The most common problem with the rear lights on the Trailblazers are faulty circuit boards which should have been replaced as part of the recall. There are separate fuses for the left and right tail lights. Check the rear fuse box, fuses 14 and 17. If they are OK then check the brown wire at the circuit board plug for power. If you have power there then the fault is with the board or the bulb. If the fuses are good and there is no power on the brown wire then there is fault between the fuse box and the plug at the tail light.