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 over an algebraically closed field , over finite-dimensional local -algebras. Then:
- The “infinitesimal automorphisms” of —that is, automorphisms of the trivial deformation over —are given by where is the tangent bundle (i.e., vector fields).
- The isomorphism classes of deformations of over the dual numbers are given by .
- There is an obstruction theory with . Specifically, given a square-zero extension of finite-dimensional local -algebras
and given a deformation of over , there is a functorial obstruction in to extending the deformation over the inclusion .
- In the previous item, if the obstruction vanishes, then the isomorphism classes of extensions of over are a torsor for .
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…)