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…)