This is a question of mathematical writing. Let me know if it would be better suited to academia.SE.
I am writing a paper in invariant theory. It uses some slightly heavy commutative algebra. There are a few points where I use facts of which I am convinced, and I believe they are widely known, but I am not sure how to look for a print reference. For example:
1) "flat of relative dimension $n$" is preserved under arbitrary base change
2) the functor of invariants commutes with flat base change
I learned (1) from Ravi Vakil's algebraic geometry notes (exercise 24.5.L). I learned (2) from the thesis of my coauthor (the proof is easy). There are other results like this I'm not thinking of right now that I probably learned from the Stacks Project.
I guess a thesis can be cited in print if needed, but it seems inappropriate to cite either Vakil's notes or the Stacks Project in a print article since they have not undergone formal peer-review, as authoritative as they are. I imagine I might be able to find one or both of these things in EGA, but then again, I might not, and I would spend a lot of time looking. As a young scholar, I do not yet feel I have a beat on what is regarded as common knowledge. My question is:
What guidelines does one use to decide if results such as these require a reference in an article or can be used as common knowledge?
Best Answer
I agree with RvDdB's answer, but want to add some thoughts. To quote from another SE answer of mine:
First, my general philosophy is that one should try to make papers reasonably accessible to young people who have not spent months or years working on this specific problem. I generally feel that most math papers should do a better job of providing references than they do.
There I also espouse the view that writing (and reading) papers and getting feedback (via reviews, discussions, questions during/after talks) provides a process whereby you learn what is more common knowledge to other people in your field and what is less common. When you are young (and sometimes even when you're not), you probably don't have a good sense of this, so it's good to err on the safe side of including a citation if you're not sure, and often a referee will help you out by suggesting a citation is needed (or less often, not needed).
But one rule of thumb is: think about what you knew before you started working on this specific problem. You may need to modify this depending on your situation (e.g., if you started working on the problem before you knew anything about the field), but perhaps thinking about this sentiment is still helpful.
Anyway, the main thing I wanted to add to the already existing answers is: think about your intended audience. I would give this advice for writing the paper in general, and it's also applicable to your particular question. For instance, if I'm writing a paper targeted at people who do local representation theory I might not bother to reference some basic properties of the local Jacquet-Langlands correspondence known 30-40 years ago (e.g., the correspondence for 1-dimensionals of division algebras), but if I think my paper should be of interest to, say, people in classical modular forms who aren't all experts in local representation theory, then I definitely will. (I'm not saying that the amount of time passed since a result was known should be the only factor here--the older less well known facts I would also cite.)
Do you think your paper will/should be of interest to people who aren't intimately familiar with properties of base change? If so, try to provide a citation.