In 1108, the Archbishop of York was Thomas II of York. Thomas refused to profess obedience to the Archbishop of Canterbury. The cathedral chapters of each place supported their archbishops. The lay investiture debate—whether secular lords like the king could determine religious titles—was raging, and it would have been against the church's best interests to appeal to the king for help in resolving their primacy issue.
The question had already been resolved in 1071 when Anselm's predecessor, Lanfranc, and the Archbishop of York, Thomas of Bayeaux, appealed to Pope Alexander II, who sided with Canterbury (although he refused to resolve a different dispute between the two, leaving it to the king's council).
Thomas of York's point was that he had received his pallium directly from Pope Paschal II (he had sent the dean of York Cathedral to Rome to fetch it). He therefore felt he did not owe anything to Canterbury.
An old and ill Anselm, from his sickbed, declared anathema any who failed to recognize the superiority of Canterbury. This was a sign that excommunication was on the table, and Henry stepped in, ordering Thomas to profess obedience to Canterbury. Anselm, hearing this, declared himself content. He died 21 April 1109, and was interred near Lanfranc at Canterbury Cathedral in what is now St. Thomas's Chapel. His current remains are unknown, having been relocated after a fire in the 1170s.
Anselm left a body of writing that is considered some of the sharpest intellectual discussions of Christianity in the Middle Ages. They were copied and spread far and wide, and influenced many including Bonaventure, Thomas Aquinas, Duns Scotus, and William of Ockham. He also spoke out strongly against slavery, although his declarations did not bring about any changes.
His lifelong battle to separate church authority from lay investiture had an unintended consequence, that put Pope Paschal at odds with Anselm's successor. I'll explain the Pope's problem with England and the next Archbishop of Canterbury tomorrow.