Simple obstructions and cone reduction

Abstract Let X be a Deligne–Mumford stack locally of finite type over an algebraically closed field k of characteristic zero. We show that the intrinsic normal cone C X of X is supported in the subcone 𝕍( Ω X [−1]) of its intrinsic normal sheaf N X . This leads to an alternative proof of cone reduction by cosections for C X . We also discuss vanishing of simple obstructions under the Buchweitz–Flenner semiregularity map for sheaves.