Simple obstructions and cone reduction

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 CX of X is supported in the subcone V(ΩX[-1]) (h¹/h⁰((Ω¹X)⁾) of its intrinsic normal sheaf NX. This leads to an alternative proof of cone reduction by cosections for CX. We also discuss vanishing of simple obstructions under the Buchweitz-Flenner semiregularity map for sheaves.