04 Trailblazer, Alarm, R/S trouble

The Grey/Black wire needs to be connected to the dome light wire of the vehicle.  Then you will connect the green wire to the (-)negative door trigger input on your alarm.  This will only catch the front two doors and the rear hatch, you must tap the rear doors individually,  the colors are GREEN/ black and light blue/black.  You can find each one under the rear door sills, or under the back seat behind the fuse box, under the carpet, not wires going into the fuse box.  That should fix your door trigger issues.