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



How do I fix someone's published error?
 **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.


1. 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.
2. See if the journal involved has already published a 
correction. If so, then stop.
3. 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.
4. 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.
5. 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


