[BF97, Definition 2.3] We say that \(L^\bullet\in D(\mathcal O_{X^{\text{\'et}}})\) satisfies condition (\(\star\)) if
- \(h^i(L^\bullet) = 0\) for all \(i > 0\)
- \(h^i(L^\bullet)\) is coherent for \(i=0,-1\).
Definition
Let \(E^\bullet \in \operatorname{ob}D(\mathcal O_{X^{\text{\'et}}})\) satisfy \((\star)\). A homomorphism \(\phi:E^\bullet \to L^\bullet_X\) in \(D(\mathcal O_{X^{\text{\'et}}})\) is called an obstruction theory for \(X\) if \(h^0(\phi)\) is an isomorphism and \(h^{-1}(\phi)\) is surjective. By abuse of language we also say that \(E^\bullet\) is an obstruction theory for \(X\).
We say that an obstruction theory \(E^\bullet \to L^\bullet_X\) is perfect if \(E^\bullet\) is of perfect amplitude contained in \([-1,0]\).
See also perfect-obstruction-theory-defn.