Is cremation prohibited, or is this just a custom that arose after the Holocaust?
Cremation is prohibited by the Torah. The atrocities of the Holocaust have no bearing at all on this prohibition. Our Rabbis tell us [Reference?] that someone who has requested cremation may not be buried in a Jewish cemetery, like someone who has committed suicide. In both cases, the person has defaced the body that G-d entrusted him with which was created in His own image.
It is a Mitzvah to bury a deceased person, as stated in the Torah (Devarim 21:23) "You shall surely bury him". In the Prophecy of Amos (2:1) we find that one of the unpardonable transgressions of the nation of Moab is that they cremated the remains of the King of Edom.
If a person’s relatives had him cremated, not at his request, his remains can be buried in a Jewish cemetery; but there is no Mitzvah to bury remains once they have been burnt.