Instructor pilot; with trainee; reported the cabin carbon monoxide detector activated during the pre-takeoff check due to a cracked exhaust system.

Narrative

Routine commercial final phase check; I was the evaluator on board. The student and I completed our preflight. We taxied to the runup area; and the runup passed. The student keyed up ZZZ ground frequency for taxi instructions; and while the student was reading back instructions; a loud alarm noise sounded in the airplane. Since I was looking down at my iPad to write down taxi instructions; I saw a big red ForeFlight alert about carbon monoxide being detected. I informed the student that there was a carbon monoxide alert. I then radioed ZZZ Ground to notify them about the CO annunciator; that we would be shutting down; and that I would like an Ops Truck to meet us at the runup. I requested the Ops truck in case I needed assistance moving the airplane. I instructed the student to shut the aircraft down. The alarm continued to sound; so I instructed the student to exit the plane (run-up area is on the ramp and not an active taxiway) and meet me at the plane's tail. I then called the FBO manager and asked for further instructions. He told me to either walk back to the hangar and he would taxi the airplane; or I could taxi it if I felt comfortable. I elected to taxi back with open doors to help airflow through the aircraft. We taxied back without further incident. No injuries from student or instructor.The next day; maintenance informed me that a cracked exhaust caused the carbon monoxide.