I recently found a mistake in a mathematical proof of a result I published in 2016. In a nutshell, a subtle error led me to claim that two conditions were equivalent. Unfortunately one of the two implications is not true. Luckily, the main result of our paper relies only on the correct implication so the main conclusion of the paper is correct. How should I handle this?
The most important step to take is to contact the journal and tell them that you wish to publish a corrigendum to your article. Journals have systems in place for making the existence of a corrigendum clear to readers who search for the original journal article. (So does MathSciNet, by the way—which reviews journal articles, not arXiv articles.)
When people cite articles, they cite the journal versions (when they exist, such as in this case). Therefore people who are reading a future article that cites your journal article will believe, quite reasonably, that they will be accessing the best version of the scientific record if they go to the journal article. A correction on a personal web page or the arXiv does not accomplish this important contribution to the scientific record, causing all kinds of problems (as user124672 mentioned in their answer).
user124672 also makes the valid point that publishing in a venue where somebody disinterested in the article has version-control power over the article (that is, editors of journals) is very important for establishing what is and isn't the definitive version of the article—which is again crucial for the scientific record. A corrigendum is the process by which an author can convince an editor to use their version-control power to update the definitive version in rare circumstances (and the rarity, and necessity for a disinterested party to agree, are features and not bugs of this system). Precisely because authors can update their arXiv and personal web page articles any time they want, such updates cannot ultimately be considered trustworthy in the careful, scientific sense.
By all means, also correct the arXiv version (with a clear explanation of what has changed), as well as the version on your personal web page—but only as supporting actions to the crucial one, which is to correct the scientific record by updating the journal article itself.
(One could imagine a world where later articles cite arXiv papers even when journal articles exist, and where arXiv has some structure in place to prevent frivolous author updates; in that hypothetical world, one could reevaluate the relative importance of the above steps. However, in our actual world, where citations are invariably to stable journal articles, I stand by the above statement.) (One can also reasonably criticize the limited-access and even financially predatory nature of many mathematical journals—I have done so myself—but that discussion about the optimal relationship between academia and capitalism does not alter the facts above.)
Please. I beg you. Do not update on arXiv a paper that was formally published years ago. This is a recipe for disaster.
Once a paper has been published, a significant number of people will only look at the published version and not bother downloading the arXiv version (because clicking the "DOI" link on mathscinet takes you there). A significant number does the exact opposite and just downloads the arXiv version without bothering to go on the publisher's website, maybe because they don't have access or can't bother looking for it.
So a portion of your readership will be aware that the result is incorrect, the other half will not, and they have basically no way of knowing that others are (un)aware. When they collide, e.g. when one cites the (in)correct version of your theorem and someone else only has the other version: things can get very confusing! I once had a referee essentially call me an idiot because I cited a Lemma X.Y from some paper, and they said the paper's lemma were unnumbered: well, the published version's was numbered, while the arXiv version wasn't... This took a while to sort out. If this happens for something as trivial as lemma numbering, can you imagine what would happen for mismatches in results?
I would also say that most people hold the view that the published version is the "definitive" version of a paper (I certainly do). Once it's published, the paper is done, and anything new goes into new papers. A few years ago, someone's name was floating for the Fields Medal, but they were not chosen. One of the rumors about this disappointment is that the researcher in question keeps putting out updated version of papers/books that were published, tweaking the results, the proofs... So one was never really sure what was correct and what wasn't in the papers. I wasn't on the Fields committee, obviously, so I don't know for sure, and there were certainly a truckload of politics involved, but this definitely seems like a plausible argument. (I won't say who the researcher is. Those who know also know that this researcher isn't otherwise lacking awards, anyway.)
So, what to do? As I see it, you have a few options, in order of severity:
- Do nothing. I strongly advise against that. People are going to read your paper and some will not notice the error.
- If you can find a relevant place to do it, mention in a new paper the error and write down the correct version of the theorem. At least people can cite and link to the correct version. But this is not very discoverable.
- Put out a corrigendum. This can just be a note on your webpage if you want, as you seem to have already done. You can also just post something on arXiv with the title "Corrigendum for title of your article" and explain what was wrong and how to correct it. You can also ask the journal to publish the corrigendum alongside the paper to make sure people see it. In all cases, when you do this, it is fine to update the arXiv version, not to change the text (which should stay as it) but to add a note in the comments section with a link to the corrigendum.
- If the error completely kills the paper, ask for a retraction. This is somewhat extreme, so I'll leave it up to your conscience. This doesn't seem to be the case here if I read you correctly.