TITLE: When is an irreducible scheme quasi-compact?
QUESTION [22 upvotes]: The standard examples of schemes that are not quasi-compact are either non-noetherian or have an infinite number of irreducible components. It is also easy to find non-separated irreducible examples. But are there other examples?
Question: Let $X$ be a locally noetherian scheme and assume that $X$ is irreducible (or has a finite number of irreducible components) and separated. Is $X$ quasi-compact (i.e., noetherian)?
If the answer is no in general, what conditions on $X$ are sufficient? Locally of finite type over a noetherian base scheme $S$? Fraction field finitely generated over a base? What if $X$ is regular? In general, the question is easily reduced to the case where $X$ is normal and integral.
It certainly feels like the answer is yes when $X$ is locally of finite type over $S$. Idea of proof: Choose an open dense affine $U\subseteq X$, choose a compactification $\overline{U}$ and modify $X$ and $\overline{U}$ such that the gluing $Y=X\cup_U \overline{U}$ is separated. Then, $Y=\overline{U}$ (by density and separatedness) is proper and hence quasi-compact.
Remark 1: If $X\to S$ is a proper morphism, then the irreducible components of the Hilbert scheme Hilb(X/S) are proper. The subtle point (in the non-projective case) is the quasi-compactness of the components (which can be proven by a similar trick as outlined above).
Remark 2: If $X\to S$ is universally closed, then $X\to S$ is quasi-compact. This is question 23337.

REPLY [8 votes]: Actually, Nick Proudfoot and I have been talking for years about the irreducible smooth surface constructed from countably many copies of ${\mathbb A}^2$ by gluing $(p,q)$ in the $n$th copy to $(p^2q,p^{-1})$ in the $n+1$th copy.  This even has a ${\mathbb G}_m$ action $\lambda \cdot (p,q) = (\lambda p, \lambda^{-1} q)$, a symplectic form $dp \wedge dq$, and a moment map $(p,q) \mapsto pq$ whose zero fiber is an infinite chain of projective lines.   This too can be regarded as the toric variety associated to a fan of infinite type in the plane.  It appears to be another way of describing Ekedahl's example.