What is the ruling regarding someone who swore on the Quran that he did not cheat on the exam, even though he looked at the cheating paper and had memorized the rule written on it? What is required of him?

Looking at the textbook during an exam to use it for help is forbidden cheating. If you swore not to cheat, and this is what happened from you, then your oath is considered a false oath, and it is called "al-yameen al-ghamūs" (the immersed oath), and it has no expiation except sincere repentance, according to the more authoritative opinion. It is advised to repent, seek forgiveness, and abandon cheating in the future.