What is the historical plot of Murder in the Cathedral by T. S. Eliot?

The historical events behind Eliot's verse
drama pertain to an on-going quarrel between King Henry II--who is reported to have
shouted, "Will no one rid me of this turbulent priest?"--and Thomas Beckett, who changed
his extravagant life for one of ecclesiastical devotion once he was made Archbishop of
Canterbury. The quarrel centered around a disagreement over what law had authority over
clergy who committed secular (non-religious) crimes.

King Henry said
that even clergy were subject to the law of the land of England, especially since
longstanding custom in England supported the practice of secular law and courts ruling
over clergy's secular crimes. This sounds reasonable to a modern reader, but in the
1100s, religion and the Church had great and powerful authority--and seemingly desired
more.

Archbishop of Canterbury Thomas Beckett was newly appointed by
King Henry II to be his representative and advocate in the Church of England, a decision
that garnered much disfavor because Becket, a military leader, had never been in Holy
Orders. Becket's position was that clergymen, even those who had taken lower orders of
the clerical hierarchy, were exempt from secular law and could only be tried and
punished within ecclesiastical, or Church, courts.

This disagreement,
a serious one since approximately one-fifth of the male population of England was in
clerical orders (and occasionally given to secular crime ...), was escalated because the
crime rate had increased and criminals were going unpunished because, when clergymen,
they claimed the right to be tried in Church courts.

This was
compounded by Becket's determination to reclaim previously confiscated Church lands and
by the complaints this land reclamation effort engendered from the present occupants of
the land. Becket and his family were finally exiled after long and complicated disputes
over the authority to rule in criminal offenses, a dispute that finally fell to Henry's
victory, and the restoration of the customs from his grandfather's day and the
recognized authority of the Crown's courts over all secular crime, even that committed
by clergymen.