It's been about 2 years since I've left academia, partly because of the kind of deeds that I'm about to describe to give a bit of context to the question.
About a month ago I was CCed in an email exchange involving my former PhD supervisor and one of our collaborator, a Post-doc in Mathematics (from now on referred to as the Post-doc). The email said that one of our papers, that we submitted to a journal about 2 years ago (that's right) was finally rejected (better late than never I suppose).
Recently, I have received a letter from the Journal of Functional Analysis that informed me that I have been indicated as a co-author of a publication recently submitted by my former supervisor. It is of course yet another attempt to get the paper published somewhere. Although the fact that the choice of journal is a questionable one (JFA is published by Elsevier, which I would gladly boycott, but this is a whole different matter I won't get into), what startled me was the fact that the Post-doc has been indicated as First Author.
Now, a clarification about the paper is due in order to understand the situation better. Most if not all of the original draft of the paper was written by me, and it pretty much contained most of my research work, which then turned into my PhD thesis together with some later results that I have obtained. There have been times when I got stuck on some points, in which cases a few chats with my supervisor helped overcome the problems. The contributions from the Post-doc towards the final version of the paper are, to say the least, very questionable. One way to summarise them is perhaps with the expression unintentional sabotage. The Post-doc kept changing sentences in the paper because he thought we was improving it. I'm not an English native speaker and nor is the Post-doc, but his English was definitely worse than mine and, as a consequence, I had to spend hours reverting his changes, over and over again. Not to mention his competency with the mathematical content of the paper itself. A good part of the time we spent in meetings to discuss the contents at the board, it was basically me (a PhD student in Pure Math with a degree in Physics at that time) recollecting basic school maths facts (literally) for the sake of the Post-doc who didn't seem to remember them (or know them altogether).
Don't get me wrong on this though. I am in no way implying that there is an voluntary attempt at bad deeds here. Knowing my former supervisor, this is what I think it is going on with the paper submission. Given that I am no longer in academia, it benefits nobody if the paper is submitted with me as first author. Hence it would make more sense to indicate the Post-doc as first author since he is still somehow making his way in the academic world.
Whilst I would totally support this decision in general as very sensible, I find it hard to go by it in this particular case, given the scenario I have described above. By allowing the Post-doc to be first author, my former supervisor is (I believe inadvertently) helping someone to be where he's probably not supposed to be, taking the chance away from someone who is more qualified. You wouldn't believe how many successful applications the Post-doc has had, given his actual knowledge on the subject. The only explanation that I could come up with is that, somehow, people are happy to offer the Post-doc a place at their departments to enjoy of his buffoonery (possibly his only positive aspect).
Now, after this long introduction, here comes the question: Should I notify the journal that I am the actual first author?