# Finding citations for 'well-known' results

I am almost ready to submit my most recent paper, and I find myself in a problem that has already occurred multiple times in my short publishing career. In this paper, I wish to state a result which I consider 'well-known', but a skimming of all the likely textbooks and survey articles doesn't yield a nice statement that I can cite. For reference, the result in question is the following:

Let $X$ be a smooth, affine variety over $\mathbb{C}$, with coordinate ring $\mathcal{O}_X$. Then there is a natural isomorphism of $\mathcal{O}_X$-modules from the Kahler differentials $\Omega(\mathcal{O}_X)$ of $\mathcal{O}_X$ to the global 1-forms on $X$ with regular coefficients.

This is a result whose proof I know, and is homework-level difficulty, but including the proof in my short paper would require terminology and techniques I'd rather not introduce and consume precious space. It's also not a necessary result for the paper; I am including it to justify the study of Kahler differentials to an audience which might include differential geometers.

So what does one do in this situation? The lazy solution is to include some weasel words to avoid finding a citation ("it is a straight-forward exercise to show that..."), but this seems like a dangerous policy to employ in general. However, finding a citation is proving unreasonably time-consuming, since it's not in the books I know (Hartshorne, Eisenbud, Kunz), and each new book/article I skim has its own notation and assumptions.

Also, while I'd be extremely grateful for a citation for the specific result above, my question is about what to do in this kind of situation. I'm trying not to get the answers mixed up.

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Well, I believe the answer now is to ask MathOverflow. Though, I have to admit, I would have guessed this is in Hartshorne. –  Ben Webster Apr 11 '11 at 20:37
What is so precious about space? –  darij grinberg Apr 11 '11 at 21:03
Darij: from the view of exposition, not much. From the point of view of getting published, perhaps more. –  Yemon Choi Apr 11 '11 at 21:30
Write weasel words, then post a honest version on arXiv. –  darij grinberg Apr 11 '11 at 22:05
It's not just a matter of physical space in journals. Readers are likely to be turned off by a paper which looks too bulky and wordy. (Of course, they will also be turned off by a paper which is too sketchy!) Good exposition requires knowing what to omit, even when paper is no limit. –  David Speyer Apr 11 '11 at 23:42

I've seen "details are presented in the arXiv version of this paper" several times. The only down side I see to this is that you do need to write up the details.

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Better this than "details will be in a forthcoming paper" - especially when people only cite the paper without the details... –  Yemon Choi Apr 11 '11 at 22:53
I really don't like the idea of circulating substantially different versions of a paper under the same name. Instead, I'd prefer for the arXiv version to be as close as possible to the published version (with an exception for typo fixes and the like, but with a note made of these changes). If more details need to be provided online but not published, a companion paper would be a better place for them. Failing that, I'd recommend putting them in an appendix that is clearly labeled as not being part of the published paper. –  Henry Cohn Apr 11 '11 at 23:16
This could also be a problem if you're publishing in a journal that eventually deposits published versions on the arXiv. Like G&T. (They still do that, right?) –  Richard Kent Apr 12 '11 at 2:17
I've run across articles on ArXiv titles "Supplement to [published article name before]" before. Seems like a nice way to avoid @Henry's concern. –  BSteinhurst Apr 12 '11 at 3:12

For your specific result, I have no suggestions or references.

In general, I suggest including a Proof Hint: or Proof Sketch:, along with a note to the editor that you would like assistance in finding a proper reference. The hint should occupy little more space than a full bibliographic citation, and you can prepare a (possibly never to be published) appendix which contains sufficient details of the proof and a summary of your efforts in finding a reference, should someone call you on the (truth of the) statement. This should be doable in a short time and allow you to delay/defer/circumvent this issue. If this is not doable in a short time, then you might rethink its use as a motivating statement, as it may be more of a distraction than motivation. (You can also solicit the editor's opinion on how to handle this issue.)