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Since the current consensus (not including me) is that this question is too elementary, I wanted to hear from people here where this is usually covered in one's education. I only found the answer because I had a suspicion of what it should be, and had some idea of what to type into a search engine -- or what I should have been looking up in my copy of Lang, or what I should have asked one of my colleagues were it not the weekend.
Cf. this older question, though I acknowledge that was back in the Old Days.
I was under the impression when MO started that it had a place for questions one honestly ran into, which one thought an appropriate "expert" or "specialist" would know about.
(Disclaimer: I have met the OP and think he is a serious researcher. This of course colours my judgement.)
What I find slightly problematic about the question is that it seems to me that at least the first part 'is there a basis' seems easy enough to find, IMO. (This is also mentioned as problematic on the old question, btw).
However, I agree with Yemon Choi that this is definitely something one might not know. Perhaps ironically precisely for the reason Fernando Muro mentions. One could learn it very early on but I think most people do not. And then "later" it might well never come up.
Added: I should perhaps answer Yemon Choi's question which I somehow over-looked, regarding how to find it. I'd say it depends a bit how one thinks about Z^n. But since the question ask for subgroups, let us folllow the group notion. So it is also abelian. So perhaps the wikipedia page on abelian groups. Then there one finds the structure theorem for finite abelian groups and a link to struct thm for finitely generated abelian groups (which looks promising), so follow this and there it gives the structure of fin gen and also mentions that a subgroup of a finitely generated one is fin gen, which gives the result.
Added 2: Sorry a bit distracted, still an addition. Regarding where one usually learns it (as I said I think perhaps not at all). Some people might teach/learn this in linear algebra; but I think this is rare. Then from a systemetic point of view, I tend to consider this as a resutl of module theory. Structure Theorem of Fin Gen Modules over a PID (or even Dedekind domain). However, this is admittedly already specialised.
It is a bit unclear to me why people are voting to reopen an essentially answered question, that was vote to close with contribution from OP.
I voted to reopen because I thought that Fernando Muro's comments, combined with the closure of the question, reflect badly on Math Overflow. Smith normal form may be elementary, but it seems ridiculous (if not rude) to marvel at a colleague for not knowing it. Perhaps this just reflects the skewed distribution on MO towards the algebraic; I imagine it must take a lot of restraint for the probabilists (PDE theorists, numerical analysts, etc.) on Math Overflow to refrain from pointing out how "elementary" similar questions about their field are.
As another data point, I've known about subgroups of free abelian groups since I first encountered it in an undergraduate independent study, and in the finitely generated case I'd have to consider this very standard indeed. But I've barely heard of Smith normal form, and asking for an algorithm seems to me like a very reasonable question for MO.
That said, I don't understand the call to reopen if it is the OP's wish to have it closed.
Let's face it; must of us are pretty ignorant outside of the area(s) in which we work. I am glad to learn about Smith normal form.
It's an elementary question but I think it's suitable for MO. Specifically, the question involves enough specific knowledge from a research perspective that it's not very common knowledge. I think questions like this are well within the margins of what the purpose of MO is.
Regarding Yemon's question, the first part of the question I saw answered in my 3rd/4th year rings and modules course (where things like Smith Normal Form came up, and the classification of f.g. modules over PIDs was presented). The second part of the question, about an algorithm to find a basis for the intersection, that wasn't directly addressed in any of my undergrad courses but I could imagine it as an "involved homework problem" in the rings and modules course.
Not to pick needlessly further on the question (as can be inferred, though this is not so visible anymore, from the fact that I did not vote to close while I could have, I am not even so 'against' it), I would still like to make explicit that there are two things OP did (or rather neglected to do) that IMO contributed quite a bit to the rapid closure:
There is no context or motivation whatsoever. (This was already mentioned by Yemon Choi that this could have helped).
There is no (visibile) sign of any effort to find an answer before asking here. As I pointed out above, no matter the background, it cannot be too hard to find some reference on abelian groups. Perhaps one does not find a complete answer, though I think the first part should basically always be addressed. (Yet at least one would get a quite clear idea what the answer should be; since what other type of abelian group should even be contained in there but a free one).
Now, the algorithm and the SNF this is somewhat less clear. Yet, if one reads the first comment, one gets a very clear sense that the closure is upon the question for the subgroups (and the rest is basically already ignored).
A question on this might be suitable, but the precise present incarnation at least has some serious shortcomings as an MO question. That this is now voted at +6 is a bit strange in my opinion (but then there are stranger things happening, regarding voting so maybe I just should stop wondering about such things).
@bsteingerg: okay, but isn't that a bit a strange thing to do. (Now at ten!) So to say to "comment" on comments via votes on the question. By contrast the first comment mentioning the SNF, which it seems is of interest to some, and is also not in the least rude, dismissive, or anything does not have a single vote.
As a general aside, does any one finally intend to give a proper answer as this got reopened. Perhaps some of the reopeners would be natural canditates to do so.
Much of the point of MO is that you can get answers to well-known questions that aren't in your subfield. I think this question is totally reasonable, and I don't see the point of closing it.
This is a remark on a different question, namely Is every projective Z[X] module free
Neither did I want to start a new thread nor did I want to fill up the comments there, and it is similar in subject and in some sense though not only a reply to Yemon Choi, so it sort of fits here [please do not link this thread on that question it will be confusing and useless, IMO):
So, why would somebody think to close this question. I can see two main reasons:
One might vote to close it as no longer relevant, after the question was answered in a comment and this was verbally accepted as answer by OP, to avoid a question staying formally unanswered while having an answer in the comments. This of course depends on the precise time of the vote and might not apply to all or any of the votes (I did not follow this in detail).
This is a general reason, it says this under 'how to ask'
Do your homework Before asking your question, try to solve it. Search Google, Wikipedia, and nLab, check any references you can think of, and try to figure the problem out yourself (maybe even sleep on it).
Now, this is perhaps only a suggestion and the later parts might be slightly too much to ask ('any reference' I mean in particular), but in the current case the first hit I get when typing "projective module polynomial ring" into Google is the Wikipedia page on Projective Modules, more specifically the subsection on Quillen--Suslin which mentions the result, and among the first couple hits is the paper by Quillen with a quite descriptive title the first lines of which gives the result, and the hits in between seem also related though I did not check.
Okay, this is a rather new OP, and one should not be too strict, perhaps; but then they got an answer very quickly and while in a terse way it was a friendly way or at least not an unfriendly one.
So, again, while this is not too simple a question by content, it is perhaps too simple because it seems trivial to find the answer elsewhere (for anybosy trying to find it, IMO).
I should perhaps add, I did not vote to close this (except I completely misremember, but I really doubt that).
I didn't vote to close either, although 'Quillen-Suslin' immediately popped into my mind when I saw it. I agree with Steve Landsburg that many people might not be aware of this result.
It's even possible OP knew of this result, but only in the form that projective modules over a polynomial ring $k[x_1, \ldots, x_n]$ are free, where $k$ is a field (which is the form I knew it by, before I began investigating it more closely yesterday). That's how it's stated in Lang's Algebra for instance. It takes a somewhat close reading of the relevant sections of Lang to see that the arguments show $k$ could be a p.i.d., and it's forgiveable that someone not expert in the area would have some lingering uncertainty that even if they suspected it would hold in that generality and that Lang's arguments prove it, that they were nevertheless missing some subtlety.
Lingering uncertainty can also attach to anything written in Wikipedia, which is IMO a tremendously valuable resource but still rife with errors.
I'm glad the question remained open long enough to get informative answers (even more informative than Fernando's answer in his comment).
@Todd Trimble:
I agree with Steve Landsburg that many people might not be aware of this result.
Yes, me too. However, I would like to reiterate, this is besides the point in my opinion. And, I repeat this, as this is really a key issue in my opinion in some of these discussions.
Whether asking somebody for a certain piece of information is reasonable/justified, does not only or even mainly depend on how well-known this piece of information is, but how hard or easy it is to get it without asking anybody.
It's even possible OP knew of this result, but only in the form that projective modules over a polynomial ring [...]
If this were so, this should be mentioned in the question.
Lingering uncertainty can also attach to anything written in Wikipedia, which is IMO a tremendously valuable resource but still rife with errors.
Yet, it is said, for example, right at the start of Quillen's paper (and this is free, else I could not have checked at the moment). As soon as one has the keyword Quillen-Suslin it is really a non-issue to sort this out. And, again, if OP suspects something but has some doubts they should explain this.
My general point is that, once again, this is a question that has no context and no motivation. In this case really strictly nothing.
It is good OP (and others) got the information, regarding this I agree. However, I also think it is important to make clear in on form or another that this is not the way to write an MO question.
In my opinion more attention should be paid to communicating issues asking questions in this form can create, for everybody involved. So, I am also glad the question got into some "trouble"; as this seems perfectly justified to me.
Having now looked at the opening lines of Quillen's paper (which I can see despite a paywall), I have to admit that my possible counter-objections to quid's arguments were shot down. So, thanks quid.
So in a strict reading of "the rules", I concede that due diligence might not have been exercised in this case (although I'd cut the OP, who is a newcomer, a bit of slack -- and avoid coming across as too vehement in 'correcting' him or her).
@Todd Trimble: Thank you for following up on this. I think in the end we do not have that much of a different opinion. In my first comment I also acknowledge the relative inexperience on the site and that (thus, or perhaps also in general) one should not be too strict. The boldface in the second one seems perhaps stronger than intended, and I should likely have said 'a good way' rather than 'the way' inn it. What I mainly wanted to state is some form of response to the comment of Steve Landsburg and the subsequent ones by Yemon Choi. There are reasons why one reasonably might consider closing the question, IMO. Maybe one should not in the end, but still it is not that surprising somebody might consider doing so, in particular after the comment-answer but before the actual answer (if this was the timing, I do not know) and the positive reception of it by OP: short question, short answer, and close also for practical reasons, since if there would never have been an answer this would have resulted in the unfortuante situation of having a formally unanswered question that has an aswer. (I considered such a vote but did not cast it right away as I did not consider it urgent and for practical reasons, and then it became obsolete since an answer appeared).
@bsteinberg: I do not consider it as really bad either. The response to Todd Trimble should explain my motivation. Yes, I agree sadly many question lack motivation. But if noone points out this lack it will never get better or even it will get worse. Indeed, there was a thread on this http://tea.mathoverflow.net/discussion/1197/interesting-questions-easily-answered-by-searches/ I also thought of that one but did originally not look it up. Actually I should have, then I could just have quoted myself from their (some details of course do not apply). Except it seems I was rather stricter then.
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