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I am trying to write a review on a published paper for Mathematical Reviews. The paper proposes a generalization of K-theory. Its main definition is fundamentally problematic. So the rest of the results are worthless. Therefore, I cannot write a brief summary of their results, because it means that I did not understand the mistake. On the other hand, the abstract of the paper is very concise and misleading, so I cannot recommend it as the review to MR either. Besides, I do not want to spend too much time to explain all the mistakes and errors in the paper (it is not my duty as a reviewer!). Therefore I was thinking to write to Mathematical Reviews and let them know about the situation and deny writing any review for this paper. But before doing that, I was wondering if there is a better solution for this problem?

P.S. For those who are not familiar with Mathematical Reviews, I should add that MR asks mathematicians to write brief reviews on papers (book, etc) that are already published and these reviews are available at mathscinet. So, these reviews are different than referee reports.

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    I fail to see the problem. Can't you still write up a summary of the main results and then indicate that they don't hold up because of the problem in the main definition? (e.g., "the main claim of the paper is that the Moon is made of blue cheese, but this result is incorrect because the author has not bothered to look through a telescope") – Koldito Jul 3 '14 at 11:01
  • @Koldito: I think it is kind of unreasonable to write half page summary of the paper and then add about one page to show that the main definition is problematic, and so the reader is advised to not take this paper seriously. – user4511 Jul 3 '14 at 11:09
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    "Besides, I do not want to spend too much time to explain all the mistakes and errors in the paper (it is not my duty as a reviewer!)." What?! It is your main duty as a reviewer! (Except for "too much": once you get enough errors to sink a paper, you can stop.) – Piotr Migdal Jul 3 '14 at 11:17
  • "Its main definition is fundamentally problematic." You mean you dislike it, or is it ill-defined oo self-contradictory? – Piotr Migdal Jul 3 '14 at 11:22
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    I know of one paper containing a major `result' that was published some years ago in one of the top journals, but where the proof was fatally flawed, and this was carefully explained in a lengthy review on MR. While it makes for rather uncomfortable reading, it has the virtue of saving the time of the would-be reader. – Shane O Rourke Jul 3 '14 at 12:31
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Your review should do three things (not necessarily in this order). It should tell the reader what the topic of the paper is, perhaps including the authors' main "theorem". (This presupposes that the paper is clear enough to have a topic; I've reviewed garbage that didn't have a discernible topic, but I gather from your question that this is not the case here.) It should make it clear that you think it's wrong. And it should give enough information about the error to allow people in the same area to understand why you think it's wrong. (Once you've explained the essential error, it's not necessary to list a lot of other errors, unless you think that doing so would help the reader or make it clearer that the paper is wrong.)

It is especially important to indicate accurately just how bad the paper is. For example, is it nonsense, or is it just wrong, or does it give a possibly correct theorem but with inadequate or erroneous proof? You can save potential readers of the paper a lot of trouble by making the situation clear.

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    Someone once published a paper whose main theorem was the negation of one of my (earlier) theorems. I was very pleased when a reviewer (Peter Johnstone, if I remember correctly) didn't just summarize that paper but pointed out the error(s) very emphatically. I can't find the review now, but I believe it contained the phrase "howlers of which an undergraduate should be ashamed". – Andreas Blass Jul 4 '14 at 5:34
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Your job as a reviewer is primarily to give a summary of the results to help other researchers find the papers they are interested in; you are not expected to evaluate the quality of the manuscript. However, there's the following passage in the Guide to reviewers:

Evaluative reviews. Your review may include a positive or negative evaluation of the item. Critical remarks should be objective, precise, documented and expressed in good taste. Vague criticism offends authors and fails to enlighten the reader. If you conclude that the item duplicates earlier work, you must cite specific references. If you believe there is in error in the item, please describe it precisely in your review and provide evidence validating your claim (e.g., a counterexample, an exact reference which supports your assertion, or an indication where the error arises in the paper). You should bear in mind that the MR Database does not include author responses to critical reviews.

This means that a critical review, as opposed to a summary, will likely be more work than you seem to think the manuscript is worth. In this case, there's the "nuclear option":

Two other treatments of items are possible, but should be used sparingly. You may recommend that the item be listed without a published review, or you may recommend that the author's summary be used as the review. If you decide to recommend one of these options, simply put your request in the Review text box (e.g., "Publish without a review", or "Use the summary as my review"). However, in most cases, the mathematics community would prefer an insightful review to either of these two treatments.

If you choose "Publish without review", the paper will be listed as "This item will not be reviewed". For a regular paper, any seasoned user of MathSciNet will understand this as "do not bother to read".

EDIT: This used to be the case; now there's no such remark anymore, but the icon next to the MR number will say "Indexed" instead of "Reviewed" -- less strong of a signal, but a signal nonetheless (especially to people who remember the old remark.)

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