Strange events with new alternator - ideas?

[aatos]

My battery light on the 2002 4.4 came on a few weeks ago - checked voltage with engine running - about 11.5. I bought a refurbished Bosch alternator on-line, installed it, and the light went away, voltage is around 14.0.

But now, if I run for quite a while, turn off the engine and restart, the battey light comes on again, and stays on until the engine is turned off and has time to cool down. If I'm driving and don't stop, the light doesn't come on. Or if I drive only a short distance, stop and restart, no ligght. Seems to be a problem related to heat build up, and only if I turn off the engine and then restart. Very hard to get gas this way.

Any experiences like that? I would suspect the regulator on the alternator, but here is no problem if I just keep driving, so I don't know.