Suppose that I found a technical flaw in one of the papers coauthored by my previous advisor. This mistake will substantially change their major result, but their contribution will be still worthwhile after the correction. I am 90% sure about my findings. It is not a difference of ideologies or subjective beliefs, but a mistake in purely technical analysis.
My previous advisor is among the top-three worldwide in his narrow field, and that paper was published on the top journal. He was neither the corresponding nor the first author. The first and corresponding author was also his previous student. So here is my possible options (any other suggestions are also welcomed):
Write a short paper and submit to the same journal. The journal welcomes this specific type of submissions. Best case is, if my previous advisor is happy to see his previous students are still contributing to this field, then I will get a top publication. Worst case is, since many of the referees are also his previous students, they could possibly defend our teacher and deny my submission, and I lost all the precious time redoing the analysis and writing the paper.
Talk to my previous advisor. Since he keeps constant contact with the journal’s editor, he can definitely update that mistake, much faster than me; that way I will lose a worthy publication. He can also decide to ignore my findings or even try to prevent my findings to be published because they significantly challenge his result.
How can I choose between these, or is there a third way?
Update: I am preparing to send a friendly inquiry, politely say things like:
Dear XX, It was a pleasure to read your exciting work titled XXX. Could you do me a favor by explaining your analysis in ABC? I was hoping to do analysis XYZ such that your results can be refined and here are my preliminary findings. Do you feel this approach promising? Would you like to help? Thanks for sharing thoughts, again.
How do you feel about the tone? Shall I be more explicit?
Update: after talking to multiple mathematicians about erratum, I find that even the most honest, objective, and rigorous mathematician will be deeply embarrassed in heart with new mistakes found in his paper.