Schemes

Consider a regular embedding ${\displaystyle I\colon Y\to W}$ fitting into a cartesian square

${\displaystyle {\begin{matrix}X&{\xrightarrow {j}}&V\\g\downarrow &&\downarrow f\\Y&{\xrightarrow {i}}&W\end{matrix}}}$

where ${\displaystyle V,W}$ are smooth. Then, the complex

${\displaystyle E^{\bullet }=[g^{*}N_{Y/W}^{\vee }\to j^{*}\Omega _{V}]}$ (in degrees ${\displaystyle -1,0}$)

forms a perfect obstruction theory for X.^[1] The map comes from the composition

${\displaystyle g^{*}N_{Y/W}^{\vee }\to g^{*}i^{*}\Omega _{W}=j^{*}f^{*}\Omega _{W}\to j^{*}\Omega _{V}}$

This is a perfect obstruction theory because the complex comes equipped with a map to ${\displaystyle \mathbf {L} _{X}^{\bullet }}$ coming from the maps ${\displaystyle g^{*}\mathbf {L} _{Y}^{\bullet }\to \mathbf {L} _{X}^{\bullet }}$ and ${\displaystyle j^{*}\mathbf {L} _{V}^{\bullet }\to \mathbf {L} _{X}^{\bullet }}$. Note that the associated virtual fundamental class is ${\displaystyle [X,E^{\bullet }]=i^{!}[V]}$

Deligne–Mumford stacks

The previous construction works too with Deligne–Mumford stacks.