You are Peter...

the largest plurality of the Fathers that comment on this say that Peter's confession is the rock, and we are all Peter if we confess Christ. now, while some do say that Peter is the rock himself (others say Christ refers to the Apostles as a whole, others to Himself), none tie in universal jurisdiction or infallibility to that claim.

Even if they did, Peter is not exclusively connected with Rome?

It is no problem to admit that Peter is the rock. In fact, it would be silly to deny this since Christ changed Simon's very name to Peter. The real question is what it means that Peter is the rock. Roman Catholics take it for granted that this proves Papal Supremacy, though this fact alone is a few steps removed from actually accomplishing what Roman Catholics think it does.

It is not enough for Roman Catholics to demonstrate that Peter is the rock. They must demonstrate that the Bishop of Rome is the unique successor of Petrine authority, and they must demonstrate that this Petrine authority includes making infallible statements and being the necessary point of unity for the Church.

Kylissa said:

Even if they did, Peter is not exclusively connected with Rome?

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true, he also founded Antioch and actually functioned as a bishop there (unlike Rome).

If I recall correctly, the RCC celebrates (celebrated?) the Feast of the Chair of St. Peter in Antioch on February 22, so they do or did at one time acknowledge his time as a bishop in that city, as well. When I've asked my RC friends about that, they've tended to minimize it by saying that while it is true, Rome is still more important because that's where he was martyred.

ArmyMatt said:

true, he also founded Antioch and actually functioned as a bishop there (unlike Rome).

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That's what I was thinking. I knew he founded Antioch, and I thought the claim actually made more sense, that "authority" if you want to press it, is more reasonably connected with being the bishop of a see, rather than being martyred at a particular location.

Οn this rock= Ταυτη πετρα.

Both the noun and the article is feminine. The word "ταυτη' is the same word used to end the paragraph; "and the gates of Hades shall not prevail against her"(αυτης)".
So both words- petra and ekklesia are in the feminine. So Christ is saying "upon her" He will build his Church, That is a foundation based upon Peter's confession.