2001 Grand Prix Starting issue

I have a '01 Grand Prix that I am having an issue with. The other day I went out and let the car warm up (Wisconsin) in the morning. I then went back and shut the car off. When I returned later the car would start and then run for about 4 seconds. The accelerator would work fine for that short time but then it would just quit. I have heard about problems with the security chip on the key but the car does the smae thing with both sets of keys I have. Battery posts are clean with good contacts and there are no security codes. Any ideas? Thanks, Eric

When it wont start/stay running is the security lite staying on

it sounds like it could be a security issue, but more likely it is a fuel relay. if it is a gtp, it has a two speed fuel pump, thus having two relays. if it is a the high speed relay, it is probably just the resistor:
http://rmcgp.clubgp.com/fpresistor.html

which are very common to go out especially in extreme weather temps. this procedure is definitely a DIY, and your symptoms definitley point to it.