How is it possible to teach something as fact to people when you have no details to verify it as fact??? Here is a statement from a Catholic author that is closer to the truth regarding the legend of Peter in Rome:
"All the essential claims of the modern papacy, it might seem, are contained in this Gospel saying about the Rock, and in Irenaeus' account of the apostolic pedigree of the early bishops of Rome. Yet matters are not so simple. The popes trace their commission from Christ through Peter, yet for Irenaeus the authority of the Church at Rome came from its foundation by two Apostles, not by one, Peter and Paul not Peter alone. The tradition that Peter and Paul had been put to death at the hands of Nero in Rome about the year AD 64 was universally accepted in the second century, and by the end of that century pilgrims to Rome were being shown the 'trophies' of the Apostles, their tombs, or cenotaphs, Peter's on the Vatican Hill, and Paul's on the Via Ostiensis, outside the walls on the road to the coast. Yet on all of this the New Testament is silent. Later legend would fill out the details of Peter's life and death in Rome--his struggles with the magician and father of heresy, Simon Magus, his miracles, his attempted escape from persecution in Rome, a flight from which he was turned back by a reproachful vision of Christ (the 'Quo Vadis' legend), and finally his crucifixion upside down in the Vatican Circus in the time of the Emperor Nero. These stories were to be accepted as sober history by some of the greatest minds of the early Church--Origen, Ambrose, Augustine. But they are pious romance, not history, and the fact is that we have no reliable accounts either of Peter's later life or of the manner or place of his death. Neither Peter nor Paul founded the Church at Rome, for there were Christians in the city before either of the Apostles set foot there. Nor can we assume, as Irenaeus did, that the Apostles established there a succession of bishops to carry on their work in the city, for all the indications are that there was no single bishop at Rome for almost a century after the deaths of the Apostles. In fact, where ever we turn, the solid outlines of the petrine succession at Rome seem to blur and dissolve...
" Neither Paul, Acts nor any of the Gospels tells us anything direct about Peter's death, and none of them even hints that the special role of Peter could be passed on to any single 'successor'. There is, therefore, nothing directly approaching a papal theory in the pages of the New Testament [Duffy, Eamon. Saints & Sinners: A History of the Popes. Yale University Press, New Haven (CT), 2002, pp.2,6).
No PRECISE details.