A non-trivial example of a retrocompact open subset of a separated scheme

Grothendieck defined retrocompact subsets of a topological space in EGA III-1. p.12. The notion of retrocompact open subsets is used in the definition of constructible subsets of a scheme which is important in algebraic geometry.

Let $X$ be a topological space. We say a subset $Z$ of $X$ is retrocompact if $Z \cap U$ is quasi-compact for every quasi-compact open subset $U$ of $X$.

If $X$ is a separated scheme, every quasi-compact open subset is retrocompact.

Suppose $X$ is a locally Noetherian scheme. Let $U$ be an open subset of $X$. Let $V$ be a quasi-compact open subset of $X$. Since $V$ is a Noetherian topological space, $U \cap V$ is quasi-compact. Hence $U$ is retrocompact.

So let us suppose $X$ is a separated scheme which is not locally Noetherian. I would like to know an example of an open subset $U$ of $X$ satisfiying the following conditions.

(1) $U$ is not quasi-compact.

(2) $U$ is retrocompact.

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1 Answer

The retro-compactness is a property relative to $U\to X$ and it is not very restrictive on $U$ itself. In particular, $X$ is always retro-compact in $X$.

Let $X$ be a quasi-affine (hence separated) scheme which is not quasi-compact (e.g. the complement in $\mathrm{Spec}\mathbb C[x_1,\dots, x_n, \dots]$ of the closed point defined by the maximal ideal $(x_1,\dots, x_n, \dots)$), then $X$ is retro-compact in $X$ but is not quasi-compact.

To have an example with $U\ne X$, take $U$ be the above quasi-affine scheme and take $X$ be the disjoint union of two copies of $U$.

Edit Any affine open immersion $U\to X$ is trivially retro-compact. So let $X$ be the affine punctured scheme as above, let $U=D(x_1-1)\cap X$. Then $U\to X$ is affine, but $U$ is not retro-compact. This gives an example with $X$ irreducible and $U\ne X$.

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Thanks. I should have added a condition $U \neq X$. – Makoto Kato Dec 12 '12 at 20:16
This will be harder if you ask further that $X$ be irreducible. – user18119 Dec 12 '12 at 20:44