The Stacks project

90.1 Introduction

This chapter develops formal deformation theory in a form applicable later in the Stacks project, closely following Rim [Exposee VI, SGA7-I] 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:

1. $F$ has a “hull” if and only if (H1)-(H3) hold.

2. $F$ 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:

1. The functor $F$ is replaced by a category $\mathcal{F}$ cofibered in groupoids over $\mathcal{C}_\Lambda $, see Section 90.3.

2. 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 [Exposee VI, SGA7-I] 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 90.4. 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 $\underline{R} = \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }(R, -)$ to $\mathcal{C}_\Lambda $ where $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$.

Categories cofibred in groupoids are dual to categories fibred in groupoids; we introduce them in Section 90.5. A smooth morphism of categories cofibred in groupoids over $\mathcal{C}_\Lambda $ is one that satisfies the infinitesimal lifting criterion for objects, see Section 90.8. This is analogous to the definition of a formally smooth ring map, see Algebra, Definition 10.138.1 and is exactly dual to the notion in Criteria for Representability, Section 97.6. 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 94.16 and 94.17. 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 90.10, 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 90.13.4. 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 90.14 that if a predeformation category has a versal formal object, then it always has a minimal one (as defined in Definition 90.14.4) which is unique up to isomorphism, see Lemma 90.14.5. But it can happen that the minimal versal formal object does not induce an isomorphism on tangent spaces! (See Examples 90.15.3 and 90.15.8.)

Keeping in mind the differences pointed out above, Theorem 90.15.5 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 90.16, 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 90.21.1, 90.22.1, and 90.23.1. This notion of a presentation takes into account the groupoid structure of the fibers of $\mathcal{F}$. In Theorem 90.26.4 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 90.27 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 90.16.6 and 90.10.5). 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 93.1. 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 90.16.7). These observations suggest that (RS) should be regarded as the fundamental deformation theoretic glueing condition.