What was the relationship between the Catholic Church and the Carolingian emperors?

For Charlemagne's feelings about being crowned emperor, I quote from The Civilization of the Middle Ages by Norman F Cantor, chapter six "The Making of Carolingian Kingship":

On Christmas day, 800, as Charlemagne rose from prayer before the tomb of St. Peter, Pope Leo suddenly placed the crown on the king's head, and the well-rehearsed Roman clergy and people shouted, "Charles Augustus, crowned great and peace-giving emperor of the Romans, life and victory!" Charlemagne was so indignant and chagrined that, according to Einhard, "he said he never would have entered the church on that day, even though it was a very important religious festival, if he had known the intention of the pope." Charlemagne did whatever he could to mollify the outraged Byzantines, who claimed that their imperial title had been stolen from them. He hardly ever used the title emperor of the Romans, which the pope had given him, but satisfied himself with the phrase, "Emperor, king of the Franks and Lombards" to indicate the real and effective basis of his power.

The imperial coronation of Charlemagne has engendered considerable controversy among historians, many of whom have dismissed Einhard's statement as excessive modesty on Charlemagne's part. The fact is that Charlemagne did not want to be crowned emperor of the Romans because first, "Roman" meant "Byzantine" to him, and he had no desire to emulate the ruler in Constantinople, and second, because he understood the constitutional implications of a papal coronation and had no intention of placing himself in a position of debt or weakness to the bishop of Rome. What makes the situation more complex, however, is that an imperial ideal was coming to the fore among the churchmen of the Carolingian realm, but it was not the same concept of the empire that prevailed either at Constantinople or at Rome. The letters of Alcuin in particular are full of references to the "Christian Empire" and to "Europa", the area contiguous with Latin Christianity whose leader was Charlemagne. In view of Charles' contributions to the welfare of Europe and in view of his position as the greatest king in Europe, Alcuin and other court churchmen were beginning to think that Charlemagne ought to take the title of emperor. This view, however, had little to do with emulation of the old Roman emperor or the ruler in Constantinople; rather, it was intended to be the apotheosis of Charlemagne's position as the leader of Christendom. It is likely that an imperial coronation of Charlemagne would have taken place had not the pope forestalled the Frankish king and his advisers on Christmas day 800. Certainly Charlemagne would not have allowed himself to be crowned by the pope; the coronation he preferred was the one used in 813 when he crowned his son and heir, Louis, emperor.

Having been crowned by the pope, Charlemagne chose to interpret his imperial title in the way delineated by Alcuin. He refused to think of himself as a Roman emperor, ignored the sanctions that were implied in his coronation by the pope, continued to call himself king of the Franks and Lombards, and regarded the title of emperor as the expression of his position as Christian war hero, theocratic monarch, and leader of the Frankish church.

The imperial ideal played a much more important role in the policies of Charles' son and grandson, Louis the Pious and Charles the Bald, and it became a concept whose content was much more heavily influenced by the original papal ideology. The ninth-century Carolingian churchmen moved away from the Christian empire of Charlemagne and in the direction of a political antiquarianism that sought the full revival of Roman imperial ideas by imitating the ornate court ceremony of the Byzantine emperors and by using the full title, emperor of the Romans. Already in 816 Louis the Pious allowed himself to be anointed by the pope with this title. To the ninth-century Carolingian rulers and their ecclesiastical supporters, emphasis on the imperial title and the association of the Carolingian ruler with the Roman emperors were a buttress against the progressive decline of royal power after Charlemagne's death. Ideology became a substitute for Charlemagne's fame as a Germanic war leader. But ideology could do nothing to stem the advancing tide of localism and the rise of feudal lordship. The ninth-century bishops composed treatises on the glories of empire and kingship and the Carolingian emperors elaborated their court ceremonial, but they were unable to maintain effective leadership in their kingdom.

The papacy, over the long run, gained no more than the Carolingians from the revival of the imperial title in the West and from the acceptance by the Carolingians of the Romanist ideology. The mid-ninth-century pope Nicholas I aggressively asserted the radical doctrine of the Donation of Constantine, and the popes were adept at using their control over the imperial title to harass the later Carolingians, but this did not save the papacy from disaster in the late ninth century. For the popes needed a strong Carolingian ruler to protect them from the gangster Roman nobility. With the decline of Carolingian power, the papacy entered one of its darkest periods, in the late ninth and first half of the tenth century, in which it became the puppet of the ruling Roman nobility and completely lost its position as a leader in European society.

If the history of the ninth century is one of failure on all sides, it should not blind us to the fact that a new element had been introduced into the political life of western Europe. In the latter part of the tenth century the title was taken up again by the German monarchy, which rose out of the ruins of the east Carolingian kingdom. The German kings were to make the imperial title an essential part of their policy until the middle of the thirteenth century, and their successors were to preserve the title until 1806.