{U}, \widehat{R}, \widehat{s}, \widehat{t}, \widehat{c})\). By Remark 06HB completion as a functor on \(\text{CofSet}(\mathcal{C}_\Lambda)\) is a right adjoint, so it commutes with limits. In particular, there is a canonical isomorphism \[\widehat{R \times_{s, U, t} R} \longrightarrow \widehat{R} \times_{\widehat{s}, \widehat{U}, \widehat{t}} \widehat{R},\] so \(\widehat{c}\) can be regarded as a functor \(\widehat{R} \times_{\widehat{s}, \widehat{U}, \widehat{t}} \widehat{R} \to \widehat{R}\). Then \((\widehat{U}, \widehat{R}, \widehat{s}, \widehat{t}, \widehat{c})\) is a groupoid in functors on \(\widehat{\mathcal{C}}_\Lambda\), with identity and inverse morphisms being the completions of those of \((U, R, s, t, c)\).

Definition

Let \((U, R, s, t, c)\) be a groupoid in functors on \(\mathcal{C}_\Lambda\). The completion \((U, R, s, t, c)^{\wedge}\) of \((U, R, s, t, c)\) is the groupoid in functors \((\widehat{U}, \widehat{R}, \widehat{s}, \widehat{t}, \widehat{c})\) on \(\widehat{\mathcal{C}}_\Lambda\) described above.

Remark

Let \((U, R, s, t, c)\) be a groupoid in functors on \(\mathcal{C}_\Lambda\). Then there is a canonical isomorphism \((U, R, s, t, c)^{\wedge}|_{\mathcal{C}_\Lambda} \cong (U, R, s, t, c)\), see Remark 06SJ. On the other hand, let \((U, R, s, t, c)\) be a groupoid in functors on \(\widehat{\mathcal{C}}_\Lambda\) such that \(U, R : \widehat{\mathcal{C}}_\Lambda \to \textit{Sets}\) both commute with limits, e.g. if \(U, R\) are representable. Then there is a canonical isomorphism \(((U, R, s, t, c)|_{\mathcal{C}_\Lambda})^{\wedge} \cong (U, R, s, t, c)\). This follows from Remark 06HE.

Lemma

Let \((U, R, s, t, c)\) be a groupoid in functors on \(\mathcal{C}_\Lambda\).

\((U, R, s, t, c)\) is prorepresentable if and only if its completion is representable as a groupoid in functors on \(\widehat{\mathcal{C}}_\Lambda\).

\((U, R, s, t, c)\) is prorepresentable if and only if \(U\) and \(R\) are prorepresentable.

Proof

Part (1) follows from Remark 06KF. For (2), the “only if” direction is clear from the definition of a prorepresentable groupoid in functors. Conversely, assume \(U\) and \(R\) are prorepresentable, say \(U \cong \underline{R_0}|_{\mathcal{C}_\Lambda}\) and \(R \cong \underline{R_1}|_{\mathcal{C}_\Lambda}\) for objects \(R_0\) and \(R_1\) of \(\widehat{\mathcal{C}}_\Lambda\). Since \(\underline{R_0} \cong \widehat{\underline{R_0}|_{\mathcal{C}_\Lambda}}\) and \(\underline{R_1} \cong \widehat{\underline{R_1}|_{\mathcal{C}_\Lambda}}\) by Remark 06HE we see that the completion \((U, R, s, t, c)^\wedge\) is a groupoid in functors of the form \((\underline{R_0}, \underline{R_1}, \widehat{s}, \widehat{t}, \widehat{c})\). By Lemma 06SB the pushout \(\underline{R_1} \times_{\widehat{s}, \underline{R_1}, \widehat{t}} \underline{R_1}\) exists. Hence \((\underline{R_0}, \underline{R_1}, \widehat{s}, \widehat{t}, \widehat{c})\) is a representable groupoid in functors on \(\widehat{\mathcal{C}}_\Lambda\). Finally, the restriction \((\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda}\) gives back \((U, R, s, t, c)\) by Remark 06KF hence \((U, R, s, t, c)\) is prorepresentable by definition.

Smooth groupoids in functors on the base category

The notion of smoothness for groupoids in functors on \(\mathcal{C}_\Lambda\) is defined as follows.

Definition

Let \((U, R, s, t, c)\) be a groupoid in functors on \(\mathcal{C}_\Lambda\). We say \((U, R, s, t, c)\) is smooth if \(s, t: R \to U\) are smooth.

Remark

We note that this terminology is potentially confusing: if \((U, R, s, t, c)\) is a smooth groupoid in functors, then the quotient \([U/R]\) need not be a smooth category cofibred in groupoids as defined in Definition 06HP. However smoothness of \((U, R, s, t, c)\) does imply (in fact is equivalent to) smoothness of the quotient morphism \(U \to [U/R]\) as we shall see in Lemma 06KL.

Remark

Let \((\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda}\) be a prorepresentable groupoid in functors on \(\mathcal{C}_\Lambda\). Then \((\underline{R_0}, \underline{R_1}, s, t, c)|_{\mathcal{C}_\Lambda}\) is smooth if and only if \(R_1\) is a power series over \(R_0\) via both \(s\) and \(t\). This follows from Lemma 06HL.

Lemma

Let \((U, R, s, t, c)\) be a groupoid in functors on \(\mathcal{C}_\Lambda\). The following are equivalent:

The groupoid in functors \((U, R, s, t, c)\) is smooth.

The morphism \(s : R \to U\) is smooth.

The morphism \(t : R \to U\) is smooth.

The quotient morphism \(U \to [U/R]\) is smooth.

Proof

Statement (2) is equivalent to (3) since the inverse \(i: R \to R\) of \((U, R, s, t, c)\) is an isomorphism and \(t = s \circ i\). By definition (1) is equivalent to (2) and (3) together, hence it is equivalent to either of them individually.

Finally we prove (2) is equivalent to (4). Unwinding the definitions:

Smoothness of \(s: R \to U\) amounts to the following condition: If \(f: B \to A\) is a surjective ring map in \(\mathcal{C}_\Lambda\), \(a \in R(A)\), and \(y \in U(B)\) such that \(s(a) = U(f)(y)\), then there exists \(a' \in R(B)\) such that \(R(f)(a') = a\) and \(s(a') = y\).

Smoothness of \(U \to [U/R]\) amounts to the following condition: If \(f: B \to A\) is a surjective ring map in \(\mathcal{C}_\Lambda\) and \((f, a) : (B, y) \to (A, x)\) is a morphism of \([U/R]\), then there exists \(x' \in U(B)\) and \(b \in R(B)\) with \(s(b) = x'\), \(t(b) = y\) such that \(c(a, R(f)(b)) = e(x)\). Here \(e : U \to R\) denotes the identity and the notation \((f, a)\) is as in Remarks 06GK (06GN); in particular \(a \in R(A)\) with \(s(a) = U(f)(y)\) and \(t(a) = x\).

If (4) holds and \(f, a, y\) as in (2) are given, let \(x = t(a)\) so that we have a morphism \((f, a): (B, y) \to (A, x)\). Then (4) produces \(x'\) and \(b\), and \(a' = i(b)\) satisfies the requirements of (2). Conversely, assume (2) holds and let \((f, a): (B, y) \to (A, x)\) as in (4) be given. Then (2) produces \(a' \in R(B)\), and \(x' = t(a')\) and \(b = i(a')\) satisfy the requirements of (4).

Deformation categories as quotients of groupoids in functors

We discuss conditions on a groupoid in functors on \(\mathcal{C}_\Lambda\) which guarantee that the quotient is a deformation category, and we calculate the tangent and infinitesimal automorphism spaces of such a quotient.

Lemma

Let \((U, R, s, t, c)\) be a smooth groupoid in functors on \(\mathcal{C}_\Lambda\). Assume \(U\) and \(R\) satisfy (RS). Then \([U/R]\) satisfies (RS).

Proof

Let \[\xymatrix{ & (A_2, x_2) \ar[d]^{(f_2, a_2)} \\ (A_1, x_1) \ar[r]^{(f_1, a_1)} & (A, x) }\] be a diagram in \([U/R]\) such that \(f_2: A_2 \to A\) is surjective. The notation is as in Remarks 06GK (06GN). Hence \(f_1: A_1 \to A, f_2: A_2 \to A\) are maps in \(\mathcal{C}_\Lambda\), \(x \in U(A)\), \(x_1 \in U(A_1)\), \(x_2 \in U(A_2)\), and \(a_1, a_2 \in R(A)\) with \(s(a_1) = U(f_1)(x_1)\), \(t(a_1) = x\) and \(s(a_2) = U(f_2)(x_2)\), \(t(a_2) = x\). We construct a fiber product lying over \(A_1 \times_A A_2\) for this diagram in \([U/R]\) as follows.

Let \(a = c(i(a_1), a_2)\), where \(i: R \to R\) is the inverse morphism. Then \(a \in R(A)\), \(x_2 \in U(A_2)\) and \(s(a) = U(f_2)(x_2)\). Hence an element \((a, x_2) \in R(A) \times_{s, U(A), U(f_2)} U(A_2)\). By smoothness of \(s : R \to U\) there is an element \(\widetilde{a} \in R(A_2)\) with \(R(f_2)(\widetilde{a}) = a\) and \(s(\widetilde{a}) = x_2\). In particular \(U(f_2)(t(\widetilde{a})) = t(a) = U(f_1)(x_1)\). Thus \(x_1\) and \(t(\widetilde{a})\) define an element \[(x_1, t(\widetilde{a})) \in U(A_1) \times_{U(A)} U(A_2).\] By the assumption that \(U\) satisfies (RS), we have an identification \(U(A_1) \times_{U(A)} U(A_2) = U(A_1 \times_A A_2)\). Let us denote \(x_1 \times t(\widetilde{a}) \in U(A_1 \times_A A_2)\) the element corresponding to \((x_1, t(\widetilde{a})) \in U(A_1) \times_{U(A)} U(A_2)\). Let \(p_1, p_2\) be the projections of \(A_1 \times_A A_2\). We claim \[\xymatrix{ (A_1 \times_A A_2, x_1 \times t(\widetilde{a})) \ar[d]_{(p_1, e(x_1))} \ar[rr]_-{(p_2, i(\widetilde{a}))} & & (A_2, x_2) \ar[d]^{(f_2, a_2)} \\ (A_1, x_1) \ar[rr]^{(f_1, a_1)} & & (A, x) }\] is a fiber square in \([U/R]\). (Note \(e: U \to R\) denotes the identity.)

The diagram is commutative because \(c(a_2, R(f_2)(i(\widetilde{a}))) = c(a_2, i(a)) = a_1\). To check it is a fiber square, let \[\xymatrix{ (B, z) \ar[d]_{(g_1, b_1)} \ar[rr]_{(g_2, b_2)} & & (A_2, x_2) \ar[d]^{(f_2, a_2)} \\ (A_1, x_1) \ar[rr]^{(f_1, a_1)} & & (A, x) }\] be a commutative diagram in \([U/R]\). We will show there is a unique morphism \((g, b) : (B, z) \to (A_1 \times_A A_2, x_1 \times t(\widetilde{a}))\) compatible with the morphisms to \((A_1, x_1)\) and \((A_2, x_2)\). We must take \(g = (g_1, g_2) : B \to A_1 \times_A A_2\). Since by assumption \(R\) satisfies (RS), we have an identification \(R(A_1 \times_A A_2) = R(A_1) \times_{R(A)} R(A_2)\). Hence we can write \(b = (b'_1, b'_2)\) for some \(b'_1 \in R(A_1)\), \(b'_2 \in R(A_2)\) which agree in \(R(A)\). Then \(((g_1, g_2), (b'_1, b'_2)) : (B, z) \to (A_1 \times_A A_2, x_1 \times t(\widetilde{a}))\) will commute with the projections if and only if \(b'_1 = b_1\) and \(b'_2 = c(\widetilde{a}, b_2)\) proving unicity and existence.

Lemma

Let \((U, R, s, t, c)\) be a smooth groupoid in functors on \(\mathcal{C}_\Lambda\). Assume \(U\) and \(R\) are deformation functors. Then:

The quotient \([U/R]\) is a deformation category.

The tangent space of \([U/R]\) is \[T[U/R] = \Coker(ds-dt: TR \to TU).\]

The space of infinitesimal automorphisms of \([U/R]\) is \[\text{Inf}([U/R]) = \Ker(ds \oplus dt : TR \to TU \oplus TU).\]

Proof

Since \(U\) and \(R\) are deformation functors \([U/R]\) is a predeformation category. Since (RS) holds for deformation functors by definition we see that (RS) holds for [U/R] by Lemma 06KT. Hence \([U/R]\) is a deformation category. Statements (2) and (3) follow directly from the definitions.

Presentations of categories cofibered in groupoids

A presentation is defined as follows.

Definition

Let \(\mathcal{F}\) be a category cofibered in groupoids over a category \(\mathcal{C}\). Let \((U, R, s, t, c)\) be a groupoid in functors on \(\mathcal{C}\). A presentation of \(\mathcal{F}\) by \((U, R, s, t, c)\) is an equivalence \(\varphi : [U/R] \to \mathcal{F}\) of categories cofibered in groupoids over \(\mathcal{C}\).

The following two general lemmas will be used to get presentations.

Lemma

Let \(\mathcal{F}\) be category cofibered in groupoids over a category \(\mathcal{C}\). Let \(U : \mathcal{C} \to \textit{Sets}\) be a functor. Let \(f : U \to \mathcal{F}\) be a morphism of categories cofibered in groupoids over \(\mathcal{C}\). Define \(R, s, t, c\) as follows:

\(R : \mathcal{C} \to \textit{Sets}\) is the functor \(U \times_{f, \mathcal{F}, f} U\).

\(t, s : R \to U\) are the first and second projections, respectively.

\(c : R \times_{s, U, t} R \to R\) is the morphism given by projection onto the first and last factors of \(U \times_{f, \mathcal{F}, f} U \times_{f, \mathcal{F}, f} U\) under the canonical isomorphism \(R \times_{s, U, t} R \to U \times_{f, \mathcal{F}, f} U \times_{f, \mathcal{F}, f} U\).

Then \((U, R, s, t, c)\) is a groupoid in functors on \(\mathcal{C}\).

Proof

Omitted.

Lemma

Let \(\mathcal{F}\) be category cofibered in groupoids over a category \(\mathcal{C}\). Let \(U : \mathcal{C} \to \textit{Sets}\) be a functor. Let \(f : U \to \mathcal{F}\) be a morphism of categories cofibered in groupoids over \(\mathcal{C}\). Let \((U, R, s, t, c)\) be the groupoid in functors on \(\mathcal{C}\) constructed from \(f : U \to \mathcal{F}\) in Lemma 06KY. Then there is a natural morphism \([f] : [U/R] \to \mathcal{F}\) such that:

\([f]: [U/R] \to \mathcal{F}\) is fully faithful.

\([f]: [U/R] \to \mathcal{F}\) is an equivalence if and only if \(f : U \to \mathcal{F}\) is essentially surjective.

Proof

Omitted.

Presentations of deformation categories

According to the next lemma, a smooth morphism from a predeformation functor to a predeformation category \(\mathcal{F}\) gives rise to a presentation of \(\mathcal{F}\) by a smooth groupoid in functors.

Lemma

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). Let \(U : \mathcal{C}_\Lambda \to \textit{Sets}\) be a functor. Let \(f : U \to \mathcal{F}\) be a smooth morphism of categories cofibered in groupoids. Then:

If \((U, R, s, t, c)\) is the groupoid in functors on \(\mathcal{C}_\Lambda\) constructed from \(f : U \to \mathcal{F}\) in Lemma 06KY, then \((U, R, s, t, c)\) is smooth.

If \(f : U(k) \to \mathcal{F}(k)\) is essentially surjective, then the morphism \([f] : [U/R] \to \mathcal{F}\) of Lemma 06KZ is an equivalence.

Proof

From the construction of Lemma 06KY we have a commutative diagram \[\xymatrix{ R = U \times_{f, \mathcal{F}, f} U \ar[r]_-s \ar[d]_t & U \ar[d]^f \\ U \ar[r]^f & \mathcal{F} }\] where \(t, s\) are the first and second projections. So \(t, s\) are smooth by Lemma 06HM. Hence (1) holds.

If the assumption of (2) holds, then by Lemma 06HN the morphism \(f : U \to \mathcal{F}\) is essentially surjective. Hence by Lemma 06KZ the morphism \([f] : [U/R] \to \mathcal{F}\) is an equivalence.

Lemma

Let \(\mathcal{F}\) be a deformation category. Let \(U : \mathcal{C}_\Lambda \to \textit{Sets}\) be a deformation functor. Let \(f: U \to \mathcal{F}\) be a morphism of categories cofibered in groupoids. Then \(U \times_{f, \mathcal{F}, f} U\) is a deformation functor with tangent space fitting into an exact sequence of \(k\)-vector spaces \[0 \to \text{Inf}(\mathcal{F}) \to T(U \times_{f, \mathcal{F}, f} U) \to TU \oplus TU\]

Proof

Follows from Lemma 06L5 and the fact that \(\text{Inf}(U) = (0)\).

Lemma

Let \(\mathcal{F}\) be a deformation category. Let \(U : \mathcal{C}_\Lambda \to \textit{Sets}\) be a prorepresentable functor. Let \(f : U \to \mathcal{F}\) be a morphism of categories cofibered in groupoids. Let \((U, R, s, t, c)\) be the groupoid in functors on \(\mathcal{C}_\Lambda\) constructed from \(f : U \to \mathcal{F}\) in Lemma 06KY. If \(\dim_k \text{Inf}(\mathcal{F}) < \infty\), then \((U, R, s, t, c)\) is prorepresentable.

Proof

Note that \(U\) is a deformation functor by Example 06JB. By Lemma 06L6 we see that \(R = U \times_{f, \mathcal{F}, f} U\) is a deformation functor whose tangent space \(TR = T(U \times_{f, \mathcal{F}, f} U)\) sits in an exact sequence \(0 \to \text{Inf}(\mathcal{F}) \to TR \to TU \oplus TU\). Since we have assumed the first space has finite dimension and since \(TU\) has finite dimension by Example 06IC we see that \(\dim TR < \infty\). The map \(\gamma : \text{Der}_\Lambda(k, k) \to TR\) see (06SV) is injective because its composition with \(TR \to TU\) is injective by Theorem 06JM for the prorepresentable functor \(U\). Thus \(R\) is prorepresentable by Theorem 06JM. It follows from Lemma 06KG that \((U, R, s, t, c)\) is prorepresentable.

Theorem

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). Then \(\mathcal{F}\) admits a presentation by a smooth prorepresentable groupoid in functors on \(\mathcal{C}_\Lambda\) if and only if the following conditions hold:

\(\mathcal{F}\) is a deformation category.

\(\dim_k T\mathcal{F}\) is finite.

\(\dim_k \text{Inf}(\mathcal{F})\) is finite.

Proof

Recall that a prorepresentable functor is a deformation functor, see Example 06JB. Thus if \(\mathcal{F}\) is equivalent to a smooth prorepresentable groupoid in functors, then conditions (1), (2), and (3) follow from Lemma 06KU (1), (2), and (3).

Conversely, assume conditions (1), (2), and (3) hold. Condition (1) implies that (S1) and (S2) are satisfied, see Lemma 06J7. By Lemma 06IW there exists a versal formal object \(\xi\). Setting \(U = \underline{R}|_{\mathcal{C}_\Lambda}\) the associated map \(\underline{\xi} : U \to \mathcal{F}\) is smooth (this is the definition of a versal formal object). Let \((U, R, s, t, c)\) be the groupoid in functors constructed in Lemma 06KY from the map \(\underline{\xi}\). By Lemma 06L1 we see that \((U, R, s, t, c)\) is a smooth groupoid in functors and that \([U/R] \to \mathcal{F}\) is an equivalence. By Lemma 06L7 we see that \((U, R, s, t, c)\) is prorepresentable. Hence \([U/R] \to \mathcal{F}\) is the desired presentation of \(\mathcal{F}\).

Remarks regarding minimality

The main theorem of this chapter is Theorem 06L8 above. It describes completely those categories cofibred in groupoids over \(\mathcal{C}_\Lambda\) which have a presentation by a smooth prorepresentable groupoid in functors. In this section we briefly discuss how the minimality discussed in Sections 06T0 and 06IL can be used to obtain a “minimal” smooth prorepresentable presentation.

Definition

Let \((U, R, s, t, c)\) be a smooth prorepresentable groupoid in functors on \(\mathcal{C}_\Lambda\).

We say \((U, R, s, t, c)\) is normalized if the groupoid \((U(k[\epsilon]), R(k[\epsilon]), s, t, c)\) is totally disconnected, i.e., there are no morphisms between distinct objects.

We say \((U, R, s, t, c)\) is minimal if the \(U \to [U/R]\) is given by a minimal versal formal object of \([U/R]\).

The difference between the two notions is related to the difference between conditions (06IM) and (06T6) and disappears when \(k' \subset k\) is separable. Also a normalized smooth prorepresentable groupoid in functors is minimal as the following lemma shows. Here is a precise statement.

Lemma

Let \((U, R, s, t, c)\) be a smooth prorepresentable groupoid in functors on \(\mathcal{C}_\Lambda\).

\((U, R, s, t, c)\) is normalized if and only if the morphism \(U \to [U/R]\) induces an isomorphism on tangent spaces, and

\((U, R, s, t, c)\) is minimal if and only if the kernel of \(TU \to T[U/R]\) is contained in the image of \(\text{Der}_\Lambda(k, k) \to TU\).

Proof

Part (1) follows immediately from the definitions. To see part (2) set \(\mathcal{F} = [U/R]\). Since \(\mathcal{F}\) has a presentation it is a deformation category, see Theorem 06L8. In particular it satisfies (RS), (S1), and (S2), see Lemma 06J7. Recall that minimal versal formal objects are unique up to isomorphism, see Lemma 06T5. By Theorem 06IX a minimal versal object induces a map \(\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}\) satisfying (06T6). Since \(U \cong \underline{R}|_{\mathcal{C}_\Lambda}\) over \(\mathcal{F}\) we see that \(TU \to T\mathcal{F} = T[U/R]\) satisfies the property as stated in the lemma.

The quotient of a minimal prorepresentable groupoid in functors on \(\mathcal C_\Lambda\) does not admit autoequivalences which are not automorphisms. To prove this, we first note the following lemma.

Lemma

Let \(U: \mathcal{C}_\Lambda \to \textit{Sets}\) be a prorepresentable functor. Let \(\varphi : U \to U\) be a morphism such that \(d\varphi : TU \to TU\) is an isomorphism. Then \(\varphi\) is an isomorphism.

Proof

If \(U \cong \underline{R}|_{\mathcal{C}_\Lambda}\) for some \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\), then completing \(\varphi\) gives a morphism \(\underline{R} \to \underline{R}\). If \(f: R \to R\) is the corresponding morphism in \(\widehat{\mathcal{C}}_\Lambda\), then \(f\) induces an isomorphism \(\text{Der}_\Lambda(R, k) \to \text{Der}_\Lambda(R, k)\), see Example 06ST. In particular \(f\) is a surjection by Lemma 06SD. As a surjective endomorphism of a Noetherian ring is an isomorphism (see Algebra, Lemma 06RN) we conclude \(f\), hence \(\underline{R} \to \underline{R}\), hence \(\varphi : U \to U\) is an isomorphism.

Lemma

Let \((U, R, s, t, c)\) be a minimal smooth prorepresentable groupoid in functors on \(\mathcal{C}_\Lambda\). If \(\varphi : [U/R] \to [U/R]\) is an equivalence of categories cofibered in groupoids, then \(\varphi\) is an isomorphism.

Proof

A morphism \(\varphi : [U/R] \to [U/R]\) is the same thing as a morphism \(\varphi : (U, R, s, t, c) \to (U, R, s, t, c)\) of groupoids in functors over \(\mathcal{C}_\Lambda\) as defined in Definition 06K3. Denote \(\phi : U \to U\) and \(\psi : R \to R\) the corresponding morphisms. Because the diagram \[\xymatrix{ & \text{Der}_\Lambda(k, k) \ar[dr]_\gamma \ar[dl]^\gamma \\ TU \ar[rr]_{d\phi} \ar[d] & & TU \ar[d] \\ T[U/R] \ar[rr]^{d\varphi} & & T[U/R] }\] is commutative, since \(d\varphi\) is bijective, and since we have the characterization of minimality in Lemma 06KN we conclude that \(d\phi\) is injective (hence bijective by dimension reasons). Thus \(\phi : U \to U\) is an isomorphism by Lemma 06KP. We can use a similar argument, using the exact sequence \[0 \to \text{Inf}([U/R]) \to TR \to TU \oplus TU\] of Lemma 06L6 to prove that \(\psi : R \to R\) is an isomorphism. But is also a consequence of the fact that \(R = U \times_{[U/R]} U\) and that \(\varphi\) and \(\phi\) are isomorphisms.

Lemma

Let \((U, R, s, t, c)\) and \((U', R', s', t', c')\) be minimal smooth prorepresentable groupoids in functors on \(\mathcal{C}_\Lambda\). If \(\varphi : [U/R] \to [U'/R']\) is an equivalence of categories cofibered in groupoids, then \(\varphi\) is an isomorphism.

Proof

Let \(\psi : [U'/R'] \to [U/R]\) be a quasi-inverse to \(\varphi\). Then \(\psi \circ \varphi\) and \(\varphi \circ \psi\) are isomorphisms by Lemma 06KQ, hence \(\varphi\) and \(\psi\) are isomorphisms.

The following lemma summarizes some of the things we have seen earlier in this chapter.

Lemma

Let \(\mathcal{F}\) be a deformation category such that \(\dim_k T\mathcal{F} <\infty\) and \(\dim_k \text{Inf}(\mathcal{F}) < \infty\). Then there exists a minimal versal formal object \(\xi\) of \(\mathcal{F}\). Say \(\xi\) lies over \(R \in \Ob(\widehat{\mathcal{C}}_\Lambda)\). Let \(U = \underline{R}|_{\mathcal{C}_\Lambda}\). Let \(f = \underline{\xi} : U \to \mathcal{F}\) be the associated morphism. Let \((U, R, s, t, c)\) be the groupoid in functors on \(\mathcal{C}_\Lambda\) constructed from \(f : U \to \mathcal{F}\) in Lemma 06KY. Then \((U, R, s, t, c)\) is a minimal smooth prorepresentable groupoid in functors on \(\mathcal{C}_\Lambda\) and there is an equivalence \([U/R] \to \mathcal{F}\).

Proof

As \(\mathcal{F}\) is a deformation category it satisfies (S1) and (S2), see Lemma 06J7. By Lemma 06IW there exists a versal formal object. By Lemma 06T5 there exists a minimal versal formal object \(\xi/R\) as in the statement of the lemma. Setting \(U = \underline{R}|_{\mathcal{C}_\Lambda}\) the associated map \(\underline{\xi} : U \to \mathcal{F}\) is smooth (this is the definition of a versal formal object). Let \((U, R, s, t, c)\) be the groupoid in functors constructed in Lemma 06KY from the map \(\underline{\xi}\). By Lemma 06L1 we see that \((U, R, s, t, c)\) is a smooth groupoid in functors and that \([U/R] \to \mathcal{F}\) is an equivalence. By Lemma 06L7 we see that \((U, R, s, t, c)\) is prorepresentable. Finally, \((U, R, s, t, c)\) is minimal because \(U \to [U/R] = \mathcal{F}\) corresponds to the minimal versal formal object \(\xi\).

Presentations by minimal prorepresentable groupoids in functors satisfy the following uniqueness property.

Lemma

Let \(\mathcal{F}\) be category cofibered in groupoids over \(\mathcal{C}_\Lambda\). Assume there exist presentations of \(\mathcal{F}\) by minimal smooth prorepresentable groupoids in functors \((U, R, s, t, c)\) and \((U', R', s', t', c')\). Then \((U, R, s, t, c)\) and \((U', R', s', t', c')\) are isomorphic.

Proof

Follows from Lemma 06KR and the observation that a morphism \([U/R] \to [U'/R']\) is the same thing as a morphism of groupoids in functors (by our explicit construction of \([U/R]\) in Definition 06KB).

In summary we have proved the following theorem.

Theorem

Let \(\mathcal{F}\) be a category cofibered in groupoids over \(\mathcal{C}_\Lambda\). Consider the following conditions

\(\mathcal{F}\) admits a presentation by a normalized smooth prorepresentable groupoid in functors on \(\mathcal{C}_\Lambda\),

\(\mathcal{F}\) admits a presentation by a smooth prorepresentable groupoid in functors on \(\mathcal{C}_\Lambda\),

\(\mathcal{F}\) admits a presentation by a minimal smooth prorepresentable groupoid in functors on \(\mathcal{C}_\Lambda\), and

\(\mathcal{F}\) satisfies the following conditions

\(\mathcal{F}\) is a deformation category.

\(\dim_k T\mathcal{F}\) is finite.

\(\dim_k \text{Inf}(\mathcal{F})\) is finite.

Then (2), (3), (4) are equivalent and are implied by (1). If \(k' \subset k\) is separable, then (1), (2), (3), (4) are all equivalent. Furthermore, the minimal smooth prorepresentable groupoids in functors which provide a presentation of \(\mathcal{F}\) are unique up to isomorphism.

Proof

We see that (1) implies (3) and is equivalent to (3) if \(k' \subset k\) is separable from Lemma 06KN. It is clear that (3) implies (2). We see that (2) implies (4) by Theorem 06L8. We see that (4) implies (3) by Lemma 06L2. This proves all the implications. The final uniqueness statement follows from Lemma 06L3.

Uniqueness of versal rings

Given \(R, S\) in \(\widehat{\mathcal{C}}_\Lambda\) we say maps \(f, g : R \to S\) are formally homotopic if there exists an \(r \geq 0\) and maps \(h : R \to R[[t_1, \ldots, t_r]]\) and \(k : R[[t_1, \ldots, t_r]] \to S\) in \(\widehat{\mathcal{C}}_\Lambda\) such that for all \(a \in R\) we have

\(h(a) \bmod (t_1, \ldots, t_r) = a\),
\(f(a) = k(a)\),
\(g(a) = k(h(a))\).

We will say \((r, h, k)\) is a formal homotopy between \(f\) and \(g\).

Lemma

Being formally homotopic is an equivalence relation on sets of morphisms in \(\widehat{\mathcal{C}}_\Lambda\).

Proof

Suppose we have any \(r \geq 1\) and two maps \(h_1, h_2 : R \to R[[t_1, \ldots, t_r]]\) such that \(h_1(a) \bmod (t_1, \ldots, t_r) = h_2(a) \bmod (t_1, \ldots, t_r) = a\) for all \(a \in R\) and a map \(k : R[[t_1, \ldots, t_r]] \to S\). Then we claim \(k \circ h_1\) is formally homotopic to \(k \circ h_2\). The symmetric inherent in this claim will show that our notion of formally homotopic is symmetric. Namely, the map \[\Psi : R[[t_1, \ldots, t_r]] \longrightarrow R[[t_1, \ldots, t_r]],\quad \sum a_I t^I \longmapsto \sum h_1(a_I)t^I\] is an isomorphism. Set \(h(a) = \Psi^{-1}(h_2(a))\) for \(a \in R\) and \(k' = k \circ \Psi\), then we see that \((r, h, k')\) is a formal homotopy between \(k \circ h_1\) and \(k \circ h_2\), proving the claim

Say we have three maps \(f_1, f_2, f_3 : R \to S\) as above and a formal homotopy \((r_1, h_1, k_1)\) between \(f_1\) and \(f_2\) and a formal homotopy \((r_2, h_2, k_2)\) between \(f_3\) and \(f_2\) (!). After relabeling the coordinates we may assume \(h_2 : R \to R[[t_{r_1 + 1}, \ldots, t_{r_1 + r_2}]]\) and \(k_2 : R[[t_{r_1 + 1}, \ldots, t_{r_1 + r_2}]] \to S\). By choosing a suitable isomorphism \[R[[t_1, \ldots, t_{r_1 + r_2}]] \longrightarrow R[[t_{r_1 + 1}, \ldots, t_{r_1 + r_2}]] \widehat{\otimes}_{h_2, R, h_1} R[[t_1, \ldots, t_{r_1}]]\] we may fit these maps into a commutative diagram \[\xymatrix{ R \ar[r]_{h_1} \ar[d]^{h_2} & R[[t_1, \ldots, t_{r_1}]] \ar[d]^{h_2'} \\ R[[t_{r_1 + 1}, \ldots, t_{r_1 + r_2}]] \ar[r]^{h_1'} & R[[t_1, \ldots, t_{r_1 + r_2}]] }\] with \(h_2'(t_i) = t_i\) for \(1 \leq i \leq r_1\) and \(h_1'(t_i) = t_i\) for \(r_1 + 1 \leq i \leq r_2\). Some details omitted. Since this diagram is a pushout in the category \(\widehat{\mathcal{C}}_\Lambda\) (see proof of Lemma 06SB) and since \(k_1 \circ h_1 = f_2 = k_2 \circ h_2\) we conclude there exists a map \[k : R[[t_1, \ldots, t_{r_1 + r_2}]] \to S\] with \(k_1 = k \circ h_2'\) and \(k_2 = k \circ h_1'\). Denote \(h = h_1' \circ h_2 = h_2' \circ h_1\). Then we have

\(k(h_1'(a)) = k_2(a) = f_3(a)\), and

\(k(h_2'(a)) = k_1(a) = f_1(a)\).

By the claim in the first paragraph of the proof this shows that \(f_1\) and \(f_3\) are formally homotopic.

Lemma

In the category \(\widehat{\mathcal{C}}_\Lambda\), if \(f_1, f_2 : R \to S\) are formally homotopic and \(g : S \to S'\) is a morphism, then \(g \circ f_1\) and \(g \circ f_2\) are formally homotopic.

Proof

Namely, if \((r, h, k)\) is a formal homotopy between \(f_1\) and \(f_2\), then \((r, h, g \circ k)\) is a formal homotopy between \(g \circ f_1\) and \(g \circ f_2\).

Lemma

Let \(\mathcal{F}\) be a deformation category over \(\mathcal{C}_\Lambda\) with \(\dim_k T\mathcal{F} < \infty\) and \(\dim_k \text{Inf}(\mathcal{F}) < \infty\). Let \(\xi\) be a versal formal object lying over \(R\). Let \(\eta\) be a formal object lying over \(S\). Then any two maps \[f, g : R \to S\] such that \(f_*\xi \cong \eta \cong g_*\xi\) are formally homotopic.

Proof

By Theorem 06L8 and its proof, \(\mathcal{F}\) has a presentation by a smooth prorepresentable groupoid \[(\underline{R}, \underline{R_1}, s, t, c, e, i)|_{\mathcal{C}_\Lambda}\] in functors on \(\mathcal{C}_\lambda\) such that \(\mathcal{F}\). Then the maps \(s : R \to R_1\) and \(t : R \to R_1\) are formally smooth ring maps and \(e : R_1 \to R\) is a section. In particular, we can choose an isomorphism \(R_1 = R[[t_1, \ldots, t_r]]\) for some \(r \geq 0\) such that \(s\) is the embedding \(R \subset R[[t_1, \ldots, t_r]]\) and \(t\) corresponds to a map \(h : R \to R[[t_1, \ldots, t_r]]\) with \(h(a) \bmod (t_1, \ldots, t_r) = a\) for all \(a \in R\). The existence of the isomorphism \(\alpha : f_*\xi \to g_*\xi\) means exactly that there is a map \(k : R_1 \to S\) such that \(f = k \circ s\) and \(g = k \circ t\). This exactly means that \((r, h, k)\) is a formal homotopy between \(f\) and \(g\).

Lemma

In the category \(\widehat{\mathcal{C}}_\Lambda\), if \(f_1, f_2 : R \to S\) are formally homotopic and \(\mathfrak p \subset R\) is a minimal prime ideal, then \(f_1(\mathfrak p)S = f_2(\mathfrak p)S\) as ideals.

Proof

Suppose \((r, h, k)\) is a formal homotopy between \(f_1\) and \(f_2\). We claim that \(\mathfrak pR[[t_1, \ldots, t_r]] = h(\mathfrak p)R[[t_1, \ldots, t_r]]\). The claim implies the lemma by further composing with \(k\). To prove the claim, observe that the map \(\mathfrak p \mapsto \mathfrak pR[[t_1, \ldots, t_r]]\) is a bijection between the minimal prime ideals of \(R\) and the minimal prime ideals of \(R[[t_1, \ldots, t_r]]\). Finally, \(h(\mathfrak p)R[[t_1, \ldots, t_r]]\) is a minimal prime as \(h\) is flat, and hence of the form \(\mathfrak q R[[t_1, \ldots, t_r]]\) for some minimal prime \(\mathfrak q \subset R\) by what we just said. But since \(h \bmod (t_1, \ldots, t_r) = \text{id}_R\) by definition of a formal homotopy, we conclude that \(\mathfrak q = \mathfrak p\) as desired.

Change of residue field

In this section we quickly discuss what happens if we replace the residue field \(k\) by a finite extension. Let \(\Lambda\) be a Noetherian ring and let \(\Lambda \to k\) be a finite ring map where \(k\) is a field. Throughout this whole chapter we have used \(\mathcal{C}_\Lambda\) to denote the category of Artinian local \(\Lambda\)-algebras whose residue field is identified with \(k\), see Definition 06GC. However, since in this section we will discuss what happen when we change \(k\) we will instead use the notation \(\mathcal{C}_{\Lambda, k}\) to indicate the dependence on \(k\).

Situation

Let \(\Lambda\) be a Noetherian ring and let \(\Lambda \to k \to l\) be a finite ring maps where \(k\) and \(l\) are fields. Thus \(l/k\) is a finite extensions of fields. A typical object of \(\mathcal{C}_{\Lambda, l}\) will be denoted \(B\) and a typical object of \(\mathcal{C}_{\Lambda, k}\) will be denoted \(A\). We define [07W9]\[\begin{equation} \mathcal{C}_{\Lambda, l} \longrightarrow \mathcal{C}_{\Lambda, k}, \quad B \longmapsto B \times_l k \end{equation}\] Given a category cofibred in groupoids \(p : \mathcal{F} \to \mathcal{C}_{\Lambda, k}\) we obtain an associated category cofibred in groupoids \[p_{l/k} : \mathcal{F}_{l/k} \longrightarrow \mathcal{C}_{\Lambda, l}\] by setting \(\mathcal{F}_{l/k}(B) = \mathcal{F}(B \times_l k)\).

The functor (07W9) makes sense: because \(B \times_l k \subset B\) we have \[\begin{align*} [k : k']\ \text{length}_{B \times_l k}(B \times_l k) & = \text{length}_\Lambda(B \times_l k) \\ & \leq \text{length}_\Lambda(B) \\ & = [l : k']\ \text{length}_B(B) < \infty \end{align*}\] (see Lemma 06GG) hence \(B \times_l k\) is Artinian (see Algebra, Lemma 00JB). Thus \(B \times_l k\) is an Artinian local ring with residue field \(k\). Note that (07W9) commutes with fibre products \[(B_1 \times_B B_2) \times_l k = (B_1 \times_l k) \times_{(B \times_l k)} (B_2 \times_l k)\] and transforms surjective ring maps into surjective ring maps. We use the “expensive” notation \(\mathcal{F}_{l/k}\) to prevent confusion with the construction of Remark 06GU. Here are some elementary observations.

Lemma

With notation and assumptions as in Situation 07W8.

We have \(\overline{\mathcal{F}_{l/k}} = (\overline{\mathcal{F}})_{l/k}\).

If \(\mathcal{F}\) is a predeformation category, then \(\mathcal{F}_{l/k}\) is a predeformation category.

If \(\mathcal{F}\) satisfies (S1), then \(\mathcal{F}_{l/k}\) satisfies (S1).

If \(\mathcal{F}\) satisfies (S2), then \(\mathcal{F}_{l/k}\) satisfies (S2).

If \(\mathcal{F}\) satisfies (RS), then \(\mathcal{F}_{l/k}\) satisfies (RS).

Proof

Part (1) is immediate from the definitions.

Since \(\mathcal{F}_{l/k}(l) = \mathcal{F}(k)\) part (2) follows from the definition, see Definition 06GS.

Part (3) follows as the functor (07W9) commutes with fibre products and transforms surjective maps into surjective maps, see Definition 06HW.

Part (4). To see this consider a diagram \[\xymatrix{ & l[\epsilon] \ar[d] \\ B \ar[r] & l }\] in \(\mathcal{C}_{\Lambda, l}\) as in Definition 06HW. Applying the functor (07W9) we obtain \[\xymatrix{ & k[l\epsilon] \ar[d] \\ B \times_l k \ar[r] & k }\] where \(l\epsilon\) denotes the finite dimensional \(k\)-vector space \(l\epsilon \subset l[\epsilon]\). According to Lemma 06HZ the condition of (S2) for \(\mathcal{F}\) also holds for this diagram. Hence (S2) holds for \(\mathcal{F}_{l/k}\).

Part (5) follows from the characterization of (RS) in Lemma 06J5 part (2) and the fact that (07W9) commutes with fibre products.

The following lemma applies in particular when \(\mathcal{F}\) satisfies (S2) and is a predeformation category, see Lemma 06I0.

Lemma

With notation and assumptions as in Situation 07W8. Assume \(\mathcal{F}\) is a predeformation category and \(\overline{\mathcal{F}}\) satisfies (S2). Then there is a canonical \(l\)-vector space isomorphism \[T\mathcal{F} \otimes_k l \longrightarrow T\mathcal{F}_{l/k}\] of tangent spaces.

Proof

By Lemma 07WA we may replace \(\mathcal{F}\) by \(\overline{\mathcal{F}}\). Moreover we see that \(T\mathcal{F}\), resp. \(T\mathcal{F}_{l/k}\) has a canonical \(k\)-vector space structure, resp. \(l\)-vector space structure, see Lemma 06IH. Then \[T\mathcal{F}_{l/k} = \mathcal{F}_{l/k}(l[\epsilon]) = \mathcal{F}(k[l\epsilon]) = T\mathcal{F} \otimes_k l\] the last equality by Lemma 06IH. More generally, given a finite dimensional \(l\)-vector space \(V\) we have \[\mathcal{F}_{l/k}(l[V]) = \mathcal{F}(k[V_k]) = T\mathcal{F} \otimes_k V_k\] where \(V_k\) denotes \(V\) seen as a \(k\)-vector space. We conclude that the functors \(V \mapsto \mathcal{F}_{l/k}(l[V])\) and \(V \mapsto T\mathcal{F} \otimes_k V_k\) are canonically identified as functors to the category of sets. By Lemma 06I6 we see there is at most one way to turn either functor into an \(l\)-linear functor. Hence the isomorphisms are compatible with the \(l\)-vector space structures and we win.

Lemma

With notation and assumptions as in Situation 07W8. Assume \(\mathcal{F}\) is a deformation category. Then there is a canonical \(l\)-vector space isomorphism \[\text{Inf}(\mathcal{F}) \otimes_k l \longrightarrow \text{Inf}(\mathcal{F}_{l/k})\] of infinitesimal automorphism spaces.

Proof

Let \(x_0 \in \Ob(\mathcal{F}(k))\) and denote \(x_{l, 0}\) the corresponding object of \(\mathcal{F}_{l/k}\) over \(l\). Recall that \(\text{Inf}(\mathcal{F}) = \text{Inf}_{x_0}(\mathcal{F})\) and \(\text{Inf}(\mathcal{F}_{l/k}) = \text{Inf}_{x_{l, 0}}(\mathcal{F}_{l/k})\), see Remark 06JS. Recall that the vector space structure on \(\text{Inf}_{x_0}(\mathcal{F})\) comes from identifying it with the tangent space of the functor \(\mathit{Aut}(x_0)\) which is defined on the category \(\mathcal{C}_{k, k}\) of Artinian local \(k\)-algebras with residue field \(k\). Similarly, \(\text{Inf}_{x_{l, 0}}(\mathcal{F}_{l/k})\) is the tangent space of \(\mathit{Aut}(x_{l, 0})\) which is defined on the category \(\mathcal{C}_{l, l}\) of Artinian local \(l\)-algebras with residue field \(l\). Unwinding the definitions we see that \(\mathit{Aut}(x_{l, 0})\) is the restriction of \(\mathit{Aut}(x_0)_{l/k}\) (which lives on \(\mathcal{C}_{k, l}\)) to \(\mathcal{C}_{l, l}\). Since there is no difference between the tangent space of \(\mathit{Aut}(x_0)_{l/k}\) seen as a functor on \(\mathcal{C}_{k, l}\) or \(\mathcal{C}_{l, l}\), the lemma follows from Lemma 07WB and the fact that \(\mathit{Aut}(x_0)\) satisfies (RS) by Lemma 06JU (whence we have (S2) by Lemma 06J7).

Lemma

With notation and assumptions as in Situation 07W8. If \(\mathcal{F} \to \mathcal{G}\) is a smooth morphism of categories cofibred in groupoids over \(\mathcal{C}_{\Lambda, k}\), then \(\mathcal{F}_{l/k} \to \mathcal{G}_{l/k}\) is a smooth morphism of categories cofibred in groupoids over \(\mathcal{C}_{\Lambda, l}\).

Proof

This follows immediately from the definitions and the fact that (07W9) preserves surjections.

There are many more things you can say about the relationship between \(\mathcal{F}\) and \(\mathcal{F}_{l/k}\) (in particular about the relationship between versal deformations) and we will add these here as needed.

Lemma

With notation and assumptions as in Situation 07W8. Let \(\xi\) be a versal formal object for \(\mathcal{F}\) lying over \(R \in \Ob(\widehat{\mathcal{C}}_{\Lambda, k})\). Then there exist

an \(S \in \Ob(\widehat{\mathcal{C}}_{\Lambda, l})\) and a local \(\Lambda\)-algebra homomorphism \(R \to S\) which is formally smooth in the \(\mathfrak m_S\)-adic topology and induces the given field extension \(l/k\) on residue fields, and

a versal formal object of \(\mathcal{F}_{l/k}\) lying over \(S\).

Proof

Construction of \(S\). Choose a surjection \(R[x_1, \ldots, x_n] \to l\) of \(R\)-algebras. The kernel is a maximal ideal \(\mathfrak m\). Set \(S\) equal to the \(\mathfrak m\)-adic completion of the Noetherian ring \(R[x_1, \ldots, x_n]\). Then \(S\) is in \(\widehat{\mathcal{C}}_{\Lambda, l}\) by Algebra, Lemma 0316. The map \(R \to S\) is formally smooth in the \(\mathfrak m_S\)-adic topology by More on Algebra, Lemmas 07EC and 07ED and the fact that \(R \to R[x_1, \ldots, x_n]\) is formally smooth. (Compare with the proof Lemma 06SM.)

Since \(\xi\) is versal, the transformation \(\underline{\xi} : \underline{R}|_{\mathcal{C}_{\Lambda, k}} \to \mathcal{F}\) is smooth. By Lemma 07WD the induced map \[(\underline{R}|_{\mathcal{C}_{\Lambda, k}})_{l/k} \longrightarrow \mathcal{F}_{l/k}\] is smooth. Thus it suffices to construct a smooth morphism \(\underline{S}|_{\mathcal{C}_{\Lambda, l}} \to (\underline{R}|_{\mathcal{C}_{\Lambda, k}})_{l/k}\). To give such a map means for every object \(B\) of \(\mathcal{C}_{\Lambda, l}\) a map of sets \[\Mor_{\widehat{\mathcal{C}}_{\Lambda, l}}(S, B) \longrightarrow \Mor_{\widehat{\mathcal{C}}_{\Lambda, k}}(R, B \times_l k)\] functorial in \(B\). Given an element \(\varphi : S \to B\) on the left hand side we send it to the composition \(R \to S \to B\) whose image is contained in the sub \(\Lambda\)-algebra \(B \times_l k\). Smoothness of the map means that given a surjection \(B' \to B\) and a commutative diagram \[\xymatrix{ S \ar[r] & B \ar@{=}[r] & B \\ R \ar[u] \ar[r] & B' \times_l k \ar[u] \ar[r] & B' \ar[u] }\] we have to find a ring map \(S \to B'\) fitting into the outer rectangle. The existence of this map is guaranteed as we chose \(R \to S\) to be formally smooth in the \(\mathfrak m_S\)-adic topology, see More on Algebra, Lemma 07NJ.

The Grothendieck–Levelt criterion

The prorepresentability criterion in this section concerns all Artinian algebras over a Noetherian ring, with varying residue fields. This is a different category from the fixed-residue-field category \(\mathcal{C}_\Lambda\) used in the rest of this chapter.

Definition

Let \(\Lambda\) be a Noetherian ring. Denote \(\mathcal{A}_\Lambda\) the category of \(\Lambda\)-algebras which have finite length as \(\Lambda\)-modules. Given a functor \(F : \mathcal{A}_\Lambda \to \textit{Sets}\) and a ring map \(A \to B\) in \(\mathcal{A}_\Lambda\), we say that \(F\) has the Cech equalizer property for \(A \to B\) if \[F(A) \longrightarrow F(B) \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} F(B \otimes_A B)\] is an equalizer diagram. The two arrows are induced by \(b \mapsto b \otimes 1\) and \(b \mapsto 1 \otimes b\).

Lemma

Let \((A, \mathfrak m)\) be an Artinian local ring and let \(A \to B\) be an injective ring map. If \(B/A\) has length \(1\) as an \(A\)-module, then \[A \longrightarrow B \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} B \otimes_A B\] is an equalizer diagram.

Proof

Put \(Q = B/A\). We have \(\mathfrak mQ = 0\), and hence \(\mathfrak mB \subset A\). The ideal \(\mathfrak mB\) is nilpotent, so its inclusion in \(A\) implies \(\mathfrak mB \subset \mathfrak m\). The reverse inclusion is automatic, and consequently \(\mathfrak mB = \mathfrak m\). Tensoring \[A \longrightarrow B \longrightarrow Q \longrightarrow 0\] with \(B\) over \(A\) gives an exact sequence \[B \longrightarrow B \otimes_A B \longrightarrow Q \otimes_A B \longrightarrow 0,\] where the first arrow is \(b \mapsto 1 \otimes b\). Suppose \(b \otimes 1 = 1 \otimes b\). The image of \(b \otimes 1\) in \(Q \otimes_A B\) is therefore zero. Since \(Q\) is a one dimensional \(A/\mathfrak m\)-vector space, the map \[Q \longrightarrow Q \otimes_A B, \qquad q \longmapsto q \otimes 1\] is injective: after choosing a basis of \(Q\) it becomes the injective map \(A/\mathfrak m \to B/\mathfrak mB\). Thus the image of \(b\) in \(Q\) is zero, so \(b \in A\). The reverse inclusion is immediate.

Theorem

Let \(\Lambda\) be a Noetherian ring and let \(F : \mathcal{A}_\Lambda \to \textit{Sets}\) be a functor. The following are equivalent.

The functor \(F\) is strictly prorepresentable in the sense of Categories, Definition categories-definition-strict-prorepresentable-minimal-pair.

The functor \(F\) commutes with finite products and has the Cech equalizer property for every \(A \to B\) for which \[A \longrightarrow B \mathrel{\substack{\longrightarrow \\[-0.6ex] \longrightarrow}} B \otimes_A B\] is an equalizer diagram of rings.

The functor \(F\) commutes with finite products and has the Cech equalizer property whenever \(A\) is local and either

\(B\) is a nonzero finite free \(A\)-module, or

\(A \to B\) is injective and \(B/A\) has length \(1\) as an \(A\)-module.

Proof

The category \(\mathcal{A}_\Lambda\) is essentially small and has finite limits. Moreover, every nonempty collection of subobjects of an object has a minimal member: subobjects are in particular \(\Lambda\)-submodules of a finite length \(\Lambda\)-module. Categories, Proposition categories-corollary-artinian-left-exact-strict-prorepresentable therefore says that (1) is equivalent to \(F\) being left exact.

If \(F\) is left exact, then it preserves the equalizer diagrams in (2), so (1) implies (2). Suppose (2) holds. A nonzero finite free algebra over a local ring is faithfully flat. The ring equalizer in (2) is exact in this case by Descent, Proposition 023N, applied to the base ring as a module. The length one case is Lemma formal-defos-lemma-length-one-cech-equalizer. Thus (2) implies (3).

Assume (3). These are the conditions traditionally denoted \((P)\), \((L)\), and \((M)\): preservation of finite products, faithfully flat Cech equalizers out of local rings, and Cech equalizers for minimal local extensions with trivial residue extensions. For finite Artinian algebras, the maps in \((L)\) are finite free; the length one formulation in (3) is the corresponding local form of \((M)\). The corrected Grothendieck–Levelt devissage proves that these tests imply that \(F\) is left exact; see the last two references in the statement.

For clarity, the residue-field step in that devissage is as follows. After reducing an equalizer problem to a local base \(A\), one chooses a finite normal extension containing the finitely many residue fields which occur and a finite free local extension \(A \to A'\) inducing it. The functor is first shown to be left exact on the subcategory with this fixed residue field. The Cech equalizers for \(A \to A'\) and for its base changes then descend the required equalizer from \(A'\) to \(A\). The 1970 correction supplies the fixed-residue-field argument without the false minimal-extension assertion used in the original proof. Hence \(F\) is left exact, and (1) follows from the Categories proposition cited above.

Remark

The correction in the reference is essential. The last assertion of Section 3, Lemma 5 of the 1969 paper is false: a simple monomorphism of Artinian algebras which is not faithfully flat need not induce only trivial residue field extensions. The criterion itself remains valid; the 1970 paper replaces precisely the part of the proof which used that assertion.

Proposition

Let \(\Lambda\) be a Noetherian ring and let \(X\) and \(Y\) be schemes over \(\Lambda\). Assume that \(X\) is flat over \(\Lambda\). The functor \[F : \mathcal{A}_\Lambda \longrightarrow \textit{Sets}, \qquad A \longmapsto \Mor_A(X_A, Y_A)\] is strictly prorepresentable.

For \(A \in \mathcal{A}_\Lambda\), put \(A_\epsilon = A[\epsilon]/(\epsilon^2)\). If \(\xi \in F(A)\), then the fibre \[F(A_\epsilon, \xi) = \{\xi' \in F(A_\epsilon) \mid \xi'|_A = \xi\}\] is canonically isomorphic to \[\Gamma\left(X_A, \SheafHom_{\mathcal{O}_{X_A}} (\xi^*\Omega_{Y_A/A}, \mathcal{O}_{X_A})\right).\]

Proof

The functor \(F\) commutes with finite products because the spectrum of a product of rings is a disjoint union. We verify the two local equalizer tests of 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. Morphisms of schemes satisfy fpqc descent by Descent, Remark 040L; hence the required diagram for \(F\) is an equalizer.

Suppose instead that \(A \to B\) is injective and \(B/A\) has length \(1\). Lemma formal-defos-lemma-length-one-cech-equalizer gives the ring equalizer, so \(\Spec(B) \to \Spec(A)\) is a finite effective epimorphism by Descent, Lemma descent-lemma-finite-effective-epimorphism-functions. Since \(X_A\) is flat over \(A\), Descent, Lemma descent-lemma-finite-effective-epimorphism-flat-schemes gives precisely the required equalizer of sets of morphisms. The theorem cited above now proves that \(F\) is strictly prorepresentable.

For the tangent statement, regard \(\Spec(A) \subset \Spec(A_\epsilon)\) as a first order thickening and base change it to \(X\) and \(Y\). The base change of \(\xi\) is a distinguished element of \(F(A_\epsilon, \xi)\). More on Morphisms, Lemma 04FJ and Remark 0CK2 show that the sheaf of such lifts is acted on simply transitively by \[\SheafHom_{\mathcal{O}_{X_A}} (\xi^*\Omega_{Y_A/A}, \mathcal{C}_{X_A/X_{A_\epsilon}}).\] The canonical identification \(\mathcal{C}_{X_A/X_{A_\epsilon}} = \mathcal{O}_{X_A}\epsilon\) and the distinguished lift identify its global sections with the displayed group.

Remark

The tangent group in Proposition formal-defos-proposition-formal-hom-prorepresentable-tangent is in degree zero. The degree-one formula printed in the reference cannot be correct: for \(X_A = \Spec(A)\) and \(Y_A = \mathbf{A}^1_A\), the lift set is naturally \(A\), whereas the corresponding first cohomology group is zero. In a general lifting problem without the distinguished base-change lift, degree one may instead contain the obstruction to the existence of a global lift.

Lemma

Let \(\Lambda\) be a Noetherian ring. Let \(X\) and \(Y\) be schemes over \(\Lambda\). If \(X\) is flat and proper over \(\Lambda\) and \(Y\) is locally of finite type over \(\Lambda\), then the local components of a topological \(\Lambda\)-algebra prorepresenting \[A \longmapsto \Mor_A(X_A, Y_A), \qquad A \in \mathcal{A}_\Lambda,\] are Noetherian.

Proof

Proposition formal-defos-proposition-formal-hom-prorepresentable-tangent gives the prorepresentability. Fix a field \(K \in \mathcal{A}_\Lambda\) and a morphism \(\xi : X_K \to Y_K\) corresponding to a point of a local component. The scheme \(X_K\) is Noetherian and proper over \(K\). Since \(Y_K\) is locally of finite type over \(K\), the module \(\Omega_{Y_K/K}\) is coherent by Morphisms, Lemma 01V2. Hence \[\SheafHom_{\mathcal{O}_{X_K}} (\xi^*\Omega_{Y_K/K}, \mathcal{O}_{X_K})\] is coherent by Cohomology of Schemes, Lemma 01Y2. Its global sections form a finite dimensional \(K\)-vector space by Cohomology of Schemes, Lemma 02O6. By the tangent formula in the proposition, the tangent space of the local component is therefore finite dimensional. The finite-tangent criterion for a local component of a strict Artinian proobject now gives Noetherianity. In the language used above, this is the bounded-cotangent-space argument of Lemma 06SF.

Caution: We will see later that in our general setting the tangent space of an object \(A \in \mathcal{C}_\Lambda\) over \(\Lambda\) should not be defined simply as the \(k\)-linear dual of the relative cotangent space. In fact, the correct definition of the relative cotangent space is \(\Omega_{S/R} \otimes_S S/\mathfrak m_S\).↩︎
For example if \(\mathcal{F}\) satisfies (S2), see Lemma 06I0.↩︎
This may be nonstandard terminology. Many authors tie this notion in with properties of tangent spaces. We will make the link in Section 06IL.↩︎