This question boils down to whether an answer to a question should contain sufficiently free and easily accessible detail about the solution to a question.

So IMHO it would be perfectly acceptable, to provide links to online resources that contain a solution.

What should however be the netiquette regarding links to e.g. online resources behind paywalls, articles in non-english language, books out of print, etc. Is it in case of those high hanging fruits considered polite to provide just the links, book titles, etc. without a summary of the essential information, from which an answer can be recovered without undue effort?

• I think in some cases the answer must refer to these resources, especially when history and/or primary sources are involved. Until we get everything online, I think we should be grateful for any relevant answer, and grateful further for later revisions that give even more accessible sources. The etiquette should be to give relevant answers, with accessibility a secondary concern. Those who wish to add commentary to reveal more should be encouraged, but iit should not be required. Gerhard "Improved Quality Through Continued Effort" Paseman, 2018.09.02. – Gerhard Paseman Sep 3 '18 at 5:29
• I think links to multiple sources (eg published version plus arXiv) is preferable to a single link. At the very least, a human-readable reference so that someone can either recognise the paper without having memorised a DOI/arXiv number/random publisher string, and/or track it down with minimal searching if no working links are present. EuDML links are great, and if nothing else is free, the author's website copy is better than nothing. Giving just a 'it's in this book published 50 years ago, on page 57' is very unhelpful, since the answer should be here, not stuck in some physical object. – David Roberts Sep 3 '18 at 7:01
• So, an upvote means YES, such articles are acceptable. A downvote means NO. (I am going by the way the title is phrased.) – Gerald Edgar Sep 3 '18 at 12:08
• Gerald, I don't think that's how it works unless the OP instructs us to vote that way. The questions deserve more of an articulated response IMO. – Todd Trimble Sep 3 '18 at 12:11
• Many free links have disadvantage that they might be lost at some point - especially something which is on a personal website. An advantage of doi-link is that it is stable, at least in theory - although it might be paywalled. (Something similar was pointed out in Asaf Karagila's answer to edits with links to material under restricted access.) ... – Martin Sleziak Sep 3 '18 at 12:50
• ...I would not be surprised if many of the posts containing a link to Internet Archive were examples of attempts to recover a dead link in some way. – Martin Sleziak Sep 3 '18 at 12:50
• +1 from me - in the hope that this could help the post to get to score $\ge3$, so that it is displayed in community bulletin and more MO users notice this discussion – Martin Sleziak Sep 4 '18 at 6:58
• As long as it's not just a link – TRiG Sep 8 '18 at 23:13

Sometimes I see an answer in the form David Roberts mentioned, the entirety of which is "See theorem 6.3 on page 357 of [1]" linking to an article paywall or to a vendor of a book, and it will be flagged "not an answer". Actually it could very well be an answer, but it could be hard to tell if left in that form.

I think all users should be encouraged to provide content which can read and evaluated on the spot (and leaving a comment to this effect is how I typically respond to the flag). I would assume that whoever writes something like the above has access to said article or book and could therefore also write out the statement of the theorem, for the public good. Of course, MO is found so useful because a really good answer typically goes further, explaining the inner insight or what is really going on, but as far as baseline etiquette goes, I think it's reasonable to expect a poster to give some description of cited material that is not readable on the spot. (By "readable on the spot", I mean not just for professors who have institutional access to the major journals, but also for researchers who are more isolated and not close to a good university library.)

I agree that it's acceptable if an answer simply links to content elsewhere on the WWW that is readable on the spot. As I say, it would be even better if the poster adds some personal insights.

(Actually, this answer should perhaps go further to clarify "readable on the spot". For example, Wikipedia is banned in certain countries. I don't know enough about such bans to comment too much further, but I understand that some researchers have found the nLab very helpful because unlike Wikipedia, it's not banned. Google Books is another edge case because not everyone has access to the same pages.)

• I disagree that a link-only answer is acceptable if the link is readable on the spot. It may be for today, but it could vanish without any notice tomorrow, rendering the answer completely useless. Properly published papers at least don’t suddenly disappear from all libraries. – Emil Jeřábek Sep 3 '18 at 14:34
• Just a brief comment on Google Books (and similar links). I agree that when somebody includes such link, they should be aware that it may be visible only to some people (and in some cases eventually stop working). But even with the limited visibility I consider such link a useful addition. At least those users that can display the linked content can see more of the book - which may be useful both when reading the post and also if they are interested in the book as such. (They can browse at least a part of the text without finding a copy elsewhere.) – Martin Sleziak Sep 3 '18 at 16:37
• @EmilJeřábek I agree with you. I'll probably come back to amend my answer. Thanks. – Todd Trimble Sep 3 '18 at 17:08
• @MartinSleziak I agree with you too; any such link could be helpful. But a bare link to a Google Book with accompanying discussion in an answer is vulnerable to the type of drawback just mentioned by Emil. – Todd Trimble Sep 3 '18 at 17:10
• Yes, certainly. The post should be perfectly readable without the link - the link can be considered "a bonus". – Martin Sleziak Sep 3 '18 at 17:26
• About censorship of Wikipedia see en.wikipedia.org/wiki/Censorship_of_Wikipedia (Most of this censorship would not affect math articles in english wikipedia) – kjetil b halvorsen Sep 14 '18 at 11:38

I don't think that links are good answers since they can decay. Whether or not they are behind paywalls is irrelevant. But I do think that it is perfectly fine (indeed, encouraged) to give answers that are complete citations of relevant papers. For instance, my answer here. These answers are perhaps not useful to some amateurs who do not have access to academic libraries, but given that our target audience is professional mathematicians and graduate students, I don't think that is a problem. If someone wants to be involved with research mathematics, they need to find a way to access the literature (a large amount of which is not available on the internet, even behind a paywall).

EDIT: I feel a little embarrassed that I immediately got a bunch of upvotes for the answer I linked to! That wasn't my goal at all; instead, I just flipped through my answers and linked to the first one of this form I came across...

• Well, I'm not officially in academe, so I guess that makes me an amateur. I also have to make a 30 minute trip to get to a decent university library. I do have ways to get at a pretty large amount of literature, but I'd rather avoid having MO be exclusionary to people in my situation. – Todd Trimble Sep 3 '18 at 17:39
• By the way, I think the answer you linked to is highly appropriate. You stated the theorem precisely, and gave the link. To me, that is perfectly adequate. – Todd Trimble Sep 3 '18 at 17:41
• What Todd said about exclusionary. Also, I think the target audience should not be professional mathematicians and graduate students (and assume the environment they live in as available to all); I think the target audience should be everyone who is willing to talk with professional mathematicians and graduate students on something close to the latter's level and terms, including former graduate students, former professional mathematicians, researchers and workers in other fields, and people who like their amateur status. Gerhard "In Short, For Almost Everyone" Paseman, 2018.09.03. – Gerhard Paseman Sep 3 '18 at 17:46
• Also, while I am grateful (see my comment to the question) for an answer like the example you give, I would be happier with an adjunct answer that starts out like "Expanding on Andy Putnam's answer, here is what the Kan-Thurston theorem means in this context...", and continues in such a way that I gain not only a sense of what a cohomology ring is, but why one would ask the question in the first place. Thank you Andy in any case for your contributions. Gerhard "See You In St. Petersburg?" Paseman, 2018.09.03. – Gerhard Paseman Sep 3 '18 at 17:55
• Reference to the paper in question is certainly needed (or at least the author's name and the title). But including a link is a nice addition - as discussed before: Is it worth editing old posts to add links for references? – Martin Sleziak Sep 3 '18 at 17:58
• @ToddTrimble: What I don't want to see are links to weird pirate websites, or an insistence on answers that only reference freely available sources. My own take is that one should cite papers/books on MO for the same reasons that you cite them in a paper (that is, for purely intellectual reasons, e.g. citing the original source of a result or a particularly elegant exposition of it). There is an unfortunate intellectual laziness I often see among young people, who refuse to come to grips with the parts of the literature that they can't read on their screen. – Andy Putman Sep 3 '18 at 19:22
• @MartinSleziak: I am totally fine with links to legitimate sources for a paper (e.g. an author's website, the arXiv, or a journals's website), and I'll include them if I have them easily accessible. But what I don't want are links to sketchy collections of papers (e.g. researchgate or academia.edu). – Andy Putman Sep 3 '18 at 19:25
• @GerhardPaseman: It would be kind of absurd to insert a discussion of what a cohomology ring is in a question like the one I answered (and once one knows the definitions, the question is natural enough that I don't think it needs further elaboration). Indeed, I think this would add negative value: it would make it harder for an expert to parse things and find a quick crisp answer, and the question at hand is technical enough that someone trying to learn what a cohomology ring was would be unlikely to stumble across an answer that somehow ended up there. – Andy Putman Sep 3 '18 at 19:29
• Andy, if you or anyone else should ever see a link to a weird pirate website at MathOverflow, then you should report it right away; we simply can't have or endorse any legally questionable activity. Also, I too decry linking to sketchy papers; I hope that's not a huge problem here. I'm not quite sure I grasp concretely the phenomenon of intellectual laziness that you say you see; speaking personally, if I'm hunting for knowledge, I'll do what it takes to get it, but I'm much happier if I can access it off the WWW without a lot of hassle. Maybe we agree more than appears. – Todd Trimble Sep 3 '18 at 21:22
• @ToddTrimble: I suspect that we basically agree, and certainly I have never felt tempted to criticize you in any way. I have repeatedly had the following experience in "real life". A graduate student comes to me with a question. I tell them an article where their question is answered. Later on I ask if they found the article useful, and I learn that since they couldn't find a scan of it online, they didn't pursue it. I then give them my speech about how we have a wonderful library and they should learn to use it! I see indications of this attitude sometimes here on MO. – Andy Putman Sep 3 '18 at 21:37
• @GerhardPaseman: I suspect that many of our disagreements here stem from the fact that I view MO as a resource intended for professional mathematicians and graduate students (together with people who intellectually are isomorphic to those kinds of people; for instance, non-professionals like Todd and the occasional crazy-advanced undergraduate). This is what makes it such a useful resource. I would be much less likely to participate if the level or signal-to-noise ratio drifted downward too much. – Andy Putman Sep 3 '18 at 21:43
• "given that our target audience is professional mathematicians and graduate students, I don't think [access to libraries] is a problem" this attitude strikes me as 'let them eat cake'. Many university libraries around the world cannot afford to buy all the journals. The very rationale for Sci-Hub (for good or ill) is that the founder literally could not get access to the papers she needed to read for her research. I agree with all your other points. – David Roberts Sep 3 '18 at 21:59
• @AndyPutman I think we probably do agree then, fundamentally. I can picture myself making the same speech; that's what libraries are for! The graduate student is obviously making his or her life rather harder with that attitude. – Todd Trimble Sep 3 '18 at 22:30
• @ToddTrimble, I saw such a link just yesterday, and wasn't sure what to do. (Unfortunately, I now forget where.) What is the proper action? Should I flag? – LSpice Sep 5 '18 at 14:03
• given that our target audience is professional mathematicians and graduate students, I don't think that is a problem I teach at a community college in the US, and I have no access to paywalled journals. To get access to journals, I have to drive to a university. – Ben Crowell Sep 12 '18 at 3:06

IMHO it depends on the question. If somebody is interested in something very special that most MO users would, most likely, never care about (like 0-2 reputation question hanging out there for a month) and I know who he is and what resources he has access too, I wouldn't hesitate too much to go all the way down to "the proof can be found in the lecture notes of X you can get from Y" though this doesn't happen too often. On the other hand, a high interest question is best answered in full, if possible. In cases when (re)typing the full solution takes way too much time and effort, references to papers and, especially, books just "behind the paywall" are mostly OK, IMHO, because my guess is that very few people do not know how to bypass paywalls if they get really interested but I would certainly (try to) avoid obscure journals and little-known languages.

In general, my opinion is that the netiquette is a quite flexible thing in such matters. One can argue that we are creating a comprehensive database for many generations to come, and one can argue equally well that we just provide one interested person (or a small group of them) with a quick one-time help that gets washed into oblivion in a couple of weeks after the answer is given. I doubt there will ever be a full consensus about what exactly is going on here and what is the ultimate purpose of the whole enterprise. There are some common points and some boundaries, but that's about it.