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In EGA IV, Chapter 8, projective systems of schemes (and morphisms between them) are considered. Let $(S_{\lambda})_{\lambda \in L}$ be a projective system of schemes and let $S$ be the projective limit of this system.

The first version of my question I think assumes that you are somewhat familiar with this part of EGA. The second version is a special case of the first version, and doesn't require familiarity with EGA IV.8.

  1. In Théorème 8.10.5, it is stated that for every property P in a given list of properties (starting with: isomorphism, monomorphism, immersion...), and under some additional conditions that I won't mention right now, if a morphism $f : X \rightarrow Y$ of $S$-schemes with property P can be "spread out" to a morphism $f_{\alpha} : X_{\alpha} \rightarrow Y_{\alpha}$ of $S_{\alpha}$-schemes for some $\alpha \in L$, then we can choose $\alpha$ in such a way that $f_{\alpha}$ has property P. The list of properties P given in EGA for which the Théorème works, however, does not contain flatness. Is there a reason for this? I.e., is there a counter-example for the statement of the theorem if we include flatness in the list? If so, are there any reasonable extra assumptions that we can put into the theorem so that it also works for flatness?

  2. This is the situation I am most interested in. Suppose that $R$ is a commutative ring, then let $R_i$ be the directed system of its finitely generated subrings. Put $S_i := \textrm{Spec}(R_i)$, so that $(S_i)_{i \in I}$ is a projective system of affine schemes. I have a morphism $f : X \rightarrow S$ that is proper, flat and of finite presentation. Does $f$ necessarily arise as the base-change of some $f_i : X_i \rightarrow S_i$ that is also proper, flat and of finite presentation?

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I realize that Question 2 probably comes down to case where $X$ is affine, which is a riddle in commutative-algebra: Let $B$ be a finitely presented flat $A$-algebra for some commutative ring $A$; is there a finitely generated subring $A'$ of $A$ and a flat $A'$-algebra $B'$ such that $B \cong B' \otimes_{A'} A$ as $A$-algebras? But I don't see the answer to this. –  René Feb 26 '12 at 17:16
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up vote 3 down vote accepted

I think what you are looking for is in the book, only later: see EGA IV, (11.2.6).

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