The best ways to cite a theorem whose assumptions/conditions are written earlier in the source?

Question

When one cites a theorem from a source in mathematics, the following problem usually arises: some assumptions/conditions of the theorem are written much earlier in the source, which do not appear in the statement of the theorem.

What are good strategies to solve this problem so that the writing could be more reader-friendly?

On the one hand, citing the theorem simply as "By theorem 3.1.4 in [Fang 2013] ..." means the reader may miss some of the conditions of theorem 3.1.4 if they appear earlier in the source.

On the other hand, yes, restating all the conditions of the theorem is considerably the "best" way. But if one has to cite a lot of hard theorems in a proof, restating everything may distract readers from the main issues.

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Let me construct an example: if a chapter in a commutative ring theory book begins with "All rings in this chapter will be Noetherian", any theorem containing the word "ring" in this chapter would not include the word "Noetherian". If I cited one theorem in this chapter merely by its "number", a reader may be easily misguided if he/she directly read the theorem without reading the first sentence of this chapter.

Answer 1

There are a number of ways to handle situations like this, depending on how much prominence you want to give to the result you are citing.

If the cited theorem is quite important to your work, and you think the reader will benefit from seeing it written out, then you can include a complete statement:

Theorem 1.2 [Smith 1953]. Suppose $X$ is compact, Artinian, and hyperfinite, and $\sqrt{Q(X)-57} > \pi/4$. Then $U(X) \sim \gamma$.

[Oh right, we don't have MathJax here. :( ]

Then elsewhere in your paper you can say "By Theorem 1.2, ..."

If the hypotheses of the theorem are particularly complicated and you don't feel it's helpful to include them in your paper, you could write:

Theorem 1.2 [Smith 1953]. If $X$ satisfies the assumptions of Section 5 of [Smith 1953], then $U(X) \sim \gamma$.

Then you can write elsewhere "We can easily check that $Y$ satisfies the conditions in [Smith 1953], so by Theorem 1.2., $U(Y) \sim \gamma$".

If you don't care to say much about Smith's result (perhaps it's well known, or you're just using it as a black box), you can just cite it, perhaps with a brief comment about its assumptions:

Since $X$ is hyperfinite, by Theorem 5.6.7 of [Smith 1967], ...

By Theorem 5.6.7 of [Smith 1967] (which requires $X$ to be hyperfinite)...