I just looked over this article from Retraction Watch.
To make a long story short, a student used previous data from their former lab and then published it without the PI's permission or knowledge.
While the above is a rather extreme example of poor ethics, I'm curious where the line gets drawn regarding the ownership of research. I can understand if the data was produced in one environment and then get published in another. What about reagents (clones come to mind) that were produced in a previous lab and then were transported to a new lab? What about ideas that were developed in one lab and then taken to another?
I think that so long as the principal investigator is actively involved in the planning, performance, or analysis of the research being funded, it is the duty of any researcher working in that group to determine the PI's status as a co-author. However, only if there is no active intellectual activity taking place—in other words, it's an entirely self-driven initiative, then it might be possible to say that the PI doesn't merit co-author status. (Even then, though, the provision of financial support should be clearly recognized.)