Well, Peter did spend some time in Antioch, possibly in Corinth and died in Rome.
Peter spent time in Antioch of Syria...rightfully so as this was part of the old Assyrian Empire, and many descendants of the Israelites of the dispersion still resided in the area.
[1 Corinthians 1:12] Now this I say, that every one of you saith, I am of Paul; and I of Apollos; and I of Cephas; and I of Christ. There is no indication by this verse that Peter was ever in Corinth. Paul is only speaking of divisions with in the Church itself.
There is only tradition that says Peter died in Rome....and this would be suspect as he received divine instruction not to go there. [Matthew 10:5] These twelve Jesus sent forth, and commanded them, saying, Go not into the way of the Gentiles, and into any city of the Samaritans enter ye not.