Battery not holding charge

I HOPE it's the generator because if it was working, disconnecting the battery cable can destroy the many computers on the car from excessive voltage. That was a trick done many years ago by mechanics who didn't understand these systems. A better and safer test is to use an inexpensive digital voltmeter to measure battery voltage. With the engine not running, a fully charged battery will read near 12.6 volts. If it is closer to 11.0 volts, suspect it has a shorted cell. With the engine running, the voltage must be between 13.75 and 14.75 volts. If it stays near 12.0 to 12.6 volts, suspect the generator or voltage regulator.