We generalize the Bogomolov–Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne–Mumford surfaces. We work over positive characteristic \(p>0\) and generalize Langer’s method to smooth Deligne–Mumford stacks. As applications we obtain the Bogomolov inequality for semistable coherent sheaves on a Deligne–Mumford surface in characteristic zero, and the Bogomolov inequality for semistable sheaves on a root stack over a smooth surface which is equivalent to the Bogomolov inequality for the rational parabolic sheaves on a smooth surface S. In a joint appendix with Hao Sun, we generalize the Bogomolov inequality formula to Simpson Higgs sheaves on tame Deligne–Mumford stacks.
