nLab perfect obstruction theory

The notion of perfect obstruction theory is introduced by Behrend and Fantechi.

For derived schemes

Every derived scheme has a canonical perfect obstruction theory given as follows.

Let $X$ be a derived scheme. Let $j$ denote the morphism from the underlying ordinary scheme

$$j: t_0(X) \to X$$

The cotangent complex functor sends this to an arrow in the tangent (infinity,1)-category

$$L_{t_0(X)} \to j^*X$$

This morphism of quasicoherent sheaves is a perfect obstruction theory.

References

The paper

• Kai Behrend, B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88, MR98e:14022 arXiv:alg-geom/9601010

shows how to construct virtual fundamental classes, using the intrinsic normal cone?.