The Pope Is Pope, Wherever He Lives
Does the papacy have to be in Rome? Could the pope change it and be the Bishop of Paris instead?
It is not Rome that "inherited" the authority of St. Peter but St. Peter's successors. St. Peter died in Rome, and that is where he was succeeded by Pope St. Linus.The pope is still the pope, no matter where he is. At one time valid popes were stationed in Avignon, France.
Rome is where the successor of St. Peter has traditionally been bishop, because that is where Sts. Peter and Paul died, and it is historically where the pope has almost always been (except for the Avignon years). Early Church Fathers often referred to Pope St. Linus as succeeding Peter in Rome. The bishops of Antioch and Jerusalem cannot claim to have succeeded St. Peter, because Peter was still active after leaving those cities.
The bishops of Rome were seen as the high authority in the early Church, evidenced by Pope Clement's letter to Corinth in which he has no doubts that he has the authority to intervene in the local Church dispute. And by the third century St. Cyprian made no hesitation to refer to Rome as the chair of Peter and "the principle Church."