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Let $X = \mathrm{Spec}(A)$ be an affine scheme. Let $U$ be a quasi-compact open subset of $X$. Then there exist an affine scheme $Y$ and a morphism $f\colon Y \rightarrow X$ such that $f(Y) = U$, right? We are interested in a similar result(if any) on an arbitrary intersection of quasi-compact open subsets of $X$. Let $(U_i)_{i\in I}$ be a family of quasi-compact open subsets of $X$. Let $T = \bigcap_{i\in I} U_i$. Do there exist an affine scheme $Y$ and a morphism $f\colon Y \rightarrow X$ such that $f(Y) = T$?

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2 Answers 2

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I don't know what is the motivation of the question. But the answer is yes.

In an affine (hence separated) scheme $X$, any affine open subset is retro-compact hence constructible. So any quasi-compact open subset of $X$ is constructible in $X$. Any intersection of constructible subsets of $X$ is pro-constructible (EGA IV.1.9.4), and, as $X$ is quasi-compact and quasi-separated, any pro-constructible subset of $X$ is the image of an affine scheme (EGA, IV.1.9.5(ix)).

Edit Sketch of the proof of EGA, IV.1.9.5(ix):

  1. First a constructible subset of $X$ is the image of an affine scheme $X'$ (EGA, IV.1.8.3). It is enough to prove it for a locally closed subset $U\cap (X\setminus V)$, so a quasi-compact open subset of an affine scheme $X\setminus V$, write it as a union of affine open subschemes $U_1, U_2...$ and take $X'$ the disjoint union of the $U_i$'s).

  2. (EGA IV.1.9.3.2) Now for a pro-constructible subset $\cap_i C_i$, if $C_i$ is the image of $f_i: X'_i=\mathrm{Spec}(A_i)\to C_i$, define $A'$ as the direct limit of tensor products of finitely many $A_i$'s and consider the canonical morphism $f: \mathrm{Spec}(A')\to X$ (each $A_i'$ is an $A$-algebra via $f_i$), show $f(X')=\cap_i C_i$. This part is harder.

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Thanks. I've just become interested in the set theoretic image of a morphism of schemes. Could you give us a hint so that we can prove it without referring to EGA? I think EGA is somewhat intimidating for most of us. –  Makoto Kato Dec 22 '12 at 21:54
    
@MakotoKato: by the way, EGA IV.1.9.5(ix) is "if and only if". –  user18119 Dec 23 '12 at 8:20
    
"EGA IV.1.9.5(ix) is "if and only if"" This is an interesting result. I searched for internet by the word "proconstructible", but very few was hit. –  Makoto Kato Dec 23 '12 at 12:51
    
The title question is not so difficult to come up with. Let $C_1,\dots,C_n$ be quasi-compact open subsets(or constructible subsets). Let $f_i: X'_i=\mathrm{Spec}(A_i)\to C_i$ be a morhism such that $f_i(X'_i) = C_i$. Then $\bigcap C_i$ is the image of $\mathrm{Spec}(A_1\otimes\cdots\otimes A_n)$. So if $(C_i)_{i\in I}$ is a family of constructible subsets, it is natural to consider the infinite tensor product of $(A_i)_{i\in I}$. –  Makoto Kato Dec 23 '12 at 13:56

No. Delete countably many points from the affine line over an uncountable field. This set is not constructible and cannot be the image of an affine.

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Could you explain why a non-constructible subset of an affine scheme cannot be the image of an affine scheme? –  Makoto Kato Dec 22 '12 at 19:28
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It can't be the image of a scheme by a locally finite presented morphism. –  user18119 Dec 22 '12 at 21:18

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