Perfect Obstruction Theory Defn

Definition

Let \(f:X\to Y\) be a morphism of algebraic stacks over some base stack \(S\). A perfect obstruction theory for \(f\), or as it is more commonly denoted, a perfect obstruction theory \(X/Y\), is a two term complex \(E^\bullet = [E^{-1}\xrightarrow{d} E^0]\) together with a morphism

\begin{align*} \phi:E^\bullet \to L_{X/Y} \end{align*}

to the relative cotangent complex of \(f\) such that

1. \(h^0(f)\) is an isomorphism
2. \(h^{-1}(f)\) is surjective.

We call \((\phi, E^\bullet)\) an absolute perfect obstruction theory if \(Y = S\). We will sometimes call \((\phi,E^\bullet)\) a relative perfect obstruction theory for \(f:X\to Y\) when \(Y\neq S\) to contrast against the absolute situation.

When a perfect obstruction theory for \(f\) exists, we say that \(f\) is virtually smooth, or that \(X\) is virtually smooth over \(Y\).