In algebraic geometry, given a Deligne–Mumford stack X, a perfect obstruction theory for X consists of:
a perfect two-term complex in the derived category of quasi-coherent étale sheaves on X, and
a morphism , where is the cotangent complex of X, that induces an isomorphism on and an epimorphism on .
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.
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