When I ues AC, the alternator will fail why, the AC cooling is very low

Our database does not have any information on your vehicle so I do not know the exact design employed and what I have to say might not apply.

Check the crank pulley to see if it is equipped with harmonic damper. If yes, you could have a bad damper allowing the pulley to slip.

If AC and alternator shares the same serpentine belt, check the belt tension.

If everything test fine, have the alternator tested. It could be failing.

When AC is turned on, ensure the idling speed is correct and not too low.

When the AC magnetic clutch kicks in, it puts an additional load in the electrical system this is where an AC idle-up solenoid is used to increase the idle to maintain the electrical load as demanded by the magnetic clutch