Perfect obstruction theory

In algebraic geometry, given a Deligne–Mumford stack X, a perfect obstruction theory for X consists of:

1. a perfect two-term complex ${\displaystyle E=[E^{-1}\to E^{0}]}$ in the derived category ${\displaystyle D({\text{Qcoh}}(X)_{et})}$ of quasi-coherent étale sheaves on X, and
2. a morphism ${\displaystyle \varphi \colon E\to {\textbf {L}}_{X}}$, where ${\displaystyle {\textbf {L}}_{X}}$ is the cotangent complex of X, that induces an isomorphism on ${\displaystyle h^{0}}$ and an epimorphism on ${\displaystyle h^{-1}}$.

The notion was introduced by Kai Behrend and Barbara Fantechi (1997) for an application to the intersection theory on moduli stacks; in particular, to define a virtual fundamental class.