Looking at this Intel core i7 nehalem micro-architecture

it seems that each core has its own private Register File. So I have a couple of short questions, because I thought that there is only 1 set of registers not dependent on the number of cores.
- Does each core have its own private set of registers? (RAX, RBX, RSP and so on).
- Does each core have its own MMU and TLB? Not just one shared across all cores?
I know the questions are highly micro-architecture dependent but I think the majority of modern x64 Intel cpu's follow the same design principle.