The purpose of this post, the third in a series on deformation theory and DGLAs, is to describe the obstruction theory for a formal moduli problem associated to a DGLA.

1. Tangent-obstruction theories

Standard problems in classical deformation theory usually have a “tangent-obstruction theory” parametrized by certain successive cohomology groups. For example, let’s consider the problem of deformations of a smooth variety ${X}$ over an algebraically closed field ${k}$, over finite-dimensional local ${k}$-algebras. Then:

• The “infinitesimal automorphisms” of ${X}$—that is, automorphisms of the trivial deformation over ${k[\epsilon]/\epsilon^2}$—are given by ${H^0( X, T_X)}$ where ${T_X}$ is the tangent bundle (i.e., vector fields).
• The isomorphism classes of deformations of ${X}$ over the dual numbers ${k[\epsilon]/\epsilon^2}$ are given by ${H^1(X, T_X)}$.
• There is an obstruction theory with ${H^1, H^2}$. Specifically, given a square-zero extension of finite-dimensional local ${k}$-algebras

$\displaystyle 0 \rightarrow I \rightarrow A' \rightarrow A \rightarrow 0,$

and given a deformation ${\xi}$ of ${X}$ over ${\mathrm{Spec} A}$, there is a functorial obstruction in ${H^2(X, T_X) \otimes_k I}$ to extending the deformation over the inclusion ${\mathrm{Spec} A \hookrightarrow \mathrm{Spec} A'}$.

• In the previous item, if the obstruction vanishes, then the isomorphism classes of extensions of ${\xi}$ over ${\mathrm{Spec} A'}$ are a torsor for ${H^1(X, T_X) \otimes_k I}$.

One has a similar picture for other deformation problems, for example deformations of vector bundles or closed subschemes. The “derived” approach to deformation theory provides (at least in characteristic zero) a general explanation for this phenomenon. (more…)

Let ${k}$ be a field of characteristic zero. In the previous post, we introduced the category (i.e., ${\infty}$-category) ${\mathrm{Moduli}_k}$ of formal moduli problems over ${k}$. A formal moduli problem over ${k}$ is a moduli problem, taking values in spaces, that can be evaluated on the class of “derived” artinian ${k}$-algebras with residue field ${k}$: this was the category ${\mathrm{CAlg}_{sm}}$ introduced in the previous post.

In other words, a formal moduli problem was a functor

$\displaystyle F: \mathrm{CAlg}_{sm} \rightarrow \mathcal{S} \ (= \text{spaces}),$

which was required to send ${k}$ itself to a point, and satisfy a certain cohesiveness condition: ${F}$ respects certain pullbacks in ${\mathrm{CAlg}_{sm}}$ (which corresponded geometrically to pushouts of schemes).

The main goal of the series of posts was to sketch a proof of (and define everything in) the following result:

Theorem 7 (Lurie; Pridham) There is an equivalence of categories between ${\mathrm{Moduli}_k}$ and the ${\infty}$-category ${\mathrm{dgLie}}$ of DGLAs over ${k}$.

4. Overview

Here’s a rough sketch of the idea. Given a formal moduli problem ${F}$, we should think of ${F}$ as something like a small space, concentrated at a point but with lots of “infinitesimal” thickening. (Something like a ${\mathrm{Spf}}$.) Moreover, ${F}$ has a canonical basepoint corresponding to the “trivial deformation.” That is, we can think of ${F}$ as taking values in pointed spaces rather than spaces.

It follows that we can form the loop space ${\Omega F = \ast \times_F \ast}$ of ${F}$, which is a new formal moduli problem. However, ${\Omega F}$ has more structure: it’s a group object in the category of formal moduli problems — that is, it’s some sort of derived formal Lie group. Moreover, knowledge of the original ${F}$ is equivalent to knowledge of ${\Omega F}$ together with its group structure: we can recover ${F}$ as ${B \Omega F}$ (modulo connectivity issues that end up not being a problem). This relation between ordinary objects and group objects (via ${B, \Omega}$) is something very specific to the derived or homotopy world, and it’s what leads to phenomena such as Koszul duality. (more…)

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 ${x}$. (more…)

Let ${X}$ be a projective variety over the algebraically closed field ${k}$, endowed with a basepoint ${\ast}$. In the previous post, we saw how to define the Picard scheme ${\mathrm{Pic}_X}$ of ${X}$: a map from a ${k}$-scheme ${Y}$ into ${\mathrm{Pic}_X}$ is the same thing as a line bundle on ${Y \times_k X}$ together with a trivialization on ${Y \times \ast}$. Equivalently, ${\mathrm{Pic}_X}$ is the sheafification (in the Zariski topology, even) of the functor

$\displaystyle Y \mapsto \mathrm{Pic}(X \times_k Y)/\mathrm{Pic}(Y),$

so we could have defined the functor without a basepoint.

We’d like to understand the local structure of ${\mathrm{Pic}_X}$ (or, equivalently, of ${\mathrm{Pic}^0_X}$), and, as with moduli schemes in general, deformation theory is a basic tool. For example, we’d like to understand the tangent space to ${\mathrm{Pic}_X}$ at the origin ${0 \in \mathrm{Pic}_X}$. The tangent space (this works for any scheme) can be identified with

$\displaystyle \hom_{0}( \mathrm{Spec} k[\epsilon]/\epsilon^2, \mathrm{Pic}_X).$ (more…)

Let ${A}$ be an abelian variety of dimension ${g}$ over a field ${k}$ of characteristic ${p}$. In the previous posts, we saw that to give a deformation ${R}$ of ${A}$ over a local artinian ring with residue field ${k}$ was the same as giving a continuous morphism of rings ${W(k)[[t_1, \dots, t_{g^2}]] \rightarrow A}$: that is, the local deformation space is smooth on ${g^2}$ parameters.

There is another description of the deformation space in terms of the ${p}$-divisible group, though. Given ${A}$, we can form the formal scheme

$\displaystyle A[p^\infty] = \varinjlim A[p^n],$

where each ${A[p^n]}$ is a finite group scheme of rank ${p^{2ng}}$ over ${k}$. As a formal scheme, this is smooth: that is, given a small extension ${R' \twoheadrightarrow R}$ in ${\mathrm{Art}_k}$ and a morphism ${\mathrm{Spec} R \rightarrow A[p^\infty]}$, there is an extension

In fact, we start by finding an extension ${\mathrm{Spec} R' \rightarrow A}$ (as ${A}$ is smooth), and then observe that this extension must land in some ${A[p^n]}$ if ${\mathrm{Spec} R}$ landed in ${A[p^\infty]}$.

As a result, we can talk about deformations of this formal (group) scheme.

Definition 16 deformation of a smooth, formal group scheme ${G}$ over ${k}$ over a ring ${R \in \mathrm{Art}_k}$ is a smooth, formal group scheme ${G'}$ over ${R}$ which reduces mod the maximal ideal (i.e., when restricted to ${k}$-algebras) to ${G}$.

Suppose ${G(k) = \ast}$. Using Schlessinger’s criterion, we find that (under appropriate finiteness hypotheses), ${G}$ must be prorepresentable by a power series ring ${R[[t_1, \dots, t_n]]}$ for some ${n}$. In other words, ${G}$ corresponds to a formal group over ${R}$.

Suppose for instance that ${A}$ was a supersingular elliptic curve. By definition, each of the ${A[p^n]}$ is a thickening of the zero point, and consequently ${A[p^\infty]}$ is the formal completion ${\hat{A}}$ of ${A}$ at zero. This data is equivalent to the formal group of ${A}$, and as we just saw a deformation of this formal group over a ring ${R \in \mathrm{Art}_k}$ is a formal group over ${R}$ which reduces to the formal group of ${A}$ mod the maximal ideal.

The main result is:

Theorem 17 (Serre-Tate) Let ${R \in \mathrm{Art}_k}$. There is an equivalence of categories between abelian schemes over ${R}$ and pairs ${(A, G)}$ where ${A }$ is an abelian variety over ${k}$ and ${G}$ a deformation of ${A[p^\infty]}$ over ${R}$.

So, in particular, deformations of ${A}$ over ${R}$ are equivalent to deformations of ${A[p^\infty]}$ over ${R}$.

In this post, I’ll describe an argument due to Drinfeld for this result (which I learned from Katz’s article). (more…)

(This is the third in a series of posts on deformations of abelian varieties; the first two posts are here and here.)

Let ${X/k}$ be an abelian variety. In the previous post, we saw that the functor

$\displaystyle \mathrm{Art}_k \rightarrow \mathbf{Sets}$

sending a local artinian algebra ${R}$ with residue field $k$ to the set of isomorphism classes of deformations of ${X}$ over ${R}$ was product-preserving, by showing that there were no infinitesimal automorphisms. The next step is to show that it is smooth: that is, given a deformation over ${R}$ and a surjection ${R' \twoheadrightarrow R}$, it can be lifted to a deformation over ${R'}$. Together with Schlessinger’s theorem, this will yield a description of the local moduli space. (more…)

This is the second post in a series on deformations of abelian varieties. In the previous post, I described the basic outline of the goals and strategies, as well as a weak version of Schlessinger’s criterion useful in showing prorepresentability of smooth moduli problems without infinitesimal automorphisms. In this post, we’ll see that the deformation problem for abelian varieties satisfies the second condition above: that abelian varieties are rigid. The material here is very classical; I learned it from Oort’s article (from a summer school in the 1970s) and Katz’s article. Most of the material in this post comes from chapter 6 of Mumford’s GIT book, which is surprisingly readable without knowledge of any other parts of it.

Let ${R}$ be an artin ring, and let ${X/\mathrm{Spec} R}$ be an abelian scheme. Consider a morphism of ${R}$-group schemes

$\displaystyle f: X \rightarrow X$

inducing the identity on the special fiber. We would like to show that it is the identity, as in the next proposition:

Proposition 7 Such a morphism ${f}$ is necessarily the identity: that is, an infinitesimal deformation of an abelian variety has no nontrivial infinitesimal automorphisms.

This will imply prorepresentability of the deformation functor, using the general form of Schlessinger’s theorem.

I’ve been trying to learn something about deformations of abelian varieties lately. One of the big results is:

Theorem 1 The “local moduli space” of an abelian variety ${X/k}$ is smooth of dimension ${g^2}$, if ${\dim X = g}$.

Why might you care about this result? Let’s say you’re in the case ${g = 1}$, so then presumably you’re interested in the stack ${M_{1, 1}}$ of elliptic curves. This is a Deligne-Mumford stack: that is, there are enough étale maps ${\mathrm{Spec} R \rightarrow M_{1, 1}}$ from affine schemes, and as a result it makes sense to talk about the strict henselianization at a point (or the completion at a “point” from an algebraically closed field). Then, the point of the above theorem is that you can work out exactly what that is: it’s a one-dimensional thing. This isn’t too surprising, because the isomorphism class of an elliptic curve depends on one parameter (the ${j}$-invariant). So knowing the deformation theory of elliptic curves lets you say what ${M_{1, 1}}$ looks like, very locally.

Let’s make this a bit more precise. An actual formulation of the theorem would specify what “local moduli” really means. For us, it means deformations. A deformation of an abelian variety ${X/k}$ over an artin local ring ${R}$ with residue field ${k}$ is the data of an abelian scheme (that is, a proper flat group scheme with abelian variety fibers) ${X' \rightarrow \mathrm{Spec} R}$ together with an isomorphism of abelian varieties ${X' \times_{\mathrm{Spec} R} \mathrm{Spec} k \simeq X}$.

Theorem 2 Let ${k}$ be an algebraically closed field, and let ${X/k}$ be an abelian variety. Then the functor of deformations of ${X}$ is prorepresentable by ${W(k)[[t_1, \dots, t_{g^2}]]}$ for ${W(k)}$ the ring of Witt vectors over ${k}$.

In other words, to give a deformation of ${X}$ over an artin local ${R}$ with residue field ${k}$ is the same as giving a homomorphism of local rings

$\displaystyle W(k)[[t_1, \dots, t_{g^2}]] \rightarrow R .$

The relevance of ${W(k)}$ here essentially comes from the fact that every complete (e.g. artin) local ring with residue field ${k}$ is uniquely a continuous ${W(k)}$-algebra. If we restricted ourselves to ${k}$-algebras, we could replace ${W(k)}$ by ${k}$. (more…)