One could argue that after what Tarkin did to that Imperial facility, the Emperor might instruct Vader to keep a close eye on him, lest the good Governor get any funny ideas.
Well, keep in mind that the Emperor also once (if you accept the prequels) created an entire robot army and an entire clone army, had them fight each other in a destructive civil war and then called the whole thing off in the matter of like 20 minutes killing everyone important on both sides.
I don’t think he’d have a problem with what Tarkin did.