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Let $I$ be a small filtered category. Let $F\colon I \rightarrow CRng$ be a functor, where $CRng$ is the category of commutative rings. We write $A_i = F(i)$ for $i \in I$, $A =$ colim $A_i$. Let $X$ be a scheme of finite presentation over $A$. Then there exist an element $0 \in I$ and a scheme $X_0$ of finite presentation over $A_0$ such that $X = X_0\times_{A_0} A$?

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You certainly know this is a special case of EGA IV, 8.8.2 when $X$ is quasi-separated. Do you ask whether this last condition can be removed ? –  user18119 Dec 28 '12 at 22:32
    
@QiL Dear QiL, By a morphism of finite presentation, I mean it is quasi-compact and quasi-separated and locally of finite presentation(this definition is adopted by the Stacks Project). I think it is nice to have a proof here, since this result seems not to be well-known(except by the experts). –  Makoto Kato Dec 29 '12 at 0:09
    
You are right for the definition of finite presented morphisms (already in EGA). As the answer to your question is already in EGA with full details, I don't see why to present a proof here. What could be interesting is to sketch the ideas of the proof. –  user18119 Dec 29 '12 at 10:17
    
@QiL "As the answer to your question is already in EGA with full details, I don't see why to present a proof here." Perhaps I would agree if it was written in English. Anyway, a sketch of the proof is welcome. –  Makoto Kato Dec 29 '12 at 11:18

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