Would that be the battery or alternator?

Could be either. Remove your alternator and take it to a parts house that has an alternator tester. That will tell you if its the alternator. GM alternators have a three diode assembly that keeps the voltage pulses generated by it from being pulled right back into it. If one of these diodes fails (which is fairly common) the alternator will still keep the car alive but the battery will slowy be killed trying to carry the extra load. Its just two bolts and two connectors to remove it. If you just replace one without testing it you are wasting your money because if the part replaced isnt faulty then you are gonna cook the new one.

:iagree: it could be either.