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This question evolves mostly from my surprise at the answer to a previous question of mine (Is every flat unramified cover of quasi-projective curves profinite?). My surprise was at that inverse limits of finite unramified (even etale) covers (by cover I mean surjective morphism) of a scheme, X, are not necessarily unramified. This, of course, is in complete contrast to the topological case. As I understand it (and correct me if I'm wrong), being formally unramified is closed under inverse limits.

But still - this seems very odd to me. Is there any characterization of the points over which an inverse limit of unramified covers is ramified? Why does this happen? In what way would it ramify? I'm seeking a heuristic that would make me understand the benefits of being formally unramified over being unramified (and therefore also in what way being unramified differs from the topological analogy, in this respect).

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    $\begingroup$ Am I missing something, or is is just that unramified contains in it the condition of being finitely presented? Since the inverse limit of schemes corresponds to a direct limit of rings, it doesn't seem so unusual that a direct limit of finitely presented algebras over some base ring would not itself be finitely presented. Probably others will soon weigh in on this. $\endgroup$
    – Emerton
    Commented Feb 14, 2010 at 20:42

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