How do I fix someone's published error? Paper A is in the literature, and has been for more than a decade.
An error is discovered in paper A and is substantial in that many 
details are affected, although certain fundamental properties 
claimed by the theorems are not.  (As a poor analogue, it would be 
like showing that certain solutions to the Navier-Stokes equations 
had different local properties than what were claimed, but that the 
global properties were not affected.  The error is not of the same
caliber as Russell's correction of Frege's work in logic.)
The author is notified, who kindly acknowledges the error.
Now what?
Should the remaining action lie fully on the author, or should
the discoverer of the error do more, such as contact the journal,
or publish his own correction to paper?  How long should one
wait before suitable action is taken?  And what would be 
suitable action if not done by the author?
Based on remarks from those who previewed this question
on meta.mathoverflow, I propose the following
Taxonomy:  There are various kinds of error
that could be considered.


*

*typographical - An error where a change of a character or a
word would render the portion of the paper correct.  In some
cases, the context will provide enough redundancy that the
error can be easily fixed by the reader.  Addressing these
errors by errata lists and other means have their importance,
but handling those properly is meant for another question.

*slip - (This version is slightly different from the
source; cf the discussion on meta for the source
http://mathoverflow.tqft.net/discussion/493/how-do-i-fix-someones-published-error/ )
This is an error in a proof which may be corrected, although
not obviously so.  In a slip, the claimed main theorem is either
true or can be rescued with little cost.  In my opinion, the degree
of response is proportional to the amount of effort needed to fix it
(and is often minor),
but there may be slips major enough to warrant the questions above.

*miscalculation -  Often a sign or quantity error.  In some cases
the results are minor, and lead to better or worse results
depending on the calculation.  I've included some miscalculations
in some of my work to see if anyone would catch them.  I've
also prepared a response which shows the right calculation and
still supports the main claims of the work.  (See below on
impact as a factor.)

*oversight or omission - This is stating a fact as true without
sufficient folklore to back up that fact.  In some cases the
author doesn't include the backup to ease (the reading of) the
paper and because the author thinks the audience can provide 
such backup.  More seriously,
the omission occurs because the author thought the fact was
true and that there was an easy proof, when actually the fact
may or may not be a fact and the author actually had a faulty
argument leading him to think it true.

*major blunder - This is claiming a result which is true,
and turns out not to be true in a socially accepted proof
system.  Proofs of Euclid's fifth postulate from the other
four fall into this type.
The above taxonomy is suggested to help determine the type
of response to be made by the discoverer.  Also, degree of
severity is probably not capable of objective measure, but 
that doesn't stop one from trying.  However, there are at
least two other considerations:


*

*Degree to which other theorems (even from other papers)
depend on the error in the result.  I call this impact.

*Degree to which the error is known in the community.
The case that inspired this question falls, in my mind,
into the category of a miscalculation that invalidates a 
proposition and several results in paper A following from
the proposition.  However, as I alluded to above in the 
Navier-Stokes analogy, the corrected results have the same 
character as the erroneous results.  I would walk on a 
bridge that was built using the general characteristics of 
the results, and not walk on a bridge that needed the 
specific results.  In this case, I do not know to what 
degree impact the miscalculation has on other papers, nor 
how well known this miscalculation is in the community.
If someone thinks they know what area of mathematics my case 
lies (and are sufficiently experienced in the area), and they 
are willing to keep
information confidential, I am willing to provide more
detail in private.  Otherwise, in your responses, I ask that no
confidentiality be broken, and that no names be used
unless to cite instances that are already well-enough
known that revealing the names here will do no harm.  Also,
please include some idea of the three factors listed
above (error type, impact on other results, community awareness),
as well as other contributing factors.  
This feels like a community-wiki question.  Please, one
response/case per answer.  And do no harm. 
Motivation:  Why do I care about fixing someone else's error? 
Partly, it adds to my sense of self-worth that I made a 
contribution, even if the contribution has no originality.
Partly, I want to make sure that no one suffers from the
mistake.  Partly, I want to bring attention to that area
of mathematics and encourage others to contribute.  Mostly
though, it just makes an empty feeling when one reaches
the "Now What?" stage mentioned above.  Feel free to
include emotional impact, muted sufficiently for civil
discourse.
Gerhard "Ask Me About System Design" Paseman, 2010.07.10
 A: Sometimes reviews in Math Reviews (i.e. MathSciNet) contain a discussion of errors in the paper under review.  This can be very helpful: it is a quick way to communicate the nature of the error to the mathematical community, and even to give a brief explanation of how the error can be fixed (when a fix is known).  So in situations where a fix can be described succinctly, updating the review seems like a good way to fix the error, especially since (I think) most readers of a paper see this review.  Of course this requires some work on the part of the reviewer, if there is one.  Does anyone know if this approach is ever used (after the initial posting of the review)?  
A: Some advice explicitly directed at less senior people.  I would very much advise some who does not yet have tenure to NOT take the nuclear option (e.g. posting a paper on the arXiv accusing someone of being wrong, or writing irate letters to the editors of a journal).  In the extremely rare cases in which this has to be done, it is best done by someone who is both pretty senior and very politically skilled.  This leads me to my other piece of advice.  Namely, talk to other, more senior people in your research area.  First, they might be able to convince you that it isn't really as serious an error as you think.  Second, they will probably know the personalities involved better, and be more effective at convincing an author to do the right thing if something has to be done.
The two times something like has happened to me, I had ended up proving stronger results than the erroneous papers by pretty different techniques.  I buried remarks at the ends of the introductions of my papers mentioning the wrong papers and explaining where they went wrong.  On one of those occasions the author had left math and I didn't know how to contact him, so I didn't correspond with him first (after I posted the paper the arXiv, one of his friends contacted him and we exchanged some friendly emails).  The other time, I explicitly cleared the language I used with the original author.
A: If I published a mistake, then I would be grateful to hear about it. Firstly it shows that someone has actually read my work. Far worse would be to discover your own mistake years later and realise that either nobody read it or nobody cared enough to correct it.
Of course you may be mistaken that there is a mistake, so you shouldn't just say flat out: "there's a mistake", but rather "I believe there's a mistake, and here's what I think it should be. What do you think?"
What Now? Discuss the "what now" with the person involved. If you both agree with what should happen then great. If you disagree about there being a mistake then talk privately with others to see if they agree with you or not. If all else fails then publish the corrections yourself if you think it is worthwhile to do so, but firstly tell the person that you are going to publish, as that gives them one last oppotunity to do it themselves.
A: Something I have done in a similar situation is to write about the content of the paper on the nLab wiki, including the correction.  I think this is a great solution, because putting the material on a wiki is itself a service to the community and to the original author (increasing the potential audience for his/her work and helping to disseminate it), and I think including the correction when you do so is unobjectionable, since the culture of a wiki (anyone and everyone is expected to add to it, improve it, and fix mistakes) is different from the culture of publishing refereed papers, or even arXiv preprints.
Of course, this may not be an option when working in a field where there doesn't yet exist any wiki comparable to the nLab -- but then clearly the solution is to start one!  (-:O
A: I've been in various analagous situations, and the following is a dream-list for how I wish things had worked.
In an ideal world, one should be able to make non-essential arxiv updates (typos etc.) on published papers without it generating a new version. This would encourage people to correct minor errors, which can actually seriously confuse a reader. I know that I would not put a new version of a paper on arxiv after catching typo-level errors, although I also know that this is bad for readers.
For more major errors, I wish people would keep errata for their papers on their webpage. For fatal errors, published errata may be necessary.
Politics gets in the way of this working properly, because the incentive to "be friends" is stronger than the inventive to plug minor holes in correct results. This would not be the case if we were doctors working towards a medicine to save the lives of sick loved-ones, for example... then we would care about catching the minor errors too. And so, I wish people aimed for perfect papers more seriously.
A:  UPDATE 07.24  : The set of answers for this question seem to have
stabilized.  I encourage all who visit this question to review all of the answers and comments 
posted here and posted behind the meta.mathoverflow link in the question.
This answer has an incomplete summary; you might find what you need in one of the
other posts.
 END UPDATE 07.24
Thanks to all who have contributed thus far.

* I liked
Igor Pak's notion of giving the author the same amount of
time as a referee to have the author do the fix on his own.
I also liked his list of potential responses, including
alternatives to avoid or use as a last resort.

* I appreciated Pete Clark who pointed out that the emotional
impact on the author may be considerable.

* I thought algori's
notion of substantial error (one that could not be fixed with
the methods used in the paper) was a good benchmark regarding
severity of error.

* I and others liked Andy Putman's advice
to seek out, shall we say, more experienced counsel before
acting.

* I thank Timothy Chow for offering an alternative
(doing the work for the author of publishing the error) that
may well fit my situation.

* I also thank Mike Shulman for
his notion of wikifying the correction; perhaps authors who
have been so corrected could weigh in on this so that we could 
determine what social/emotional/academic impact this method
might have.

* I thank Daniel Moskovich for his inspiration to
move this society toward a perfect world, at least with respect
to correcting errors in papers.

* Also, I want to acknowledge
the common sense in unknown(yahoo)'s suggestion to continue
discussion with the author.

Based on the input so far, I am going to suggest the following
as an answer template, to be modified at the dictates of
common sense, decency, and situational factors. 
Recall the assumption that the author has been
contacted already and acknowledges the error.


*

*Consult with one or more colleagues in the field who
can evaluate the error and suggest a course of action.
If they suggest dropping the matter, then stop.

*See if the journal involved has already published a 
correction.  If so, then stop.

*Contact the author again after a period of time
(3 to 6 months) and ask what the author thinks is
an appropriate action to take.  Offer to assist in
writing up a correction, at little or no cost to the
author.  If the author suggests a reasonable course
of action, follow it.  Then stop.

*Prepare your own version of the correction.  If the
author has not acted in good faith, and if the colleagues
encourage the idea, mail the author a copy of the correction
as well as a stated intention to post the correction in
3 to 6 months if the author has problems following up with
providing his own correction.  Keep the correction for your files.

*If a year has passed since the acknowledgment, and several
months have passed since you announced your intention of
posting the correction, then (given that it is a good thing
to do) post an announcement saying what is being corrected,
and provide a link to the details.  
In the above, do no harm.  In particular, approach the
situation with the attitude that, regardless of how poorly
the author might respond, the goal is to provide a correction
to the academic audience, with as much or more sensitivity 
and respect due to the author as you would expect for yourself.
It is possible a better answer exists out there.  If someone
can provide it or a link to it, I will acknowledge it.  If
this answer gets a substantial number of votes from the community,
then I will accept it, with the understanding that the other
posters contributed to this answer.  In any case, I believe
this question and all the answers will serve as a helpful resource
to those who find themselves close to this situation.
Gerhard "Don't Need No Stinkin' Badges" Paseman, 2010.07.12
A: No-one's made a similar comment yet, so I'll add my 2 cents.  I think that


*

*Degree to which the error is known in the community.


Should, ideally, have no effect on the original author's actions.
I say this as someone who quite likes to move between different (sub-)subject areas, and who's  a bit anti-social, and who picks up old problems etc. etc.  In other words, I'm often trying to learn things directly from papers, without much help from a "community".  It would be incredibly frustrating to find an error, to work at correcting it and/or understanding a work-around, only to find out that it was "well-known" to those in the know ten years ago.  Part of the beauty of mathematics is that it is, to some extent, eternal, and an article can remain accessible to others for many, many years.  So, please, think of this when weighing up whether to make corrections.  (My own principle is to keep a list on my webpage of minor corrections: I've also, sadly, had to issue some proper errata...)
(I should say that actually the above scenario has, thankfully, not happened to me.  But an analogous issue-- that of preprints circulating in the community and being referenced etc. without ever being published-- is a bugbear of mine).
A: Although there are many different cases to consider, in all of them I think Step 1 is the same: write to the author of Paper A.  Your message should convey the sentiment that you believe you have found the following specific mistake(s) in Paper A.  Does the author agree?  
Among all choices of phrasing which unambiguously convey this sentiment, you should strive to find the one which is maximally polite, respectful and non-confrontational.  The tone of your first message will play a large role in determining whether the author responds at all and, if so, the nature of your subsequent correspondence.  A past experience of mine amounts to conducting an experiment in this regard: a colleague of mine had pointed out a (fatal) error in Paper A, but the author felt attacked and responded but didn't really engage my colleague mathematically.  This went on for a while -- frustratingly, to my colleague -- and culminated in an amazing "J'accuse!" moment at a big math conference -- unfortunately I was in "the wrong" special session at the time so missed out on seeing it with my own eyes by about 50 meters, but I met the author at the same conference, read Paper A, and eventually came to the same conclusions as my colleague.  I wrote to the author as nicely as I possibly could, and the response was markedly better than the one my colleague had gotten.  That's not the end of the story by any means, but it illustrates my point.
Put yourself in the author's shoes: under any circumstances, it sucks to receive a "your paper is wrong" message.  I think that at least 90% of the time the author will not believe it at first, so that some collegial back-and-forth will be required.  (When these kind of messages get sent to me, my first response is almost invariably an explanation of why I am correct, whether or not that's the final verdict.)
I think the biggest branching point in the tree of all possible responses is: does the author privately agree with you that there is a mistake?  If you can't get to that point, the whole affair becomes much tougher and more unpleasant.  
A: I apologize for posting before reading the long discussion; I did a cursory check however and I noticed that Ted Hill has not yet been mentioned. His text How to Publish Counterexamples in 1 2 3 Easy Steps is a first-hand account of dealing with submitting corrections to authors and editors in a rather high-profile case. The facts of the case are intricate enough that one wants to remain circumspect in one's conclusions, but one thing is overwhelmingly clear: the process is not for the fainthearted.
A: I think the question is a bit too detailed.  A short version is this: what do people do when they discover an error in other people's papers?   Obviously, like the question explains, there is no universal rule - this all depends on the type of an error, relative importance of the results in the paper, relationship between a person who made an error (let's call her/him X) and who discovered it (Y), etc.  Let me simply list some relatively standard options.  
1) Y tells the error to X.  X finds a way to fix it, publishes an "erratum" in the journal, on the arXiv and/or on his/her own webpage.  Gives profuse thanks to Y (but only if Y is gives a permission to do so).  Occasionally this a joint (X,Y) paper.  Either way, this is the most desirable outcome.  
1)' Even if the result is false in full generality, X should still publish an "erratum" saying "such-and-such weaker version survives", or even "every hope to prove such-and-such is lost forever"... 
2) Y wants to remain anonymous, or X can't be bothered.  Then Y writes a letter to editor in chief of the journal which published X's paper.  It is their responsibility as much as X's.  Let the editor(s) deal with the mess.  This is the easiest way out (for Y).  
2)' A slightly better way to remain anonymous is for Y (by agreement with the editors)  publish a short erratum under an assumed name.  I have seen this happen, but in a long run this does not work - eventually people find out who was the author (and in a couple of cases I know, MathSciNet rather counter productively links the pen name to Y).  On the other hand, if you really want to remain anonymous, e.g. use an assumed name linked to a fake email account, your erratum submission will not be taken seriously (journals get quite a few crackpot submissions).  
3) Y is heavily involved in the field and is writing an article/book (B) on the subject.  Y doesn't know how to fix the error.  Then sometimes it is a good idea to include this piece of math in the final remarks or an appendix.  Y might want to be nice and inform X first, before making the error public.  This is a good solid option which allows others to say "error in A was pointed out in B".  
4) The error is fundamental, kills paper A, but Y knows how to fix it.  Y should publish a new paper explaining the error in full, right in the introduction or the first section.  Y should write the paper in such a way as if assuming that X will be refereeing the paper...  On rare occasions, it can happen that later Z publishes a paper acknowledging an error in Y's paper, and claiming to have "finally" found "a definite proof", etc.  Sometimes an unavoidable chaos ensues, but the good faith decision by Y to publish was still a good one.  
5) Y can prove (by different means) a result which follows easily (or even a special case) of that by X.  Y should still write a paper.  Lots of delicacy is required in trying to explain the whole story.  This is the hardest thing to do.  Consult a senior expert before making the paper available.  
6) In extreme cases, Y can just post a note on the arXiv (this happens occasionally, see the meta discussion), but let me strongly discourage this practice.  It should only be used when no other recourse is available.  When this kind of thing happens, the allegedly erroneous paper A is refereed, but the erratum is not, so the ousiders no longer know what to think.  This can undermine the credibility of the field and turn people away from the working on the problem.  
UPDATE: After reading other answers, I realized that I am answering a slightly different question.  This is only meant to catalog the possibilities, not to endorse them or explain "how to get there".  The latter is often really delicate and difficult, so don't try it if you are not sure!  Although some of these outcomes are preferable to others, this is also on case-by-case.  Finally, the order is somewhat arbitrary.  
A: I was once in a situation similar to yours.  The author acknowledged the error and in principle agreed to publish an erratum, but it gradually became clear that the author would probably never get around to doing so.  I eventually resolved the issue by offering to act as the author's secretary, TeXing up the erratum myself but putting only the author's name on it.  The author accepted my offer and everything went according to plan.  The published erratum bears no sign of my involvement and it's conceivable to me that the journal to this day thinks that I really was just the author's secretary and not another mathematician.  In my mind this was nearly a perfect solution.  Of course I got no credit but I didn't want any.
A: First of all one should agree on what a substantial error is. Of course, if some or all theorems in a paper turn out to be wrong, this is pretty substantial. But it can also happen that the main results are true as stated, but the proofs are incorrect or incomplete, and then one is confronted with the problem: where to draw the line between substantial and not. I am not proposing a solution to his problem. In fact, I don't think there can be any simple solution: this should be considered on a case by case basis. However, in my opinion the guideline should be: an error or a gap in a proof is substantial when it can't be corrected with the methods used in the paper. This may sound vague but in some cases I'm aware of it was clear what this means.
Now supposing an error has been found, it is for the author to take steps such as publish an erratum or to update the paper on the arxiv or to put a note on the arxiv explaining what is wrong. Now the really difficult case is when the author acknowledges the error but flatly refuses to anything about it. (I'm aware of a couple of such cases; no names will be named.) In this case again there is no universal recipe, I believe, but continue negotiating with the author, editor etc. As a last resort, if everything else should fail, I guess one should put a note on the arxiv explaining that the results and/or proofs of the paper are incorrect, especially when it comes to results other people are likely to use.
A: If you know that the person reads mathoverflow, then you could post a question asking what to do in these circumstance and maybe the person will take the hint.
Or if you are not the person pointing out the mistake, but the pointee, and wish the pointer had gone about things differently then again you could post a question on MO and hope the person bears this in mind the next time.
