Perfect obstruction theory

In algebraic geometry, given a Deligne–Mumford stack X, a perfect obstruction theory for X consists of: a perfect two-term complex E = [ E − 1 → E 0 ] {\displaystyle E=[E^{-1}\to E^{0}]} in the derived category D ( Qcoh ( X ) e t ) {\displaystyle D({\text{Qcoh}}(X)_{et})} of quasi-coherent étale sheaves on X, and a morphism φ : E → L X {\displaystyle \varphi \colon E\to {\textbf {L}}_{X}} , where L X {\displaystyle {\textbf {L}}_{X}} is the cotangent complex of X, that induces an isomorphism on h 0 {\displaystyle h^{0}} and an epimorphism on h − 1 {\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.

Source: Wikipedia — Perfect obstruction theory (CC BY-SA 4.0)