Deformation theory and DGLAs I

There’s a “philosophy” in deformation theory that deformation problems in characteristic zero come from dg-Lie algebras. I’ve been trying to learn a little about this. Precise statements have been given by Lurie and Pridham which consider categories of “derived” deformation problems (i.e., deformation problems that can be evaluated on derived rings) and establish equivalences between them and suitable (higher) categories of dg-Lie algebras. I’ve been reading in particular Lurie’s very enjoyable survey of his approach to the problem, which sketches the equivalence in an abstract categorical context with the essential input arising from Koszul duality between Lie algebras and commutative algebras. In this post, I’d just like to say what a “deformation problem” is in the derived world.

1. Introduction

Let ${\mathcal{M}}$ be a classical moduli problem. Abstractly, we will think of ${\mathcal{M}}$ as a functor

$\displaystyle \mathcal{M}:\mathrm{Ring} \rightarrow \mathrm{Sets},$

such that, for a (commutative) ring ${R}$ , the set ${\mathcal{M}(R)}$ will be realized as maps from ${\mathrm{Spec} R}$ into a geometric object—a scheme or maybe an algebraic space.

Example 1 ${\mathcal{M}}$ could be the functor that sends ${R}$ to the set of closed subschemes of ${\mathbb{P}^n_R}$ which are flat over ${R}$ . In this case, ${\mathcal{M}}$ comes from a scheme: the Hilbert scheme.

We want to think of ${\mathcal{M}}$ as some kind of geometric object and, given a point ${x: \mathrm{Spec} k \rightarrow \mathcal{M}}$ for ${k}$ a field (that is, an element of ${\mathcal{M}(k)}$ ), we’d like to study the local structure of ${\mathcal{M}}$ near ${x}$ .

Grothendieck’s idea is that we can do this by studying maps from small schemes into ${\mathcal{M}}$ . Namely, let ${\mathrm{Art}_k}$ be the category of artinian ${k}$ -algebras with residue field ${k}$ ; then for ${R \in \mathrm{Art}_k}$ , ${\mathrm{Spec} R}$ is a “nilthickening” of ${\mathrm{Spec} R_{\mathrm{red}} = \mathrm{Spec} k}$ . We study maps ${\mathrm{Spec} R \rightarrow \mathcal{M}}$ which agree with ${x}$ on ${\mathrm{Spec} k}$ . That is, we study diagrams

which we can think of small deformations of the point ${x}$ .

If ${\mathcal{M}}$ is a scheme, then understanding all these maps amounts to understanding the formal completion of ${\mathcal{M}}$ at the point ${x}$ . Even otherwise, a combination of tools such as Grothendieck’s formal GAGA and Artin’s approximation theorem can enable one to go from such “formal deformations” to global deformations (over schemes much bigger than a fuzzy point).

Let’s axiomatize this:

Definition 1 A deformation functor is a functor ${F: \mathrm{Art}_k \rightarrow \mathrm{Sets}}$ satisfying certain axioms:

${F(k) = \ast}$ . (This is the “formal” part of the definition.)
Given a diagram in ${\mathrm{Art}_k}$ ,

where the maps ${A \rightarrow A'', A' \rightarrow A''}$ are surjective, the diagram

is a pull-back square of sets.

The second condition is a reflection of the fact that ${\mathrm{Spec} (A \times_{A'} A') }$ is the push-out in the category of schemes, ${\mathrm{Spec} A \sqcup_{\mathrm{Spec} A''} \mathrm{Spec} A'}$ .

Example 2 Given a scheme ${X}$ and a point ${x: \mathrm{Spec} k \rightarrow X}$ , we define a functor ${F}$ on ${\mathrm{Art}_k}$ by sending ${R \in \mathrm{Art}_k}$ to the set of maps ${\mathrm{Spec} R \rightarrow X}$ that restrict to ${x}$ on the “closed point.” This is a deformation functor, called the formal completion of ${X}$ at ${x}$ . The name is because

$\displaystyle F(R) = \hom_k( \widehat{\mathcal{O}_{X, x}}, R),$

so that the deformation functor is prorepresentable—it’s realized by maps out of a complete local ring.

Example 3 As a result, the deformation problem of closed subschemes in ${\mathbb{P}^n_k}$ is always prorepresentable. That is, given a closed subscheme ${X \subset \mathbb{P}^n_k}$ , we can look at small deformations of ${X}$ : that is, flat subschemes of ${\mathbb{P}^n_R}$ that are “thickenings” of ${X}$ (restrict to ${X}$ on the closed fiber). This deformation problem is prorepresentable because of the representability of the Hilbert scheme. However, it’s possible to see this directly, using Schlessinger’s criterion; the deformation problem for closed subschemes of ${\mathbb{P}^n_k}$ can be studied “directly” and behaves well. Given this well-behaved deformation theory, a deep representability theorem of Artin can be used to show that the Hilbert scheme exists (at least as an algebraic space). As I understand, this sort of argument is very important in the derived context.

As time went on, people realized that it’s of interest to replace “sets” by “groupoids” in the definition of a moduli problem. Rather than do that, let’s take the plunge to the derived world.

2. The principal actors

Let ${k}$ be a field of characteristic zero.

Definition 2

We let ${\mathrm{CAlg}}$ be the category (throughout, ${(\infty, 1)}$ -category) of connective ${E_\infty}$ -algebras over ${k}$ . This is the replacement for “commutative rings.” Since ${k}$ is characteristic zero, ${E_\infty}$ -algebras may be modeled by commutative differential graded ${k}$ -algebras.
Let be the full subcategory of consisting of -algebras such that:
1. ${\pi_i A = 0}$ for ${i \gg 0}$ .
2. Each homotopy group ${\pi_i A}$ is a finite-dimensional vector space over ${k}$ .
3. ${\pi_0 A}$ is artinian with residue field ${k}$ .

This is the replacement for the category ${\mathrm{Art}_k}$ ; now we’re allowing a small amount of “homotopy fuzz.” Given any object ${A \in \mathrm{CAlg}_{sm}}$ , there is a canonical augmentation map (well defined up to a contractible space) ${A \rightarrow k}$ .

Let ${\mathcal{S}}$ be the category of spaces.

Definition 3 A formal moduli problem is a functor

$\displaystyle F: \mathrm{CAlg}_{sm} \rightarrow \mathcal{S},$

such that:

${ F(k) }$ is contractible.
Let ${A, A', A'' \in \mathrm{CAlg}_{sm}}$ and suppose given maps ${A \rightarrow A'', A' \rightarrow A''}$ both of which induce surjections on ${\pi_0}$ . Then the natural map
$\displaystyle F( A \times_{A'} A'') \rightarrow F(A) \times_{F(A')} F(A'')$

is a homotopy equivalence.

There is a natural category ${\mathrm{Moduli}}$ of formal moduli problems, where the morphisms are the natural transformations.

In this definition, the first condition is the reason for the word “formal.” For example, given a moduli problem, meaning a functor ${\mathcal{M}: \mathrm{CAlg} \rightarrow \mathcal{S}}$ , and a point ${x \in \mathcal{M}( \mathrm{Spec} k)}$ , we can define the formal completion of ${\mathcal{M}}$ at ${x}$ via

$\displaystyle \hat{\mathcal{M}}( A) = \mathrm{fib}( \mathcal{M}(A) \rightarrow \mathcal{M}(k)),$

where the (homotopy) fiber is taken over the point ${x}$ . This construction is not necessarily a formal moduli problem, since it does not have to satisfy the second condition.

However, the second condition is often satisfied for the following reason: given maps ${A \rightarrow A'', A' \rightarrow A''}$ inducing surjections on ${\pi_0}$ , then you should think of ${\mathrm{Spec} (A \times_{A''} A')}$ as obtained by “gluing” ${\mathrm{Spec} A}$ and ${\mathrm{Spec} A'}$ along the common closed subscheme ${\mathrm{Spec} A''}$ . Here is an example of this phenomenon.

Theorem 4 (DAG IX) Let ${A, A', A''}$ be connective ${E_\infty}$ -rings with maps ${A \rightarrow A'', A' \rightarrow A''}$ inducing surjections on ${\pi_0}$ . Let ${\mathrm{Mod}^c_A}$ denote the category of connective ${A}$ -modules. Then we have an equivalence of categories

$\displaystyle \mathrm{Mod}^c_{A \times_{A''} A'} \simeq \mathrm{Mod}^c_{A} \times_{\mathrm{Mod}^c_{A''}} \mathrm{Mod}^c_{A''} ,$

where the functor sends a connective ${A \times_{A''} A'}$ -module ${M}$ to the tuple ${( M \otimes_{A \times_{A''} A'} A', M \otimes_{A \times_{A''} A'} A'', \psi) }$ where ${\psi}$ is the natural equivalence.

In other words, to give a connective module over ${A \times_{A''} A'}$ is the same as giving a module over ${A}$ , a module over ${A'}$ , together with an isomorphism of their base-changes to ${A''}$ .

Proof: The functor ${F}$ defined in the statement of the theorem is a colimit-preserving functor, and in fact a left adjoint: the right adjoint ${G}$ is given by sending a tuple ${(M, M', \psi)}$ to the fiber product in the diagram:

Since tensoring preserves finite limits of modules, it follows that for any ${A \times_{A''} A'}$ -module ${M}$ (not necessarily connective!), it can be recovered from its base-changes to ${A, A', A''}$ via the above pull-back. In other words,

$\displaystyle G \circ F \simeq \mathrm{Id} ,$

which states that ${F}$ is a colocalization: it is fully faithful. To show that ${F}$ is an equivalence (on connective modules), it suffices that ${G}$ never annihilates a nontrivial object in ${\mathrm{Mod}^c_{A} \times_{\mathrm{Mod}^c_{A''}} \mathrm{Mod}^c_{A''}}$ . Suppose given such an object ${(M, M', \psi)}$ such that the pull-back that one gets is zero. This means that

$\displaystyle M \otimes_A A'' \simeq M \oplus M'.$

Choose the smallest index where the homotopy groups don’t vanish, say ${n}$ . Then one has:

Each of the vertical maps are surjective — this can’t happen for a direct sum unless everything vanishes. $\Box$

For nonconnective modules, the result fails. An example is given by taking $A = A' = k[x], A'' = k$ (as discrete $E_\infty$ -algebras), and with $M = \bigoplus_{i \mathrm{\ even}} k[i]$ and $M' = \bigoplus_{i \ \mathrm{odd} } k[i]$ .

3. The main result

The main result of DAG X is given by:

Theorem: There is an equivalence of $\infty$ -categories between $\mathrm{Moduli}$ and the $\infty$ -category of DGLAs (defined by localizing the ordinary category at the quasi-isomorphisms).

In the next couple of posts, I’d like to sketch the proof of this result, which gives a concrete construction of a formal moduli problem out of a DGLA.

Climbing Mount Bourbaki