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What is the definition of a profinite morphism in http://www.math.upenn.edu/~pop/Teaching/2010_Math624/2010_Math624PS08.pdf problem 5? This is not actually a homework of mine but I was unable to find the definition.

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    If possible, it's better to upload an image of the text, or to write it out, than to give a nasty-looking, long link. Even better, your link *forwards* somewhere; can you be bothered to link *directly* to what you are interested in instead of to a forwarder? Also: you are taking something out of someone's webpage; bad form not to mention whose. Finally: why not ask the author? He would be in the best position to tell you.2011-01-06
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    A Google search turns up nothing except this question and that pdf, so this is likely to be either outdated or an invention of the author. Have you tried e-mailing him/her?2011-01-07
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    @Qiaochu: Dear Qiaochu, Regarding "him/her", this is Florian Pop's web-page. *He* is a well-known arithmetic geometer at U Penn.2011-01-07

1 Answers 1

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A profinite morphism presumably means something like a projective limit of finite morphisms. Since finite morphisms are affine, we can think what this would mean in terms of maps of affine schemes. An inductive limit of finite maps $R \to S_i$ is the same as a map $R \to S$ where $S$ is integral over $R$, so this leads me to guess that a profinite morphism is a morphism $X \to Y$ which is affine, and so that on affine opens Spec $R$ in $Y$, the preimage in $X$ is of the form Spec $S$ with $S$ integral over $R$.

This is compatible with exercise 1 (b), and exercise 5, on the assignment that you link to. Note, though, that as others have already pointed out, assuming that my guess is correct, this terminology is not standard, or at least not common. More typically, such morphisms are called integral. (This is the terminology used in Ravi Vakil's notes, and presumably also in the stacks project, and I also presume it is the terminology used in EGA.)