It is important to differentiate between the transponder and central locking elements of a modern car key.
Central locking keys often use Radio Frequencies of around 433MHz and operate a rolling code system to help prevent code harvesting and use to open your vehicle. This was possible and prevalent in the UK some years ago. The rolling code prevents this type of attack and relies upon synchronising keys and the car security system.
The transponder is a totally different and separate part of the keys functionality. The communication coil passes data from the keys ID chip to the cars security system. The keys ID does not change. This is why a car is married to a particular set of keys.... it has to be programmed with all expected key ID's. The transponder in my Audi Key is a small block of plastic and contains the same type of technology that you would find in a Dog/Cat ID chip. Basically a communications coil connected to a tiny IC. The whole transponder is probably in the coil assembly on your key. Take a close look at it, as it is one of the heavier and more vulnerable components. PS The transponder does not normally require any power from the key itself. Power comes from inductive coupling between the cars steering column coil and that of the key.
If you Google 'RFID tags' you should find heaps of info on the technology
EDIT - Looking at the picture again I can see that the coils has a capacitor across it, so is a tuned circuit forming the low frequency aerial tuned to the operating frequency of the communications link. The ouput of this aerial appears to be going to the chip to it's right. This would suggest that the key uses a custom IC that handles both the central locking and the transponder functions in one package. The transponder is still a separate function however and there is no rolling code. You shoudl check all connections from the transponder aerial to the IC and ensure they are not cracked.