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…)