The Missing Collaborator:
when a collaborator stops contributing and the clock is ticking for the other
Everything I suggest here works when coauthors are getting along fine and there is a simple honest reason why the one coauthor cannot contribute. Most situations I know where this has occurred has been due to a collaborator being caught up in a job hunt or becoming the chair of their department. These people can be excellent coauthors in the future, so don't ruin the relationship over one paper. Other times the situation is more difficult but the following suggestions still may work. Assuming one has already explained to the coauthor that there is a deadline looming and the coauthor cannot or will not contribute, one possibility is to complete the paper on your own, keeping them as a coauthor and hope they don't stall the revision process. If they have no time to work, they should at least, allow you to complete the work and post it on the arxiv and submit it with their name on it. If they suggest that they don't trust your work, then they may suggest you only give them acknowledgements or put their major contribution in a coauthored appendix or break up the work into coauthored and uncoauthored pieces. If you cannot complete the paper on your own, because there is some key step you cannot handle, this is where you might ask the coauthor if an additional coauthor could be brought in to help complete the project. This additional coauthor might just write an appendix and not become a coauthor on the whole project or may join in on the whole project. If it is really only one small lemma, it goes in an appendix. I do not recommend dropping a coauthor at the write up stage. This is very dangerous as the coauthor may view you as a thief. I know situations where coauthors wrote one paper together and second related paper was sole authored. The abandoned author publicly spoke of the other author as a thief, even though the abandoned coauthor had not had time to contribute to the second paper beyond the initial formulation of ideas. The abandoned coauthor expected the first coauthor to wait. This damaged the first coauthor's reputation. The damage done depends on the power of the coauthor and the people the coauthor knows. In the very least, faculty who work at institutions with less time for research mathematics, become wary of faster coauthors who go ahead on their own. If you are resentful that you contributed too much to a project, I would recommend that you just avoid coauthoring with that person again.
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