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.