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](https://mathoverflow.net/questions/23337/is-a-universally-closed-morphism-of-schemes-quasi-compact).



When is an irreducible scheme quasi-compact?
There are smooth counterexamples. Let $S\_0$ be a smooth separated irreducible scheme over a field $k$ with dimension $d > 1$, and $s\_0 \in S\_0(k)$. Blow up $s\_0$ to get another such scheme $S\_1$ with a $\mathbf{P}^{d-1}\_k$ over $s\_0$. Blow up a $k$-point $s\_1$ over $s\_0$ to get $S\_2$, and keep going. Get pairs $(S\_n, s\_n)$ so that the open complement $U\_n$ of $s\_n$ in $S\_n$ is open in $U\_ {n+1}$ and is strictly contained in it. Glue them together in the evident manner, to get a smooth irreducible $k$-scheme. It is locally of finite type, but is not quasi-compact (since the $U\_n$ are an open cover with no finite subcover). This is separated (either by direct consideration of affine open overlaps, or by using the valuative criterion). 


